reisenberger masterthesis 2014 - institute for formal
TRANSCRIPT
Faculty of Engineeringand Natural Sciences
PBoolector: A Parallel SMT Solver for QF_BVby Combining Bit-Blasting with Look-Ahead
MASTER'S THESIS
submitted in partial fulfillment of the requirements
for the academic degree
Diplom-Ingenieur
in the Master's Program
COMPUTER SCIENCE
Submitted by
Christian Reisenberger, BSc.
At the
Institute for Formal Models and Verifcation
Advisor
Univ.-Prof. Dr. Armin Biere
Co-advisor
Dipl-Ing Aina Niemetz, BscDipl-Ing Mathias Preiner, Bsc
Linz, November 2014
Abstract
The development of parallel algorithms for the Satisfiability Modulo Theory(SMT) problem is a quite novel research field. Although there are already sev-eral parallel approaches for the related Satisfiability (SAT) problem out there,few implementations of parallel SMT solvers exist. In recent years paralleliza-tion has grown to an important topic throughout all fields of computer science.The clock frequency increase for new processor generations has reached a limit.Current processor generations are improved by embedding multiple cores usingshared memory architecture. In this thesis a parallel SMT algorithm for thequantifier free theory of fixed-size bit-vectors (QF BV) is proposed and imple-mented in the tool PBoolector. The Cube and Conquer approach, which wasalready shown to be a successful parallel SAT solving algorithm, is lifted frombit-level to the SMT term-level. The idea of Cube and Conquer is to split aformula into independent and simpler subproblems which are solved in parallelby a sequential decision procedure. The PBoolector implementation splits abit-vector formula and solves the subproblems in parallel with the bit-blastingprocedure. The search space splitting is based on the so-called look-ahead meth-ods. Look-ahead methods are well researched methods from the SAT area,which are adapted in this thesis for SMT bit-vector formulas. We show thatthe implemented algorithm can achieve clear improvements (up to super-linearspeedup) on a set of hard benchmark files, which are a bottleneck for sequentialSMT solvers. The formula structures for which the parallel SMT approach isappropriate will be identified and evaluated in detail.
Zusammenfassung
Die Entwicklung von Parallelen Algorithmen fur das Satisfiability Modulo Theo-ry (SMT) Entscheidungsproblem ist ein relativ neues Themengebiet in derheutigen Forschung. Obwohl bereits einige Ansatze zur parallelen Losung desverwandten Erfullbarkeitsproblems fur Aussagenlogik (SAT) existieren, gibt eswenige Parallelisierungsansatze fur SMT. In den letzten Jahren wurde Paral-lelisierung zu einem wichtigen Thema in samtlichen Bereichen der Informatik.Der kontinuierliche Anstieg von Taktraten neuer Prozessorgenerationen hat eineObergrenze erreicht. Prozessoren werden heutzutage verbessert indem mehre-re Rechenkerne auf einem Chip integriert werden, welche sich den verfugbarenSpeicher teilen. In dieser Arbeit wird ein paralleler SMT Algorithmus fur diequantorenfreie Theorie von Bit-Vektoren mit fixer Bitbreite (QF BV) vorgestelltund im Tool PBoolector implementiert. Der Cube and Conquer Algorithmus,ein erfolgreicher paralleler SAT Losungsansatz, wird von Bitebene auf die SMTTermebene angehoben. Die Idee von Cube and Conquer besteht darin eine For-mel in unabhangige und einfachere Teilprobleme zu teilen welche mit einemsequentiellen Eintscheidungsalgorithmus parallel gelost werden. Die PBoolectorImplementierung teilt eine Bit-Vektor Formel und lost die Teilprobleme paral-lel mittels Bit-Blasting. Die Teilung des Suchraumes erfolgt uber sogenannteLook-Ahead Methoden. Look-Ahead Methoden sind gut erforschte Methodenaus dem SAT Umfeld, welche in dieser Arbeit fur SMT Bit-Vektor Formeln ad-aptiert werden. Wir zeigen, dass der implementierte Algorithmus fur eine Mengevon harten Aufgabenstellungen, welche Probleme fur sequentielle SMT Solverdarstellen, klare Verbesserungen (bis hin zu einem superlinearen Speedup) er-reicht. Formelstrukturen, welche fur den parallelen SMT Ansatz geeignet sind,werden identifziert und im Detail erortert.
Contents
1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries 62.1 Satisfiability for Propositional Logic . . . . . . . . . . . . . . . . 6
2.1.1 DPLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Look-Ahead Based SAT Solvers . . . . . . . . . . . . . . . 72.1.3 Conflict Driven Clause Learning (CDCL) Solvers . . . . . 9
2.2 The Theory of Bit-Vectors . . . . . . . . . . . . . . . . . . . . . . 92.3 Boolector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Bit-Blasting . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Term Rewriting / Simplification . . . . . . . . . . . . . . 132.3.3 Boolector API . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Related Work 153.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Parallel SMT Solvers . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Parallel Z3 . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Parallel Interval Constraint Solver - Picoso . . . . . . . . 16
3.3 (Concurrent) Cube and Conquer . . . . . . . . . . . . . . . . . . 17
4 Implementation 204.1 Splitting QF BV SMT Formulas . . . . . . . . . . . . . . . . . . 21
4.1.1 Boolean Nodes . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 Defining New Predicates . . . . . . . . . . . . . . . . . . . 214.1.3 Setting Single Bits . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.1 Main Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Search Tree Generation . . . . . . . . . . . . . . . . . . . 244.2.3 Dynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . 274.2.4 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . 29
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4.3 Look-Ahead API Extension . . . . . . . . . . . . . . . . . . . . . 314.3.1 Look-Ahead Candidates . . . . . . . . . . . . . . . . . . . 324.3.2 Probing Method . . . . . . . . . . . . . . . . . . . . . . . 334.3.3 Reduction Approximation Method . . . . . . . . . . . . . 334.3.4 Look-Ahead Types and Heuristics . . . . . . . . . . . . . 344.3.5 Combination of Look-Ahead Types . . . . . . . . . . . . . 37
4.4 Implementation Extensions . . . . . . . . . . . . . . . . . . . . . 404.4.1 Parallel Boolector Instance . . . . . . . . . . . . . . . . . 404.4.2 Using Boolector in Non-Incremental Mode . . . . . . . . . 414.4.3 Failed Literal Reduction . . . . . . . . . . . . . . . . . . . 414.4.4 Further Implementation Extensions . . . . . . . . . . . . 42
5 Evaluation 445.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.2 Implementation Issues . . . . . . . . . . . . . . . . . . . . 45
5.2 SMT-LIB Competition Benchmarks 2014 . . . . . . . . . . . . . 465.2.1 Reference Runtimes . . . . . . . . . . . . . . . . . . . . . 475.2.2 Results for PBoolector in Incremental Mode . . . . . . . . 485.2.3 Results for PBoolector in Non-Incremental Mode . . . . . 525.2.4 Results for Different Configurations . . . . . . . . . . . . 535.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Crafted Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.1 Lazy-If-Then-Else Example . . . . . . . . . . . . . . . . . 575.3.2 Square of Sum . . . . . . . . . . . . . . . . . . . . . . . . 625.3.3 Carry Safe Adder . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Selected Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 665.4.1 Smulov1bw* . . . . . . . . . . . . . . . . . . . . . . . . . 675.4.2 MultiplyOverflow . . . . . . . . . . . . . . . . . . . . . . . 695.4.3 Calypto . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.4 Log Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4.5 Galois - IffyInterlaevedModMul-8 . . . . . . . . . . . . . . 73
6 Conclusion 756.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Appendices 78
A Table Notations 79A.1 Comparison Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.2 Detailed Result Tables . . . . . . . . . . . . . . . . . . . . . . . . 79
B Figures 82
C Detailed Results 89
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Chapter 1
Introduction
1.1 Motivation
The Satisfiability (SAT) problem for propositional logic is a very well researchedproblem in computer science. It was the first problem that was proven to beNP-complete [25]. Therefore, all problems in NP can be transformed in poly-nomial time into a SAT problem. The SAT problem is to decide for a givenpropositional formula F whether it is satisfiable (sat), or unsatisfiable (unsat).If F is sat , there exists an assignment (also called interpretation or model) of allvariables in F such that F evaluates to true. However, if F is unsat , there doesnot exist an assignment for F such that F evaluates to true. A first efficient algo-rithm for solving the problem, the Davis–Putnam–Logemann–Loveland (DPLL)procedure, was proposed in the 1960’s [27]. With increasing hardware capabili-ties, in the recent two decades the SAT problem became applicable for practicalapplications like hardware and software verification tasks. The ability to solvereal world problems lead to a renaissance in this research field, where great im-provements have been achieved over the last 20 years [14]. The most powerfulapproach is the state-of-the-art Conflict Driven Clause Learning (CDCL) algo-rithm [56], which is based on the DPLL procedure from the 60’s .
Propositional logic has a very limited expressive power. For reasoning onmany application domains more expressiveness is desirable. A first choice formore expressiveness would be to use first order logic, which is undecidable ingeneral and therefore hard for automated reasoning. First order logic can berestricted by first order theories [7]. First order theories give enough expressive-ness to be appropriate for modelling real world application and problems. Thetheories are useful especially in computer science for verification tasks. Exam-ples of first order theories are the theories of equality, integers, reals, bit-vectorsand data structures like lists or arrays. Some examples for bit-vector verificationapplications are shown in the evaluation chapter (Chap. 5). One of the advan-tages of some first order theories or fragments of theories, is that satisfiability is
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(efficiently) decidable. The Satisfiability Modulo Theory (SMT) problem is thedecision problem whether a first order logic formula is satisfiable with respectto some background theory T , i.e., it decides if F is satisfiable modulo theoryT . In this thesis, a parallel algorithm for solving the quantifier free fragment offixed-size bit-vectors (QF BV) is implemented.
In recent years, the need of parallel algorithms has increased. The increase ofprocessor clock rates every year has reached a limit through physical constraintsof the silicon chips. Modern processor generations are improved by embeddingmultiple cores using a shared memory architecture. This leads to new challengesfor SAT and SMT solvers. The Question is how the highly optimized and ofteninherently sequential algorithms can be adapted to modern multicore architec-tures. In the SAT field there already exist some approaches [43, 50], whereasparallel SMT solving is a quite novel research field. Only few approaches forparallel SMT solvers have been proposed, yet.
In this thesis a parallel SMT solving approach for QF BV formulas is imple-mented. The common solving algorithm for bit-vector formulas, the bit-blastingalgorithm, is combined with look-ahead methods, which help to split the formulainto several independent subproblems. The resulting subproblems are simplerand can be solved by the sequential bit-blasting algorithm in parallel. To givean idea about the algorithm’s basic idea, an example is given in the followingsection.
1.2 Introductory Example
The basic idea of the solving approach implemented in this thesis is illustratedwith the following example. All notations used in the example are informally de-scribed. In the preliminary chapter more formal definitions are given. ConsiderEqu. 1.1 (lazy-if-then-else example motivated by a similar example in [36]). Theformula is a first order formula in the background theory of quantifier-free bit-vectors. Using a background theory denotes that a meaning is given to the firstorder uninterpreted variables, functions and predicates. Variables and constantsare numbers of fixed-size bit-vectors, similar to data types in programming lan-guages. For instance the variable z can be an unsigned integer, a bit-vectorvariable with bit-width 32 representing numbers from 0 to 232 − 1. Commonfunctions on bit-vectors like “·” (arithmetic multiplication) or “p ? a : b” (aso-called if-then-else conditional or ite: if predicate p holds then a else b isreturned) are defined.
z · (x0 = y0 ? y0 · (x1 = y1 ? y1 : 1) : 1) =
(x0 = y0 ? x0 · (x1 = y1 ? x1 · z : z) : z) (1.1)
The left part and the right part of the equality in Equ. 1.1 represent two
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z · (x0 = y0 ? y0 · (x1 = y1 ? y1 : 1) : 1)
6= (x0 = y0 ? x0 · (x1 = y1 ? x1 · z : z) : z)
z · y0 · (x1 = y1 ? y1 : 1)
6= y0 · (x1 = y1 ? x1 · z : z)
z · y0 · y16= y0 · y1 · z
z · y0 · 16= y0 · z
x1 6= y1 x1 = y1
z · 1 6= z
x0 6= y0x0 = y0
Figure 1.1: Splitting the satisfiability version of Equ. 1.1 into simpler subfor-mulas
different models of implementations in some programming language. The imple-mentation on the right side is an optimization of the implementation on the leftside, where the outermost multiplication is propagated to the innermost terms.If we can prove that the two models are equal we can replace the original im-plementation with the optimized one. This verification method is commonlycalled equivalence checking. To prove that the two implementations are equal,the validity of Equ. 1.1 has to be shown. A formula is valid iff for all possiblevariable assignments it evaluates to true (>, 1). Since in practice SatisfiabilityModulo Theory solvers are used, the problem needs to be converted to a sat-isfiability problem. A formula is valid iff the negation of the formula is unsat .Thus, to check the validity of Equ. 1.1, it is negated and given to an SMT solversupporting QF BV. As shown in this thesis, formulas containing multiplicationswith high bit-width are often very hard to solve in reasonable time for state-of-the-art SMT solvers.
In this thesis an approach is presented to split such hard formulas into severalmore easy to solve subformulas, which can be solved independently. The gener-ation of smaller and more easy subformulas is shown in Fig. 1.1. On the top ofthe tree is the original formula to solve which is the negation of Equ. 1.1. First,the formula is split into two subformulas by asserting the predicate x0 = y0 tobe false on the left side, and true on the right side. Finding the best predicateon which the formula is split is determined by means of a look-ahead method.On the left side, with the knowledge that x0 6= y0, the outermost conditionalscan be replaced with their “else” branch and the formula can be reduced toz ·1 6= z. This subformula is trivially unsat : There is no variable assignment forz such that z ·1 6= z evaluates to true. In other words, any variable z ·1 is always
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equal to itself. On the right hand side the positive assertion allows to reducethe formula by removing the conditionals with their “if” branch. Further, x0
can be replaced with y0 because we asserted x0 = y0. The resulting reducedsubformula can again be split into two smaller subproblems by asserting theremaining predicate x1 = y1. Applying conditional and variable eliminationagain, analogously to how it was done after the first split, leads to two trivialsubproblems. On the left side, there can be no assignment for z and y0 suchthat z · y0 · 1 6= y0 · z evaluates to true (unsat). The multiplication is commuta-tive. Same argument applies on the right side with result unsat . All generatedsubproblems are unsat . Therefore it can be concluded that the original formulais unsat and Equ. 1.1 is valid. The two models are equal and the non-optimisedimplementation can be replaced with the optimized implementation. By makingsome assumptions and applying some rewriting rules the hard to solve multipli-cation operations can be removed and trivial subformulas are generated whichcan be solved quite efficiently.
1.3 Outlook
Chap. 2 briefly introduces SAT algorithm’s with a focus on look-ahead solvingapproaches. The theory of the quantifier free fragment of fixed-size bit-vectorsis defined, followed by an explanation of the basic concepts of Boolector1, anSMT Solver for the quantifier free theories of fixed-size bit-vectors and arrays(QF ABV), on which the parallel SMT solver is based on. The subsequentChap. 3 gives an overview for state-of-the-art parallel SMT and SAT solverapproaches. Chap. 4 introduces the algorithm implemented in this thesis, be-ginning from a high level view down to several implementation details. Chap. 5contains the evaluation results for different benchmark sets. We show for whichtype of benchmarks the algorithm works and for which not. The last chapterChap. 6 concludes the thesis and gives an outlook for possible future extensionsto improve the algorithm’s performance.
Tables and figures which are too big for floating text integration are addedin the appendix.
1http://www.fmv.jku.at/boolector
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1.4 Contributions
• In this thesis we show a parallel SMT approach for solving QF BV for-mulas by combining conventional bit-vector solving techniques with look-ahead methods. We lift the Cube and Conquer [39] approach from bit-levelto the SMT term-level.
• We propose look-ahead methods for bit-vector formulas. With the aid oflook-ahead a formula is split into a set of more easy to solve subformulas,which can be solved in parallel.
• A new class of rewriting could be introduced, namely the AIG level rewrit-ing, with the help of the parallel implementation. By a closer investigationof the parallel evaluation results the first AIG rewriting rule was found forBoolector (see Sect. 5.4.3 problem 14 ).
• The algorithm’s limitation will be shown. In practice the approach workswell for a certain kind of bit-vector formulas. The possible speedup isheavily related to the input formula’s structure.
• We propose several optimisation methods to improve the algorithm.
• Formula structures on which the approach works well will be identifiedand examined in more detail.
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Chapter 2
Preliminaries
This chapter will give a definition for bit-vector formulas and a brief introductioninto solving concepts on which our parallel algorithm is based on.
2.1 Satisfiability for Propositional Logic
To understand the terms used in the latter sections, here the basic conceptsfor propositional SAT solving are shown. We will take slightly closer look tolook-ahead based SAT solvers. For more details about the topic the reader isreferenced to [14].
Given a propositional formula, a SAT solver checks for a propositional for-mula whether it is satisfiable or unsatisfiable. The majority of state-of-the-artSAT solvers take a propositional formula in Conjunctive Normal Form (CNF)as input, which is defined as follows:
• A formula in CNF is a conjunction of clausesC1 ∧ C2 ∧ . . . ∧ Cn
• A clause C is a disjunction of literalsC = l1 ∨ l2 ∨ . . . ∨ lm
• Each literal l is either a Boolean variable in positive phase x or a Booleanvariable in negative phase ¬x
• A clause is called unit if it consists of one literal only.
• A clause C is satisfied if at least one literal in C is assigned to be true (forexample in clause ¬x ∨ y, if variable x assigned to be false or variable yis assigned to be true).
CNF restricts general propositional logic by just using conjunctive and dis-junctive connectives. A general propositional formula can be transformed into
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an equivalent CNF formula with exponential increase in terms of size in theworst case. To avoid this exponential increase of, a method called Tseitin trans-formation [57] is used. Any propositional formula can be Tseitin transformedinto an equisatisfiable CNF formula. Two formulas f and g are equisatisfiable,if g is satisfiable (unsatisfiable) if and only if f is satisfiable (unsatisfiable). Thefollowing equation shows an example of a satisfiable formula in CNF, e.i., itevaluates to true with variable assignments a = true, b = false, c = false:
F = (a ∨ b ∨ c) ∧ (¬a ∨ ¬b) ∧ (¬a ∨ ¬c) (2.1)
2.1.1 DPLL
The basic DPLL algorithm [14,27] is based on the so called Boolean ConstraintPropagation (BCP). If a clause is unit, i.e., it contains only one literal l, l has tobe set to true to find a satisfying assignment. Then the formula can be reducedby applying following two rules. First, if the unit clause with l is satisfied allclauses containing l are also satisfied and can be removed. Second, the literal¬l can be removed in each clause where it occurs. The second rule is a conse-quence of the unit resolution rule. New unit clauses can appear in the reducedformula (implied unit clauses). BCP is the process of applying the two rulesuntil fix-point. If no clauses are left (all clauses are removed by the first rule),the formula is satisfiable (sat), and if the empty clause is detected (all literalsof a clause are removed by the second rule) the formula is unsatisfiable (unsat).
The DPPL algorithm starts with BCP on the input formula F . If no re-sult (sat/unsat) is found by the reduction a decision variable xdec is picked.Then DPLL is recursively called two times, first with xdec assigned to be true(F ∧xdec) and second with xdec assigned to be false (F ∧¬xdec). If one recursivecall returns sat , F is sat . If both recursive calls return unsat the previous deci-sion is undone (backtracking) and set in the opposite phase. In the unsatisfiablecase the algorithm stops after “all” combination of possible assignments whereexamined.
Consider Equ. 2.1 and assume DPLL makes a decision and selects variablea. In the first recursive call it is assumed to be true and added as unit to theformula F ∧ a. Then the clause (a ∨ b ∨ c) is also true and can be removed(rule one). By applying rule two, ¬a can be removed from the clauses (¬a∨¬b)and (¬a ∨ ¬c), resulting in a reduced formula with two unit clauses ¬b and ¬c.BCP on the two new implied unit clauses leads to the empty formula and weconclude that the formula is sat . The satisfying assignment is the combinationof the decisions made (a) and the units implied by BCP (¬b,¬c).
2.1.2 Look-Ahead Based SAT Solvers
Look-ahead based SAT solvers implement a variant of the DPLL algorithm. InDPLL we need to select a variable after BCP and call DPLL recursively. To
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make this decision, however, is subject to various heuristical methods. Look-ahead based SAT solvers select decision variables based on a so-called look-aheadprocedure [40]. A look-ahead on a formula F with variable x is done as follows:x is added as unit first in positive and then in negative phase to F . BCP isperformed for both phases and the reduction is measured. This is done for eachvariable in F . The variable which leads to the highest reduction is selectedas decision variable in the current DPLL step. Finding a decision variable bylook-ahead is a costly operation but the DPLL search tree can be pruned byconcentrating on making “good” decisions instead of cheap guesses. Empirically,look-ahead solvers exhibit a good performance on hard combinatorial problems.
Heuristics
A difference or distance heuristic measures the reduction. Usually this is doneby counting the number of reduced but not satisfied clauses after BCP. A methodused in the OKsolver [52] called clause reduction heuristic weights the numberof reduced clauses with the number of literals left: Reduced clauses with twoliterals left are more important and get a higher weight than clauses with threeliterals left and so forth. More on branching heuristics for selecting a decisionvariable can be found in [48]. The reduction of a variable in positive and innegative phase have to be combined: A variable is a “good” decision variableif the reduction can be maximised in both phases. An effective heuristic tocombine the two reductions is done by multiplication: Select the variable whichmaximizes the score positive reduction · negative reduction.
The direction heuristic decides after a variable is picked whether the variableis set first in positive or in negative phase in the DPLL search tree. Making aright decision can improve performance on satisfiable instances significantly. Ifa theoretically perfect heuristic always chooses the right value, no backtrackingneeds to be done in the DPLL search tree. On unsatisfiable instances bothbranches would have to be visited anyhow. One strategy implemented in varioussolvers is to approximate the branch which is more likely to be satisfiable andselect this direction first. A simple way to do this is to select the phase whichmore often occurs in the formula. Select the positive phase first if the formulacontains the decision variable xdecision more frequently than ¬xdecision, andotherwise select the negative phase first.
Failed Literals
If the look-ahead on variable x leads to a conflict (BCP leads to an emptyclause), x is called a failed literal, and the unit clause ¬x can be added to theformula. If x and ¬x are failed literals the look-ahead procedure is aborted. Inthis case no solution can be found and the search can be stopped on the currentsearch branch. DPLL backtracks in the search tree by undoing previously madedecisions if possible.
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Further Enhancements
To reduce the cost of the look-ahead, several extensions to the base algorithmare proposed. Most of these enhancements reduce the “good” decision quality.One enhancement is to use a preselection of variables on which the look-ahead isperformed. Another improvement is the so-called Tree Based Look-Ahead [41],which builds up an implication graph and selects those variables first, on whichimplications are shared. This can efficiently reduce the cost of the look-aheadfunction.
2.1.3 Conflict Driven Clause Learning (CDCL) Solvers
For completeness sake, we also mention the Conflict Driven Clause Learning ap-proach here (for details see [56]). The method has its strength in solving hugeand real world application instances. In contrast look-ahead approaches, CDCLuses a rather cheap heuristic for selecting a decision variable. If a CDCL solverruns into a conflict, the conflict is analyzed and a conflict clause is learned. Con-flict clause learning allows non-chronological backtracking, which can effectivelyprune the DPLL search tree.
2.2 The Theory of Bit-Vectors
In this thesis we consider the quantifier free fragment of the theory of fixed-size bit-vectors QF BV. As the name implies, there are no (nested) quantifiersallowed for this restricted first order logic. First order theories are commonlydefined by its signature Σ and a set of axioms A (see [16]). In the following wedefine a subset of bit-vector arithmetic by defining the grammar as in [47]:
formula : formula ∧ formula | ¬formula | (formula) | atom
atom : term rel term | Boolean-variable | term[constant ]
rel : < | =
term : term op term | variable | ∼ term | constant | atom ? term : term |term[constant : constant ]
op : + | − | · | >> | << | & | | | ⊕ | ◦
On the top level, bit-vector formulas can be arbitrary connected by Booleanoperators. Common, useful operators are obtained by using a combination ofthe shown operators (e.g., a ∨ b ≡ ¬(¬a ∧ ¬b) or a ≥ b ≡ ¬(a < b)). Atoms canbe seen as bit-vectors with bit-width 1. Terms are bit-vectors with bit-width≥ 1. Variables are bit-vector variables with bit-width n, representing naturalnumbers in a finite domain from 0 to 2n−1. Constants are fixed natural numbersfrom 0 to 2n − 1. Terms can be produced, for instance, with binary functions(term op term). Binary arithmetic functions (+,−, ·), denote the respectivearithmetic operator on bit-vectors. Operator + for instance is the bit-vector
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addition with common overflow semantics. The operators << and >> are log-ical shift left and right. Operators &, |,⊕ are the logical bit-wise operatorsAND, OR and XOR, respectively. The operator ◦ is the concatenation of twobit-vectors. For example the concatenation of two bit-vectors with bit-width 4results in a bit-vector with bit-width 8. The unary operator ∼ is the logicalbit-wise negation of a term. The if-then-else operator p ? a : b returns term aif atom p is true, and term b otherwise. And finally there are slice operatorsdefined such that a[n] returns bit n of term a, and the slice over a range a[m : n]returns a sub-term of a, containing the bits m down to n.
The set of operators defined defined above is a subset of all possible bit-vector operators. It can be extended with any arbitrary bit-vector operator(e.g., arithmetic division or remainder functions, reduction functions like an“AND reduce” which connects each bit of a bit-vector by logical AND andreturns a Boolean result, (unsigned/signed) sign-extension functions, . . . ). Moreinformation for SMT theories and standards can be found at the SMT-LIBwebsite1.
2.3 Boolector
Boolector2 [18] is an efficient SMT solver developed at the Institute of FormalModels and Verification at Johannes Kepler University Linz. Boolector supportsformulas in the theory of QF BV. Actually it was developed to show an efficientapproach which solves bit-vector formulas in combination with the extensionaltheory of arrays QF ABV [19]. The theory of arrays especially is interesting forhardware and software verification tasks. Arrays allow to model for instancesoftware memory or hardware caches and do some reasoning on it. The mostrecent improvements in Boolector were to support lambda expressions [51, 54].Lambda expressions have a higher expressiveness which allow to model parallelupdate function on arrays like it is done in the standard C library functionsmemset() or memcpy(). The latest version (v2.0) of Boolector also supports thetheory of equality and uninterpreted functions EUF. Boolector has won severalprices at frequently held SMT solving competitions3: In 2014 Boolector won thecompetition in the QF BV and QF ABV divisions. In 2012 and 2008 Boolectorwon the first place for QF BV and QF AUFBV divisions.
As input format Boolector supports the SMT-LIB v1 and v2 formats [6],and its own BTOR format [20] an easy to parse input format for bit-vectorsand arrays. Boolector provides a C and Python API for direct tool integration.
Internally Boolector represents the formula as s Directed Acyclic Graph(DAG) with structural hashing. That means that nodes with identical sub-
1http://smt-lib.org/2http://fmv.jku.at/boolector/3http://smtcomp.org/
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1: var (8)x0
2: var (8)y0
3: eq (1)
12
4: var (8)x1
5: var (8)y1
6: eq (1)
12
7: var (8)z
8: mul (8)
2 1
9: cond (8)
1 32
10: mul (8)
1 2
11: cond (8)
1 32
12: const (8)
13: cond (8)
2 13
14: mul (8)
12
15: cond (8)
13 2
16: mul (8)
12
17: eq (1)
12
Figure 2.1: Formula 1.1 represented as DAG. Each node contains an uniqueid and its type. The node’s bit-width is notated in the parenthesis. A negatednode is indicated with a dot (node 17). The edge values represent the parameterordering.
structures are shared. Fig. 2.1 shows formula 1.1 with bit-width 8 representedas a DAG. The root node is a Boolean node and represents the whole formula.If there are several roots, each root is an assertion which all have to be satis-fied. This is semantically the same as if all root nodes where conjoined withthe Boolean ∧ operator. The roots are also called outputs. The leave nodesare variables or constants with bit-width ≥ 1. Variables are the inputs of theformula. The intermediate nodes are arbitrary nested operators defined in thetheory. Note that, for instance, conditional 15 and 11 share the same substruc-ture, the predicate y0 = x0 (node 3).
As in this thesis, a parallel algorithm for solving formulas in the theory offixed-size bit-vectors is implemented, in the next sections, the bit-vector solvingprocedure implemented in Boolector, the so-called bit-blasting approach, will beintroduced. More details for solving other SMT theories and their combinationscan be found in [16] and [47].
2.3.1 Bit-Blasting
Bit-blasting is the state-of-the-art approach for solving bit-vector formulas. Theapproach benefits from nowadays available high-performance SAT solvers andis quite efficient in combination with term-level rewriting (see Sect. 2.3.2).
Each bit-vector operation in the formula is encoded as a Boolean circuit. In
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n variables clauses8 989 2802
16 4005 116832 16181 4787464 65109 193986
128 261269 781122256 1046805 3135042
Table 2.1: The size of the CNF problem of Equ. 1.1 with bit width n afterTseitin transformation.
Boolector this is implemented by introducing an intermediate layer, the AndInverter Graph (AIG) layer. An AIG represents a circuit (or Boolean formula)consisting of only AND (∧) and NOT (¬) gates. In general, AIG implementa-tions also uses structural sharing of isomorphic sub-graphs. In Boolector, eachterm-level operatior is first encoded as an AIG. For example, an equality node(x = y) can be encoded as a circuit with a Boolean output which is true iff eachbit of x is equal to each bit of y. The circuit of the bit-vector equality node overvariables with bit-width 2 is defined as follows (note that the subscript numbersdenote the Boolean bit variables of the term at position i):
x0 = y0 ∧ x1 = y1 ⇒ ¬(x0 ⊕ y0) ∧ ¬(x1 ⊕ y1)
toAIG====⇒ (¬(x0 ∧ ¬y0) ∧ ¬(¬x0 ∧ y0)) ∧ (¬(x1 ∧ ¬y1) ∧ ¬(¬x1 ∧ y1))
Next, the AIG circuit is encoded to a propositional formula in CNF viaTseitin transformation. Finally, the resulting CNF formula is fed to the under-lying CDCL SAT which determines it to be sat or unsat .
Arithmetic operators are transformed into circuits which are similar to hard-ware arithmetic circuits. For example, an add operator is encoded as a chain offull-adder circuits. A multiplication operator is encoded as a shift-and-add mul-tiplier circuit. Fig. B.1 in the appendix shows Equ. 1.1 with bit-width 8 encodedas an AIG. The reason why multiplications are often a bottleneck for bit-blastingis shown in Tab. 2.1, which shows the number of clauses and variables of theCNF representation of Equ. 1.1 (generated by Tseitin transformation), whichcontains 4 multiplications with bit-width n. With increasing bit-width the mul-tiplication circuits become huge and therefore hard for the SAT solver. Forexample, Equ. 1.1 in the 256 bit version, the CNF formula contains 1 millionvariables and 3 million clauses (note that in addition to the exploding CNF size,the symmetry and the connectivity of bit-blasted multiplications is a burden forthe SAT solvers).
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2.3.2 Term Rewriting / Simplification
As mentioned above bit-blasting can be very powerful in combination with termrewriting (and other simplification techniques). The goal is to reduce the for-mula already on term-level to keep the later generated CNF problem as small aspossible, such that the SAT solver can solve it efficiently. In Boolector, rewrit-ing is divided into 3 levels. Level 1 applies local rewriting rules during formulaconstruction (e.g, a∧¬a ≡ ⊥). Level 2 and 3 apply global rewriting rules (e.g.,term substitutions, arithmetic normalization). Rules of quadratic worst casecomplexity are bounded by recursion depth [17]. We will illustrate some localand global rewriting rules with the following example. Reconsider one of thesubformula (z · y0 · (x1 = y1 ? y1 : 1) 6= y0 · (x1 = y1 ? x1 · z : z)) ∧ x1 = y1produced in introductory example in Fig. 1.1.
(z · y0 · (x1 = y1 ? y1 : 1) 6= y0 · (x1 = y1 ? x1 · z : z)) ∧ x1 = y1 conditional elim.
z · (y0 · y1) 6= y0 · (x1 · z) ∧ x1 = y1 variable elim.
z · (y0 · y1) 6= y0 · (y1 · z) ∧ y1 = y1 equivalence
z · (y0 · y1) 6= y0 · (y1 · z) ∧ > Boolean rewr .
z · (y0 · y1) 6= y0 · (y1 · z) normalisation
z · (y0 · y1) 6= z · (y0 · y1) equivalence
⊥
The first rule, conditional elimination was already shown in the introductoryexample: With the knowledge that the conditional’s predicate is true (x1 = y1)the conditionals can be eliminated by replacing them with the “if” branch. Thesecond rule, variable elimination, eliminates x1 by replacing every occurrenceof x1 with y1 through the knowledge that x1 = y1. The equivalence rule isimplemented with the help of structural node sharing. If a substructure (herey1) is equal to itself the equality can be replaced with >. The expression > canbe removed with the Boolean rewriting rule x ∧ > ≡ x. In the next line, arith-metic normalisation is applied on the commutative multiplication operation byreordering the variables. And finally the equivalence rule solves the subprob-lems. “The same substructure is not equal to itself” is always ⊥.
This example highlights the importance of rewriting. The formula proir torewriting contains 3 multiplications. Assume that the bit-width of the variablesis 256. Without rewriting, the formula would be bit-blasted to a very hard CNFformula containing millions of variables and clauses. Applying some “simple”arithmetic and logical rules, the formula can be solved with result unsat alreadyon the term-level, without even generating a Boolean circuit and starting theSAT solver.
2.3.3 Boolector API
This section will introduce the most important functions of the Boolector C API,which in the parallel implementation will be used. The Btor structure represent
13
Boolector ModelChecker . . . PBoolector
Boolector API Interface
Boolector Core
SAT Solver API Interface
Lingeling PicoSAT . . .
Figure 2.2: The Boolector architecture from a top level view.
an Boolector instance which contains formula in the internal representation.
• boolector sat(Btor*):intFirst the formula is simplified by preprocessing (including term rewriting),then it is bit-blasted to SAT. It returns the result sat (10) if the formulais satisfiable and unsat (20) otherwise.
• boolector limited sat(Btor*,int):intThis function behaves similar to boolector sat(), but the underlying CDCLSAT solver stops after the given conflict limit is reached. The functionreturns unknown (0) if the number of conflicts is exceeded, else it returnsthe result sat or unsat .
• boolector clone(Btor*):Btor*Returns an exact but disjunct copy (clone) of a given Boolector instance-This method is important for producing independent subproblems. Thesearch can be independently (in parallel) continued on the clone.
A high-level view of Boolector’s internal architecture is shown in Fig. 2.2.On the bottom, a set of SAT solvers can be plugged into Boolector via theSAT solver API interface. The back end SAT solver is exchangeable. EitherLingeling, PicoSAT or any other supported solver can be used. The core ofBoolector implements the lemmas on demand approach for lambdas, it main-tains the AIG layer and is further responsible for bit-blasting the formula toSAT (see [19, 51, 54])). On top of the Boolector API several tools are imple-mented. For instance the Boolector front end a standalone SMT solver tool anda model checker implementation are based on the API. The parallel approachimplemented in this thesis, namely PBoolector, is a further tool based on theBoolector API.
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Chapter 3
Related Work
3.1 Overview
Parallelization is a rather novel research field for SMT solving. For SAT solversseveral parallelization approaches have been proposed. An overview on thestate-of-the-art in parallel SAT solving is given in [43,50]. SMT solvers use SATsolvers as back end procedure (e.g., Boolector [19], Z3 [28], STP [35]). Simpleparallel SMT solvers integrate parallel SAT solver implementations to run onmulticore architectures (simplifyingSTP [5]). Parallel SAT/SMT solvers can beroughly classified into three types. The types differ in the level of abstractionwhere parallelization is introduced. The most abstract level 2 is a competitiveapproach, whereas level 1 and 0 are cooperative approaches:
• Level 2: The most abstract layer is called portfolio approach. Differentsequential algorithms or the same algorithm with different parameters (aportfolio of solvers) are started on the same input formula. The solverinstances are run in parallel. The fastest algorithm returns the resultand all other running instances are stopped. This simple approach hasbe shown to be very successful. Different kinds of formulas exhibit quitedifferent performance behaviours on different algorithms or parameter set-tings. Hence, overall it is likely that one of the started algorithms performsbetter than others on a given formula.
• Level 1: The second approach at an intermediate level of abstraction isbased on search space partitioning. It tries to split the problem intosmaller and more easy to solve portions. The subproblems can be solvedin parallel by the sequential algorithms.
• Level 0: And finally, the third approach is an attempt to parallelize thesolving algorithms. This is a more classical approach of parallelization. Inpractise, it is quite hard to find appropriate ways to distribute the workof the highly optimized and inherently sequential algorithms. For SAT
15
solvers, implementations have been proposed which share the work of themost intensive part, the Boolean Constraint Propagation (see Sect. 2.1.1).
Some implementations share information whereas others run completely in-dependent. Sharing learned information introduces synchronisation overheadwhich possibly has negative influence to the parallelization capabilities. On theother hand, information sharing prevents another worker to perform work whichwas already done by someone else. Parallel CDCL SAT solver usually exchangelearned conflict clauses to share information (see e.g., ManySat [37]).
3.2 Parallel SMT Solvers
There are few approaches which directly address parallelization for SMT al-gorithms. The introduction of the concurrent track in the SMT competition2010 was designed to advance the research for parallel implementations [5]. In2010, two parallel solvers where submitted: The solver test pmathsat runs sev-eral instances of the sequential MathSat 5 [22] solver in parallel with differentparameters (portfolio). The solver simplifyingSTP is a variant of the STP [35]solver. It supports QF BV formulas. First the bit-vector formula is simplifiedon the term level, then it is converted to propositional logic which is solvedby the portfolio based parallel SAT solver ManySat [37]. For the competition2012 no solvers were submitted to the parallel track [24]. Two more advanced(and well analyzed) parallel SMT implementations using information sharingare introduced in the following sections.
3.2.1 Parallel Z3
One of the few parallel SMT solvers is a parallel version of Z3 [28]. Z3 is an SMTsolver supporting a variety of theories. It integrates a DPLL-based SAT solver,different theory solvers and an E-matching abstract machine for quantifiers. Theparallel implementation [59] follows a portfolio approach. Different portfoliosare shown to solve formulas in the theory of QF IDL (quantifier free integerdifference logic). The first portfolio runs 4 different arithmetic theory solvers.Another portfolio chooses different parameters for the SAT solver. Best speedupwas achieved by using the same theory solver but initialising them with differentrandom seeds. Compared to sequential Z3 the portfolio approach achieves anaverage speedup of up to 3.5 on 4 cores on the selected benchmarks. In theimplementation the produced lemmas of the theory solvers are shared over theparallel instances, which was shown to be crucial for good performance.
3.2.2 Parallel Interval Constraint Solver - Picoso
Picoso [45] is a parallel SMT solver for the undecidable theory of reals extendedwith transcendental functions. It is based on the iSat algorithm [32], a combi-nation of the DPLL search algorithm with interval constraint propagation, en-riched with common CDCL methods like conflict learning and non-chronological
16
backtracking. Initially the real variables are assigned to intervals. To find a sat-isfying assignment the search space is traversed by splitting a variable’s intervalinto two disjoint intervals. This results into two, more constraint subproblems.New constraints are propagated by updating the remaining variable’s intervals.The search space is split until either a solution is found, or each subproblemleads to a conflict. Picoso introduces parallelization by continuing the search(after generating the subproblems) on different machines or CPU cores (work-ers). Picoso is a typical application of the search space partitioning approach(Level 1). Learned conflicts are exchanged over different workers. Search spacesplitting is done on demand: If a worker becomes idle a running worker is askedto get one of its unresolved (already split) subproblems. Compared to sequentialiSat, Picoso achieves an average speedup on selected benchmark files of ∼ 3.8on a 4 core machine. In [44] an extension for Picoso is proposed, where thesearch space partitioning approach is combined with portfolios.
3.3 (Concurrent) Cube and Conquer
Cube and Conquer (CC) [39] is a parallel approach for solving propositionallogic formulas. Parallelization is introduced by search space partitioning (level1) without information sharing. As it is the base concept for the implementationpresented in Chap. 4, we will give a more detailed description in the following.
CDCL SAT solvers nowadays are very successful for solving huge industrialinstances. They are using inexpensive local heuristics to decide which variableis picked next in the search tree. Look-ahead SAT solvers, on the other hand,are able to solve small hard combinatorial problems. They try to make goodand global decisions which variable is picked next for assignment by using moreexpensive heuristics. CC, introduced by Heule et al. [39] tries to combine thestrengths of both methods.
A look-ahead solver is used to split the formula into smaller, more easy tosolve subproblems (cube phase). This is done by adding look-ahead decisionsin the form of cubes to the formula. A cube is a conjunction of literals. Eachliteral in the cube is a decision made by the look-ahead procedure. The formulais split into thousands of subproblems which are small enough that they can besolved efficiently by a CDCL procedure (conquer phase). Due to the indepen-dence of the subproblems the CDCL solver procedures can be run in parallel.The sum of the subproblem’s runtimes should be similar to the runtime of theoriginal problem. The runtime of individual subproblems should be comparable.
Fig. 3.1 shows a simple example of how a CNF formula F is split. Each noderepresents a subproblem on which the look-ahead procedure makes a decision toassign a variable. On the root node, the original formula F , it is decided to as-sign x5 to true. In the next step variable x7 is assigned and so on. If the problemis small enough that it efficiently can be solved with a CDCL procedure a cutoff
17
x5
x2
F5x3
F4F3
f t
f tx7
F2F1
t f
t fF1 := F ∧ (x5 ∧ ¬x7)
F2 := F ∧ (x5 ∧ ¬x7)
F3 := F ∧ (¬x5 ∧ ¬x2 ∧ ¬x3)
F4 := F ∧ (¬x5 ∧ ¬x2 ∧ x3)
F5 := F ∧ (¬x5 ∧ x2)
Figure 3.1: The lookahead solvers search tree on the left. The leaves shows thelookahead’s cut off. The smaller subproblems F1 . . .F5 are solved by a CDCLprocedure
heuristic decides to stop the look-ahead phase. The resulting leaves F1, F2, . . . ,F5 represent the conjunction of F with all previous made decisions, which canbe solved in parallel by the CDCL solver. If one subproblem evaluates to sat theoriginal formula is sat . If all subproblems are unsat the original formula is unsat .
There are two crucial points to make CC work well. First, a good look-ahead heuristic has to be chosen. The look-ahead procedure makes the decisionon which variables the formula is split. For the decision heuristic a slightly mod-ified version of the too costly standard look-ahead heuristic is used: Instead ofweighting the reduced clauses, the number of assigned variables during the look-ahead is counted. The variable which leads to the most variable assignmentsduring BCP in positive and negative phase is selected first.
Second, a suitable cutoff heuristic has to be chosen. This is a metric indicat-ing that the problem is small enough to stop splitting. For the cutoff, a dynamicmetric was proposed, which depends on the number of the made decisions andimplied literal assignments during BCP. The solver ILingeling [39] which im-plements CC, employs a modified version of the march look-ahead solver incombination with the CDCL solver Lingeling [11]. In [39] it was shown that theCC approach can outperform other parallel solvers like ManySat especially onvery hard (industrial) SAT instances.
An enhancement of CC was proposed by the Concurrent Cube and Con-quer (CCC) approach [58]. One problem of CC is that on many benchmarks ithad slower runtimes than a sequential CDCL solver. For CCC it was tried toimprove the lack of inappropriate cutoff heuristics. This was done by runninga CDCL solver concurrently to the cube phase. After each decision made bythe look-ahead solver, a CDCL solver (assuming the new decision) is startedin parallel. If the CDCL solver is able to solve the problem before the look-
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ahead solver makes a new decision, the current search branch can be closed.Then, look-ahead can continue search on other open search branches. Usingthis method the search branch is implicitly cutoff if the problem is small enoughfor CDCL solving. The hard part here is to implement an appropriate back-tracking mechanism to continue the look-ahead search in another branch. Ifthe CDCL solver can not solve the problem within the time limit, look-aheadselects a variable to continue splitting. After each split the parallel CDCL solveris restarted with the respective look-ahead decisions. Further, in CCC a metricwas introduced to predict if a formula is appropriate for cube and conquer. Ifthe formula seems not appropriate, the predictor aborts the search procedureafter a couple of seconds and runs the problem on a sequential CDCL solver.The CCC approach is implemented in the tool CLingeling [10].
The approaches shown in this section are implemented based on the CDCLsolver Lingeling1. Some further (parallel) tools based on Lingeling [8–12] arementioned in the following: A reduced version of CLingeling was done in thetool Flegel. Flegel recursively splits the search space with look-aheads and runsa limited CDCL search procedure before doing the next decision. There alsoexists a portfolio version of Lingeling called PLingeling. The tool Treengelingcombines the positive aspects of Lingeling’s parallel tool family: PLingeling,Flegel, CLingeling and ILingeling. Mainly the concepts of the Treengeling im-plementation are used in this work to develop a parallel SMT solver. The detailsare explained in Chap. 4. The tools from the Lingeling solver family are veryefficient and have won several prices over the last years in the annually heldSAT competitions2.
1http://fmv.jku.at/lingeling/2http://www.satcompetition.org/
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Chapter 4
Implementation
The algorithm proposed in this thesis is mainly motivated by the (Concurrent)Cube and Conquer (CC) approach [39,58] (see Sect. 3.3). CC is a parallel solv-ing method for propositional formulas by combining look-ahead methods with aConflict Driven Clause Learning (CDCL) solver. The method was implementedin a tool called Treengeling [12], which is a parallel SAT solver based on theCDCL solver Lingeling. The main goal of this thesis is to adapt the conceptsof CC to the SMT world. It further evaluates if look-ahead methods are alsoapplicable in combination with the bit-vector solving procedure of Boolectorwhere powerful term-rewriting rules are combined with bit-blasting.
The implemented tool PBoolector is a parallel SMT solver for QF BV for-mulas. It is written in C and build up on the top of the Boolector’s C-API. Thisallows to keep the Boolector implementation unchanged. For missing function-alities needed in PBoolector the API had to be extended. The most importantextension is the boolector lookahead() method, which is explained in more de-tail in Sect. 4.3. Smaller implemented auxiliary API extensions, needed for thealgorithm, are explained directly in the corresponding implementation sections.
A high level view of the basic idea of our algorithm is sketched as follows:
1 Try to solve the formula(s) within some given limit.
2 If this limit is exceeded, split the formula(s) into smaller subformulas and
continue with 1
3 Terminate with sat if one of the subproblems is sat .
4 Terminate with unsat if all subproblems are unsat .
The main goal of our algorithm is to split the a formula into smaller andmore easy to solve subformulas (subproblems). The decision on how the formulais split is made by boolector lookahead(). Splitting is continued as long as the
20
subproblems can not be solved by Boolector’s decision procedure within somegiven limit (c.f. cutoff heuristic in CC). Such a limit could be for instance atime limit, where a formula, which cannot be solved within a certain time, issplit into several subproblems. In our actual implementation, we will use theunderlying SAT solver’s conflict limit as a limit for cutoff. Step 1 and 2 are
repeated until one of the termination criterias 3 or 4 is fulfilled.
4.1 Splitting QF BV SMT Formulas
To reduce a hard to solve formula into many smaller subproblems, the formulais split into easier to solve subproblems. For example, in the propositional case,Treengeling picks a Boolean variable and assumes it to be true/false to gettwo and more simpler subproblems. For SMT formulas in the theory of bit-vectors it is slightly different how partitioning is done. Instead of choosing onlyvariables, PBoolector selects any arbitrary suitable node in the formula’s DAGrepresentation as follows.
4.1.1 Boolean Nodes
In the simplest case boolecor lookahead() selects a Boolean node on which theformula is split by asserting it to be true and false. Boolean nodes are alloperations resulting in a term with bit-width one. This includes all bit-vectoroperations (+, ·,∼, ..) having operands of bit-width one and all predicates withoperands of bit-width ≥ 1 (=, <, . . . ). Further, all if-then-else nodes containa Boolean condition node as first parameter on which the formula can be split.For example in the introductory example (Fig. 1.1) look-ahead decides to selectthe if-then-else node and split the formula by asserting its condition to be true(x0 = y0) on one branch and to be false (x0 6= y0) on the second branch.
4.1.2 Defining New Predicates
Another option is to select any non-Boolean node and define new predicates onit. Variables or any other arbitrary term with bit-width > 1 can be used tosplit the formula. Reconsider formula F in the introductory example on topof Fig. 1.1. F could also be split by picking the bit-vector variable x1 (withbit-width > 1) and asserting a newly defined predicate x1 = 0 on one branchand its negation x1 6= 0 on the other branch. Note that not only equality, butany other predicate can be used (e.g. x1 < 0 and x1 ≥ 0).
4.1.3 Setting Single Bits
Another approach for splitting is to set single bits of arbitrary bit-vector terms.This is especially interesting for hard operations like multiplication. In [49] itwas shown that the solving effort can be reduced by setting the least significant
21
bit (LSB) of a multiplication to true in one branch, and to false in the secondbranch.
4.2 Main Algorithm
This section will show our implemented PBoolector algorithm on top of theBoolector API. First the implementation of the main loop will be introduced(Sect. 4.2.1) followed by some more detailed explanation of the procedures usedin the main loop (Sect. 4.2.2, Sect. 4.2.3). Finally implementation details forthe parallelization strategy are given (Sect. 4.2.4).
4.2.1 Main Loop
A more detailed view of the PBoolector’s main loop (step 1 - 4 sketched atthe beginning of this chapter) is given in List. 4.1 as pseudo-code.
Listing 4.1: The main loop of PBoolector.
1 struct sp {2 F, lkhd, res;3 };4
5 pboolector (F) {6 res = 0;7 sp[ ] = {(F,0,0)}; //list of subproblems initially containing F8 search(sp[ ], lim); //initial search9 while ((res = flush(sp[ ]))==UNKNOWN) {
10 lookahead(sp[ ]); //look−ahead phase11 split(sp[ ]); //split phase12 lim = upd(lim);13 search(sp[ ], lim); //search phase14 }15 return res;16 }
Subproblems are organized in a structure sp (line 1). The structure containsa formula instance F, a look-ahead node lkhd and an intermediate result res.The generated subproblems are maintained in the list sp[ ]. Initially the listcontains one subproblem, which is the original formula F (line 7). The con-tinuous splitting and searching is implemented in the while loop (line 9-14).The loop consists of following 3 phases: look-ahead (line 10), the split (line 11)and search phase (line 13). Each phase performs the according operation foreach subproblem in the list. For instance the search-phase makes a boolector-limited sat(F,lim) call for each subproblem F in the list sp[ ].
The algorithm starts with an initial limited search on the original formula(line 8). The initial search allows to solve easy instances immediately withoutsplitting the formula. However, if the initial search cannot solve the original
22
F UNKNOWN
F0UNKNOWN
x1 = y1
F1 UNKNOWN
x1 6= y1
��HHF00UNSAT
x1 = 0
F01UNKNOWN
x1 6= 0
���HHHF010UNSAT
x0 = y0
���HHHF011 UNSAT
x0 6= y0
��HHF10 UNSAT
x0 6= y0
��HHF11 UNSAT
x0 = y0
Figure 4.1: Sample execution represented as binary search tree
formula the main loop is entered. In each iteration (round), the subproblemsare tried to be solved as follows: First a look-ahead is performed to split thesubproblems. The split phase produces for each subproblem, on which look-ahead is performed two new subproblems using the boolector clone() method.List sp[ ] is updated and now contains all new subproblems after splitting. Next,the limit is dynamically updated. The strategy for updating the limit is shownin Sect. 4.2.3. Finally, the actual work is performed in the search phase, whereeach subproblem is tried to be solved within a certain limit. The loop’s exitcondition is implemented in the flush method shown in List. 4.2.
Listing 4.2: The flush method implements the main loop’s exit condition.
1 flush(sp[ ]) {2 for (i = 0; i<length(sp[ ]); i++) {3 if (sp[i].res == UNSAT)4 removefromlist(sp[ ], i);5 closed++;6 else if (sp[i].res == SAT)7 return SAT;8 }9 return length(sp[ ])==0 ? UNSAT : UNKNOWN;
10 }
Flush deletes (closes) each solved and unsatisfiable subproblem in list sp[ ].If sp[ ] is empty it returns unsat . If one subproblem is evaluated to sat itreturns sat and unknown otherwise to stay in the loop and perform the nextround. Note that global variable closed counts the number of deleted nodes foreach round. It is used for dynamically updating the limit shown in Sect. 4.2.3.
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The execution of the algorithm can be visualized as a binary search tree. Apossible example execution to solve the satisfiable version of Equ. 1.1 is shownin Fig. 4.1. The execution is similar to Fig. 1.1, but now with a closer look to theactual implementation. To visualize the expansion of the search tree, differentnodes are selected by the look-ahead for splitting. Initially, the original formulaF is tried to be solved within the given limit. The search results into unknown.No node can be closed by flush() and the loop is entered. The look-ahead on Freturns the predicate x1 = y1. The formula is split by asserting the predicateto be true and false. The list of subproblems is updated and contains now F0
and F1. After updating the limit, in the search phase F0 and F1 are tried to besolved without success (result = unknown). The method flush() can not closeany subproblem and the next loop iteration is performed. In the next iterationF0 and F1 are split into 4 subproblems F00, F01, F10 and F11. Note that thelook-ahead for each subproblem is independent: The look-ahead on F1 decidesto split on the next predicate x0 = y0 and the look-ahead on F0 proposes todefine a new predicate x0 = 0 (x0 6= 0) to split the formula. The search phasewith the limited sat call is now able to evaluate F00, F10 and F11 to be unsat .The method flush() deletes these three subproblems from the list. There is stillone unresolved subproblem F01 left. In the next round F01 is split into F010 andF011 which are tried to be solved with result unsat . The method flush() deletesF010 and F011. The list is empty, flush() aborts the while loop and the resultunsat is returned from PBoolector.
To keep the algorithm simple it is implemented in a way, such that thesearch tree is a binary tree. This gives better control over the tree generation(see next section). In general, it would be possible to split a problem intothree or more subproblems. For instance on F0, three new predicates (x1 = 0,x1 = 1 and x1 6= 0 ∧ x1 6= 1) could be defined to do a multiple split. In ourcurrent implementation, however, these three subproblems are generated byperforming two binary splits resulting in four subproblem where one is triviallyunsat (F ∧ x1 = 0 ∧ x1 = 1 ≡ ⊥).
4.2.2 Search Tree Generation
In the previous section it was proposed that each subproblem is split into twonew subproblems. Generating the search tree in such a breadth-first manner,the increase of subproblems would be exponential if subproblems can not besolved. In practice breadth-first tree generation can lead quite fast to a mem-ory overflow: If an instance of a formula occupies 100MB memory (which is anot unusual value for big (bit-blasted) formulas), doing a binary split until treedepth 10 leads to a very high memory usage of 210 ·100MB = 102.4GB. On theother hand, generating the tree in a less memory consuming depth-first manneris not appropriate for the algorithm neither. By splitting only one subprob-lem each round, less formulas are generated. The problem here is that the listof subproblems will contain formulas with very different workloads for solving.One formula would be split several times whereas other formulas are not split
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at all. Different workload for solving would be bad for the later parallelization(Sect. 4.2.4). We want to produce several subproblems each round with similarsolving effort which should be solved in parallel. Hence the actual implementa-tion combines both, the breath-first and depth-first method. The search tree isgenerated in a depth-first manner but with some breadth. Each round a limitednumber (maxactive) of subproblems is produced. The following listings examinethe methods used in the main algorithm (List. 4.1) in more detail. The searchtree generation is controlled in the look-ahead and split phase.
Listing 4.3: Lookahead procedure.
1 maxactive = 82
3 lookahead(sp[ ]) {4 sortlist(sp[ ]);5 for (i = min(maxactive, length(sp)) − 1; i>=maxactive/2 && i>=0; i−−) {6 sp[i].lkhd = boolector lookahead(sp[i].F);7 }8 }
List. 4.3 shows the implementation of the look-ahead phase. Since the num-ber of generated subproblems should be restricted to be not greater than max-active (default value 8), a look-ahead is performed on maximum maxactive/2subproblems. Doing a look-ahead on 4 instances results in 8 additional sub-problems generated in the split phase. The question is, which subproblems inthe list should be selected to split first. The chosen strategy is to select sim-ple subproblems but not the simplest ones. It is assumed that the formula’shardness is proportional to its size (in reality this assumption is not alwaystrue but it will fit for our purposes). The Boolector API was extended withboolector get size() which returns the formula size (the number of unique nodesof the formula’s DAG representation). In line 4 the subproblems are first sortedby increasing size. Simpler subproblems are at the beginning and harder sub-problems at the end of the list. Next, simple formulas are selected to performa look-ahead on. In the case that the number of current subproblems is ≥ 8, alook-ahead on sp[7] to sp[4] is performed. The idea behind this selection strategyis explained at the end of the current section.
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Listing 4.4: Split procedure.
1 split(sp[ ]) {2 for (i = 0; i<length(sp); i++) {3 if (sp[i].lkhd != NULL) {4 pred = defpredicate(sp[i].lkhd); //define a predicate5 clone = boolector clone(sp[i].F);6 boolector assert(sp[i].F, pred);7 boolector assert(clone, boolector not(clone, pred));8 addtolist(sp[ ], (clone,0,0))9 added++;
10 sp[i].lkhd = 0 //reset lookahead node11 }12 }13 }
Each formula with a valid look-ahead node is split (List. 4.4). The pseu-docode uses C style pointers for look-ahead nodes. The lkhd field of structure spcontains a valid look-ahead node if the pointer is not NULL. First, a predicatehas to be defined on the look-ahead node (line 4). In the case that sp[i].lkhdis a Boolean node the predicate is the node itself. In the case that sp[i].lkhdhas bit-width > 1 a predicate is generated (e.g, x = 0). In line 5 the originalformula is cloned. Then the predicate is asserted to be true in the original for-mula (subproblem 1) and to be false in its clone (subproblem 2). The clone isadded to the list. The added counter is later used for the dynamic limit update(Sect. 4.2.3).
Listing 4.5: Search procedure.
1 search(sp[], lim) {2 sortlist(sp[ ]);3 for (i = 0; i < maxactive && i < length(sp); i++) {4 sp[i].res = boolector sat limit(sp[i].F, lim);5 }6 }
In the search phase (List. 4.5) the subproblems are finally tried to be solvedwithin the given limit lim. This is the most work intensive part. Usually 99%of the total runtime is spent in this phase. First, the list is again sorted byhardness (line 2). Next, the smallest maxactive subproblems are chosen andtried to be solved with boolector limited sat() (line 3-5). To recap, Boolector’sSAT call internally first simplifies the formula on term-level and second solvesthe bit-blasted formula with the SAT solver.
Fig. 4.2 visualizes the abstract form of the search tree after some rounds. Thetree is generated in an asymmetric way by selecting small but not the smallestsubproblems for splitting (see List. 4.3). The small subproblems are split intosmaller and smaller subproblems until they are solved. If they are solved, thenext larger subproblems are chosen for splitting. Note that if look-ahead would
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depth
size
lkhd/split
maxactive(search)
listof subproblems
Figure 4.2: The abstract form of the search tree after some rounds. The filledarea shows the current subproblems to be solved. The list is sorted by size(it is assumed that size is correlated to hardness). On the left side are smallsubproblems split several times (high depth in the search tree). On the rightside are bigger subproblems which are split less (low depth in the search tree).The look-ahead phase selects the upper part of maxactive subproblems to splitin the current round. The search phase tries to solve the maxactive smallestsubproblems.
take the smallest subproblems (most left part of the list), the splitting would betoo greedy, e.g., subproblems are produced which are too easy for search and theabstract sorted tree would look even more skewed. If the largest subproblemswould be taken for splitting (most right part of the list) the tree would begenerated in a balanced form. The algorithm would not be greedy enough tosolve many subproblems as early as possible. As a consequence, we choose anintermediate, greedy but not too greedy, version. For unsat instances the wholetree has to be expanded anyhow, until all subnodes are closed. Hence, theselection strategy will rather influence sat instances, where the main loop canbe immediately aborted by making the right look-ahead decisions.
4.2.3 Dynamic Limit
In the search phase of the main loop, subproblems are tried to be solved withinsome limit. If the limit exceeds, the subproblems are split until they are smallenough to be solved within the limit. To abort the SAT call, the back endCDCL solver’s conflict limit n is used. The SAT solver, and therefore alsoboolector sat limit(), returns with result unknown if n conflicts are exceededwithout result. The abortion leaves the SAT solver in a consistent state. Af-ter incrementally adding some further constraints, the search can be continued
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where it was aborted. Work already done does not need to be redone in thenext round of the main loop.
In each round, upd(lim) updates the limit in a dynamic way. Changing thelimit allows to control PBboolector’s interplay between the split and the searchphase. If the limit is low, less time is given to the SAT solver and the algorithmis forced to concentrate on the look-ahead and splitting. If the limit is high thealgorithm is forced to concentrate on the search phase by giving more time tothe SAT solver.
Listing 4.6: Update limit function.
1 closed, added; // number of closed and added nodes in current round2 lim=5000; //intial limit3 lmin=1000, lmax=50000; //min and max limit4
5 upd(lim) {6 if (!closed) lim ∗= 0.5;7 else if (closed > added) lim ∗= 2;8 else if (closed < added) lim ∗= 0.9:9 if (lim < lmin) lim = lmin;
10 if (lim > lmax) lim = lmax;11 added = closed = 0;12 return lim;13 }
List. 4.6 shows the update function. The variable added maintains the num-ber of added subproblems in the current round. This value is updated by thesplit() procedure. The variable closed maintains closed subproblems in the cur-rent round updated by flush(). If no subproblem can be closed, it is assumedthat the subproblems are too hard. Then the algorithm should concentrate onsplitting. The limit is halved (line 6). If PBoolector is able to close more sub-problems than added, subproblems may be split too much. It could be possiblethat the SAT solver would have been able to solve the problem more efficientlybefore splitting and with a higher limit. Therefore, more time is given to thesearch phase, by doubling the limit (line 7). If the algorithm is in an inter-mediate phase where more subproblems are added than closed the algorithm isforced to slightly concentrate more on splitting. The limit is reduced by 10%(line 8).
The look-ahead and split phases can be seen as the global search part of thealgorithm. The search phase (bit-blasting and CDCL SAT solving) is the localsearch part. On hard to solve formulas we perform a global search and split intosmaller subproblems. After enough reduction we switch to a local search. If thesubproblems become hard again the algorithm is forced back to global search.
The limit is bound by lmin and lmax. A too small minimal limit can producea lot of subproblems. A too high maximal limit can cause the SAT solver to
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spend too much time for individual subproblems. The options “-l <n>” (initiallimit), “-lmin <n>” (minimal limit) and “-lmax <n>” (maximal limit) can beset for PBoolector to configure the limits. The default values for the optionsare selected similar to the Treengeling default values.
4.2.4 Parallelization
Subproblem’s are generated via the boolector clone() interface, which makes adeep copy of a given solver instance. The resulting instance is an exact copy ofthe solver’s current state. No memory is shared by the individual subproblems,which allows to run all operations in each phase in parallel. After finishing aphase the parallel operations are synchronized in the main loop and the opera-tions of the next phase are started in parallel.
For parallelization the POSIX thread API Pthreads is used. The used APIcalls are briefly described in this section. We refer to the Linux man pages fora detailed API reference. Threads are created with pthread create(). Createtakes the function to be executed in the new thread and the corresponding ar-guments as parameters. For maintaining the thread, a thread instance of struc-ture type pthread t is returned. The pthread join() method waits for a certainthread instance until termination. Pthread supports mutexes and conditionalvariables for synchronisation. Mutexes are of type pthread mutex t. To pro-tect shared data. pthread mutex (un)lock() is used. Condition variables (typepthread cond t) allow to block the current thread until a certain condition ismet. The pthread cond wait() method blocks the current thread. The waitingthread can be woken up by another thread with pthread cond signal() when thecondition is fulfilled.
In List. 4.7 the parallelization of the most work intensive search procedureis shown. Parallelization of the other phases is done analogously by exchangingthe Boolector operation (e.g., the look-ahead call is used instead of the limitedSAT call). Note that the pseudocode only sketches the parallelization task andthe Pthread API calls are shown in a simplified form.
The subproblems to solve are maintained in the initial empty list jobs[ ](line 10). The job structure contains the thread, to which it is assigned, and thesubproblem to solve. The number of maximal workers is by default the numberof cores available on the machine (line 5). Each CPU core should try to solve 4subproblems in each round (line 6). So on a 4 core machine in each round thereare 16 active subproblems to be solved. First, maxactive jobs are scheduled to berun in parallel (line 13 -16). Second, the jobs are distributed and run as workerthreads on the available cores (line 18-25). A created thread (line 24) executesthe searchsp() method with the corresponding parameters (sp, lim), where theactual work (line 34) is performed. If all workers are busy (line 20) the threadcreation is stopped by the conditional wait (line 21) and the main thread isblocked. If a thread finishes its work, the number of current workers is decre-
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Listing 4.7: Parallelization of search phase.
1 struct job{2 thread, sp3 };4 mutex, cond;5 maxworkers = number of cores;6 maxactive = maxworker ∗ 4;7 numworkers;8
9 searchparallel(sp[ ], lim) {10 jobs[ ]={}; numworkers=0;11 sortlist(sp[ ]);12 //schedule jobs;13 for(i=(length(sp[ ])<maxactive)?length(sp[ ])−1:maxactive−1; i>=0; i−−) {14 job.sp = sp[i];15 addtolist(jobs[ ], job);16 }17 //run jobs in worker threads18 for (i=0; i<length(jobs[ ]); i++) {19 pthread lock(mutex);20 while (numworkers>=maxworkers)21 pthread cond wait(cond, mutex);22 numworkers++;23 pthread unlock(mutex);24 job.thread = pthread thread(searchsp, jobs[i].sp, lim);25 }26 //join remaining workers27 for(i=0; i<length(jobs); i++) {28 pthread join(jobs[i].thread);29 deletefromlist(jobs, job[i]);30 }31 }32 //limted search on subproblem sp33 searchsp(sp, lim) {34 boolector limited sat(sp.F, lim);35 lock(mutex);36 numworker−−;37 pthread cond signal(cond);38 unlock(mutex);39 }
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mented (line 36) and the blocking main thread is woken up (line 37). In themain thread, the while condition (line 20) becomes false and the next job is dis-tributed to a new worker thread on the free CPU slot. If all jobs are distributedthe remaining running threads are synchronised (line 27-30). Shared access tothe variable numworkers is avoided by locking. The method pthread cond wait()internally unlocks the mutex and then blocks the thread. Therefore, a finishingworker thread is able to enter the protected area (line 35-38) and wake up theconditional wait. Before pthread cond wait() returns, according to the PthreadAPI model, internally mutex is locked again and the main thread continues fromline 21.
In short: The procedure searchparallel() schedules a certain number of jobsand executes them one after the other on the next free CPU slot. On each coreat most one SAT call is done at the same time. This prevents thread switch-ing. Modern architectures use hardware CPU virtualisation (hyper-threading).In the current implementation the default value of maxworkers is equal to thenumber of real cores plus number of virtual cores. The number of maximalworkers can also be set with option “-t <n>” for PBoolector. Additionally it ispossible to set the number of maximal active nodes with option “-a <n>”. Thenumber of maximal active nodes has to be always greater equal to the numberof maximal workers.
By assigning more than one job to a worker (line 6), unbalanced workloadcan be compensated. In practice it is often impossible to split a formula intotwo parts with the same workload (even if the same limit is used for the SATcall). For instance assigning some variable x = 0, the formula may be solvedmuch faster than by assigning x 6= 0 on the other branch. The predicate x 6= 0is in general a very weak restriction and usually cannot speed up the SATcall. If there is unbalanced workload, searchparallel() does occupy one CPU corewith the work intensive subproblem, whereas more and smaller subproblemsare assigned to the other CPU slots. It is preferable to schedule the most workintensive jobs first, in order to be able to compensate them with less workintensive jobs. To do so, initially the subproblems are ordered by ascendinghardness (line 11). Harder jobs are scheduled first by running the schedule loopfrom maxactive-1 to 0 (line 13).
4.3 Look-Ahead API Extension
Beside some utility functions the look-ahead method is the only implementation,which had to be integrated into the Boolector core implementation. Therefore,a new API extension for Boolector was implemented:
• boolector lookahead(Btor*):BoolectorNode*Returns a node of the formula’s DAG (BoolectorNode*) on which the for-mula can be split.
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The look-ahead method decides on which node to split. The “quality” ofthat decision is crucial for the algorithm. This is demonstrated by the follow-ing example: Reconsider the search tree in Fig. 4.1. Look-ahead uses as firstdecision x1 = y1(x1 6= y1). After splitting, look-ahead chooses for F0 to spliton x1 = 0 (x1 6= 0) and for F1 to split on x0 = y0 (x0 6= y0). For this example“bad” decisions are made to demonstrate the expansion of the search tree. Any-how, in the introductory example in Fig. 1.1 it was shown, that F can be solvedfaster by making “better” decisions. If x0 = y0 (x0 6= y0) is selected first, theleft branch can be closed immediately. The subformula gets trivially unsat byapplying term rewriting rules. On the remaining subformula the next conditionpredicate x1 = y1 (x1 6= y1) is selected. The two generated subproblems becometrivially unsat too, which concludes the problem to be unsat . In the introduc-tory example (Fig. 1.1) 4 subproblems are created, whereas in the executionexample with the “bad” decisions (Fig. 4.1) 8 subproblems are generated forthe same formula F .
The question is which node of the formula DAG should be selected by boolec-tor lookahead(). To give an idea which node would be the best, Sect. 4.3.2shows an expensive implementation, the Probing Method. Note that the prob-ing method uses the same concepts as methods which are usually associatedin literature with the term look-ahead [40]. Look-ahead SAT solvers select adecision variable by probing the reduction. In this thesis, the term look-aheadis used in a slightly different way. If not explicitly stated otherwise, the termlook-ahead is always associated with a faster and for PBoolector more appropri-ate implementation called the Reduction Approximation Method. This methoddoes not measure the real reduction by costly simplification operations, but ap-proximates reduction by means of cheaper heuristics. Implementation detailsfor approximation are explained in Sect. 4.3.3.
The process of making a look-ahead decision is sketched in the followingpoints, and explained in detail in the according sections:
1 Find look-ahead candidates (Sect. 4.3.1).
2 Approximate or measure reduction for each candidate (Sect. 4.3.2 andSect. 4.3.3).
3 Return the most appropriate node (Sect. 4.3.4).
4.3.1 Look-Ahead Candidates
The look-ahead method first searches for look-ahead candidates. A look-aheadcandidate is a node on which the formula can be split. Candidates are all nodeswhich have an asserted (constrained) node as antecessor. If a node has noconstrained parent it is no look-ahead candidate. Further, the constraint nodesare no candidates itself. If for example predicate p and formula G are constraintnodes (F := G ∧ p) the formula F can not be split on p: asserting p to be true
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does not change F (G ∧ p ∧ p ≡ G ∧ p) and asserting the negation is triviallyunsat (G∧ p∧¬p ≡ ⊥). In the implementation the formula’s DAG is traversedonce by starting at the constraint root nodes and storing each child in a hashtable lkhd candidates.
4.3.2 Probing Method
The goal is to select a look-ahead candidate which maximizes the simplificationreduction for both subproblems, which are generated in the main loop’s splitphase. The probing method implementation simply finds this candidate by trial(probing). For each candidate the real reduction is measured. To measure thepositive reduction, the node (or a new defined predicate to split) is asserted tobe true and the rewrite engine (simplification) is called. The positive reductionpred is the difference of the formula’s size before and after the simplification.The negative reduction nred is measured by asserting the node or the predi-cate to be false. After measuring all nodes, the node with the highest scorepred ·nred is returned by the look-ahead method. The multiplication is used inthe score to return nodes that have high reduction in positive and in negativephase. Splitting should produce two subproblems, which should be able to besimplified in the main loop. For example a node with pred = 5 and nred = 5(score = 25) is preferable to a node which would produce unbalanced subprob-lems with pred = 18 and nred = 1 (score = 18).
The probing method is very cost intensive. Simplification is a time linearoperation in number of nodes contained in the DAG. Calling the linear simpli-fication function for each node in the DAG leads boolector lookahead() to havea quadratic time complexity in number of nodes.
For testing and evaluation purposes the probing method was implemented inPBoolector. With option “-pr” probing is enabled. With the additional option“-rp” the look-ahead method returns the the node with highest score pred·nred.If the simplification during probing directly leads to a result (sat/unsat) inpositive and/or in negative phase the look-ahead is aborted and the node isdirectly returned for splitting. This is similar to find failed literals in look-ahead SAT solvers. In practise, if option “-pr” is enabled most of the runtimeis spent in the look-ahead phase.
4.3.3 Reduction Approximation Method
Due to the overhead of the probing method, in PBoolector look-ahead was im-plemented without exactly measuring the reduction by using cheaper approx-imation heuristics. In the implementation the formula DAG is traversed onesand statistical features are calculated. The features are used by the heuristic toapproximate the reduction.
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It is assumed that the number of occurrences is proportional to the formula’sreal reduction. As the formula is internally represented as a DAG using struc-tural hashing, nodes are referenced several times from their parents. The mostbasic statistical feature to approximate the reduction is to count the number ofpositive occurences (pocc) and negative occurrences (nocc) for each candidate.The node with the highest score pocc · nocc is returned. Equal to the probingmethod, this method returns the node with maximal (approximated) reductionin both subproblems.
In the implementation Boolector’s unique table is used to count the occur-rences. The unique table is a hash table containing each unique node of theDAG. For each unique node the children are traversed. If a child is referencedpositive and it is contained in the lkhd candidates table a corresponding positivereference counter is incremented. If the child is referenced negative analogously,its negative reference counter is incremented. Each node has to be visited onlyonce. Therefore, this implementation is close to linear in the number of nodes.The term “close” is used because hashing with collision chains has only close toconstant time access.
4.3.4 Look-Ahead Types and Heuristics
In the implementation several different look-ahead types are defined. For eachlook-ahead type a separate heuristic can be applied. The look-ahead methodgenerates a table containing for each type the node with the maximal score. Thissection discusses in detail each look-ahead type, the corresponding heuristic andthe expected reduction. All types are summarized in Tab. 4.1 at the end of thissection. Sect. 4.3.5 shows the implementation of how the individual look-aheadtypes are combined. The desired look-ahead type is selected in PBoolector withoption “-lt <id>”.
Equality Nodes
First bit-vector equalities are examined. An equality is always a Boolean nodeon which the formula can be split by asserting it to be either true or false.The used heuristic is pocc · nocc. Bit-vector equalities are separated into threelook-ahead types:
BEQ1VN are bit-vector equalities containing at least one parameter whichis a variable with bit-width > 1. By asserting an equality with a variable tobe true (x = term) simplification can trigger basic Boolean rewriting and alsovariable elimination. By asserting x = term every occurrence of x can bereplaced with the node term, which results in a formula with x eliminated. Onthe second branch where the equality is asserted negatively (x 6= term) onlyBoolean rewriting rules can be triggered.
BEQ1V1 are equality nodes containing at least one variable with bit-widthone. An inequality with Boolean parameters (x 6= term) is rewritten into anequality with one parameter negated (x = ¬term). Splitting leads on both
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branches to Boolean rewriting and, different to BEQ1VN, also in both branchesto variable elimination.
BEQ1OTHER are all remaining bit-vector equality nodes without vari-ables. Here only Boolean rewriting rules can be triggered after splitting.
Inequality Nodes
Inequality (unsigned less than <) nodes have look-ahead type ULT1N. Thedisadvantage is that no powerful rewriting rules like variable elimination orconditional rewriting can be triggered by asserting the node. Asserting an in-equality only Boolean rewriting rules can be applied to reduce the formula. Theused heuristic is pocc · nocc.
If-Then-Else Nodes
A good candidate for splitting are if-then-else nodes (look-ahead type BCONDN).If the conditional predicate is asserted to be true or false the conditional canalways be rewritten by replacing it with the corresponding branch. Condition-als often occur uniquely in a formula but with high reduction potential. If forexample a condition has only one positive reference pocc = 1 (nocc = 0). Thescore with heuristic pocc · nocc would be 0. Nodes with score 0 are not selectedby the look-ahead procedure. In order to select also unique referenced condi-tionals the heuristic pocc + nocc is used. In the evaluation chapter (Chap. 5) itis shown that some formulas containing hard operations (like multiplications)with conditional parameters can be solved with very good speedup by splittingon conditionals.
Other Boolean Nodes
Any other node with bit-width one can be chosen as look-ahead, which aredefined as follows: Variables (VAR1), slice operations (SLICE1), and oper-ations (AND1) and all remaining operations (except constants) with bit-withone (OTHER1). For AND1 and OTHER1 types the pocc·nocc heuristic is used.For VAR1 and SLICE1 the heuristic pocc + nocc is used. Like for conditionalsthe “·” heuristic would not select uniquely referenced variables or slices. The“+” heuristic is used, because in practice it was observed that fixing one-bit-slices or variables with only one reference, sometimes can speedup the solvingprocedure. An example was found in the SMT-LIB benchmarks, the problem 19from category calypto (see Sect. 5.4.3).
Defining New Predicates for Hard Operators
It is especially desired to be able to rewrite hard operators like multiplicationsor additions. Look-ahead type MULNV0 selects multiplications that containat least one variable (x · term). The formula is split by defining a new predicate(x = 0) and asserting it to be true (x = 0) and false (x 6= 0), respectively.Setting x = 0 the multiplication can be reduced by variable elimination and
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arithmetic rewriting to a zero constant (x · term ∧ x = 0 ≡ 0). The problemis that on the second search branch with the constraint x 6= 0 nothing can bereduced or rewritten. The subproblem is mostly as hard as the original problem.The constraint x = 0 is very strong whereas x 6= 0 is a very weak constraint(especially if x has high bit-width). Therefore, to split on multiplications bydefining new predicates on its variables can only speed up satisfiable instances.If a solution (sat) can be found in the easy partial search tree with reduced mul-tiplications the algorithm immediately concludes with result sat . On the otherhand, if there is no solution, a set of subformulas with similar hardness to theoriginal formula is generated (all subformulas with constraints x 6= 0, y 6= 0, . . .)and has to be solved.
Note that on all previously described look-ahead types the node on which issplit becomes a constraint or is eliminated by rewriting. Thus, it is not possiblethat in a following round the look-ahead method returns the same node again,because a constraint node is not a look-ahead candidate. By asserting x = 0the variable and the multiplication is eliminated too (x · term ≡ 0). However,asserting x 6= 0 does not trigger any rewriting and can not elimate the mul-tiplication (x · term). In order to avoid returning the same node again in asubsequent round, it needs to be checked if variable x is not constrained withx 6= 0. If the constraint exists the node is not considered as look-ahead node.
The look-ahead type MULNV1 is the same as MULNV0 except that itsvariable x is set to 1 to trigger the arithmetic rewriting rule 1 · term ≡ term.For additions the type ADDNV0 is defined where the variable is set to 0(0 + term ≡ term). Similar types can be defined for all other hard arithmeticoperations (division, modulo, . . . ). It is left for future work to implement andevaluate the types for the remaining arithmetic operations.
Bit-Splitting on Hard Operations
Another way to speed up hard operations is to split on single bits. Bit-splittingis implemented for the look-ahead types MULNBIT0 and ADDNBIT0.MULNBIT0 selects an arbitrary multiplication node in the DAG and splits onthe LSB. ADDNBIT0 does the same for adders.
In the bit-blasted CNF encoding the term structure of the formula is lost.The SAT solver should examine in the DPLL search tree first the Booleanvariable which corresponds to a multiplier’s/adder’s LSB. But without termstructure knowledge it is not possible to select the right variable. In PBoolectorthe operator’s LSB can be fixed already on term-level. Bit-splitting is differentthan other look-ahead types proposed in this section. For all other look-aheadtypes the aim was to facilitate formula reduction for the rewrite engine, prior tobit-blasting. However, the aim of bit-splitting is not a reduction of the formulabut to guide the search of the SAT solver.
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For bit-splitting a slice of the operation’s LSB is made. The Boolean slicenode is asserted one time in positive and one time in negative phase. Notethat the assigned context bit is not only available in the CNF Encoding it isalso available in the intermediate AIG layer, which gives future opportunitiesto use the information also for new AIG rewriting rules. Similar to MULNV*and ADDNV*, in the look-ahead method it needs to be checked if a operation’sLSB is already asserted. If the bit is already asserted the respective node is notconsidered anymore as look-ahead.
4.3.5 Combination of Look-Ahead Types
PBoolector can be run for each look-ahead type independently. For exampleoption “-lt 4” only considers nodes with look-ahead type BCONDN. This sec-tion shows how different look-ahead types are combined. After finding the nodewith maximal score for each look-ahead type one of the types has to be selectedand returned by the look-ahead method.
Default Fixed Order Combination
The types are selected by a fixed order. Look-ahead types which are mostappropriate for splitting are selected first. The defined order was a result oftheoretical consideration and observed behaviour in the experiments. The look-ahead types are selected as follows:
1 If a node of type BEQ1VN exists return node with maximal score.
2 Else if a node of type BCONDN exists return node with maximal score.
3 Else if a node of type SLICE1 exists return node with maximal score.
4 Else if a node of type VAR1 exists return node with maximal score.
5 Else if a node of type MULNBIT0 exists return node with maximal score.
6 Else if a node of type BEQ1OTHER exists return node with maximalscore.
7 Else if a node of type AND1 exists return node with maximal score.
8 Else if a node of type ULT1N exists return node with maximal score.
9 Else if a node of type OTHER1 exists return node with maximal score.
10 Else if a node of type BEQ1V1 exists return node with maximal score.
11 Else return null (no look-ahead node found).
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Id Type Description Heuristic0 BEQ1VN Bit-vector equalities containing a vari-
able with bit-width > 1.pocc · nocc
1 BEQ1V1 Bit-vector equalities containing a vari-able with bit-width = 1.
pocc · nocc
2 BEQ1OTHER Every bit-vector equality which is notin look-ahead type 0 or 1.
pocc · nocc
3 ULT1N Unsigned less than bit-vectors. pocc · nocc4 BCONDN Bit-vector conditions with bit-width >
1.pocc + nocc
5 VAR1 Bit-vector variables bit-width = 1. pocc + nocc6 SLICE1 Bit-vector slices with bit-width = 1. pocc + nocc7 AND1 And bit-vector operations with bit-
width = 1.pocc · nocc
8 OTHER1 All other nodes with bit-width = 1 (ex-cept constants) which are not in look-ahead type 0 - 7.
pocc · nocc
9 MULNV0 Multiplication containing a variable xwith bit-width > 1. Split by setting x =0 and x 6= 0.
pocc + nocc
10 MULNV1 Multiplication containing a variable xwith bit-width > 1. Split by setting x =1 and x 6= 1.
pocc + nocc
11 MULNBIT0 Multiplication m with bit-width > 1.Split by setting m[0] = true and m[0] =false.
pocc + nocc
12 ADDNV0 Addition containing a variable x withbit-width > 1. Split by setting x = 0and x 6= 0.
pocc + nocc
11 ADDNBIT0 Addition a with bit-width > 1. Split bysetting a[0] = true and a[0] = false.
pocc + nocc
Table 4.1: Summary of defined look-ahead types.
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First, nodes of type BEQ1VN and BCONDN are selected. They have highrewriting potential (variable elimination and conditional reduction) and haveshown some good results in the experiments. Next, SLICE1 and VAR1 are se-lected. These 2 types are selected with some higher priority because on someinput formulas very good speedup can be achieved. Next, MULNBIT0 is se-lected. Here again splitting on MULNBIT0 nodes achieved good performance onsome experiments. Type ADDNBIT0 is not added, as no improvements couldbe achieved for this type. Also the remaining types could not achieve perfor-mance gain on any benchmark. Somehow interesting is the type BEQ1V1. Forthis type good reduction by variable elimination was expected. But experimen-tal results have shown that splitting on this type the algorithm gets very slowon several benchmarks. Therefore, this type is ranked at the end of the selection.
In general it is hard to define a fixed order. It seems that the best orderstrategy depends on the input formula’s structure. With the implemented so-lution it was tried to find a good general strategy. The fixed order combinationis done by running PBoolector with look-ahead type option “-lt 14”. Type 14is PBoolector’s default option.
Fixed Order Combination for Satisfiable Instances
Another combination for selecting look-ahead types is defined by selecting thoselook-ahead types first that should be able to improve satisfiable instances (MULNV0,MULNV1, ADDNV0). Again, a fixed order selection is done by first trying toreduce the hard operators. The strategy is as follows:
1 If a node of type MULNV0 exists return node with maximal score.
2 Else if a node of type MULNV1 exists return node with maximal score.
3 Else if a node of type ADDNV0 exists return node with maximal score.
4 Else if a node of type MULNBIT0 exists return node with maximal score.
5 Else select by the default fixed order combination.
This strategy is used by running PBoolector with look-ahead type option “-lt 15”.
Other Combination Methods
Two other methods to combine look-ahead types are implemented: Option“-lt 16” selects the look-ahead type randomly. Option “-lt 17” uses probingfor selecting the best look-ahead type. For these look-ahead types the reductionof the node with maximal (heuristic) score is measured. The look-ahead typewith maximal reduction pred · nred is returned.
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4.4 Implementation Extensions
The evaluation in Chap. 5 shows that on several benchmarks PBoolector per-forms slower than the sequential reference implementation. This section willpresent some extensions which aim to improve the results. Sect. 4.4.1 shows amethod similar to portfolio approaches to make PBoolector competitive to thesequential solver. The optional features proposed in Sect. 4.4.2 and Sect. 4.4.3are expected to have great impact on the results because similar (but more ad-vanced) approaches improved the performance of the Treengeling solver. Sect. 4.4.4shows some smaller implementation extensions to improve PBoolector.
4.4.1 Parallel Boolector Instance
To make PBoolector competitive to other solvers, the implementation was ex-tended to run a parallel sequential Boolector instance. The parallel solver in-stance tries to solve the formula without a limit in a separate thread. If a so-lution is found by the parallel instance, the interleaving split and search in themain loop is aborted. However, if a solution is found by splitting and searchingthe parallel instance is aborted. Starting PBoolector with “-sp” (start parallelinstance) option will always have faster or at least similar runtimes as plainBoolector.
The drawback with option “-sp” is that the parallel instance occupies onecore and the main loop can utilise maximal number of cores − 1 cores. Theparallel instance has just similar runtime to Boolector if it is run on a sepa-rate physical core respectively. If the instance runs on a virtual core sharedwith other computations the parallel instance always has slower runtimes thanBoolector. In practice, it turns out that the parallel instance is slightly slowerthan sequential Boolector, even if only physical cores are used. A similar prob-lem arises by running two sequential Boolector processes on the same inputformula on two cores in parallel. The two processes will need more wall clocktime than just starting one process. For instance starting Boolector on theselected QF BV benchmark 8moves mti 8-Atd 00004 bmc.smt2, it is solved in80.6s. Running two identical processes on two cores with same input, the solv-ing time increases for each process to ∼ 98s. This effect seems similar to thechallenges observed for the portfolio approach of Z3 [59]. Considerable over-head was observed running multiple identical solver instances in parallel. Cacheand memory bus congestion have negative impact on the solver’s performance.A closer investigation on this phenomena was done for SAT solvers by Aigneret al. [1]. Here it was shown that Lingeling (the default background solver ofBoolector) has lowest slowdown compared to other SAT solvers, by runningmultiple instances in parallel.
To get a clean termination and stop a parallel running boolector (limited) sat()call the SAT solver’s termination callback function is used. The callback canbe set with the new API extension boolector set term sat(). Termination is cur-
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rently only supported if Lingeling is used as back end SAT solver. The boolec-tor (limited) sat() method internally solves the formula after bit-blasting withLingeling’s lglsat() call. The method lglsat() continuously checks the termina-tion function and aborts with result unknown if the termination function returnstrue. The termination function is implemented to return true if a solution isfound either by the parallel instance or in the main loop. Since for bit-vectorformulas boolector (limited) sat() spends most time in the lglsat() call, it is suf-ficient to abort the search by aborting the SAT solver.
4.4.2 Using Boolector in Non-Incremental Mode
Boolector’s incremental mode allows to call boolector( limited) sat() (in the fol-lowing abbreviated with bsat()) on the same formula instance several times. IfBoolector is used in non-incremental mode bsat() can only be called once onan instance. Running Boolector (and therefore the back end SAT solver) innon-incremental mode allows to do several optimisations which speed up thesolving process. However, PBoolector is a natural application for incrementalsolving. In the main loop after doing a limited search, constraints are added andin a subsequent round bsat() is called again. This section proposes an optionalimplemented feature, which allows to use Boolector in non-incremental mode.In the evaluation (Chap. 5) it is shown that using non-incremental mode onsome instances perform better. On the other hand there are several instances,which perform better using the default incremental mode.
To be able to maintain non-incremental subproblems, in the search phasebefore calling bsat(), the instances are cloned. Next, on the cloned instancesbsat() is called exactly one time. The result is stored and the cloned instancesare deleted. For the original instances, which are split by adding constraints,bsat() is never called. There are two drawbacks using non-incremental mode.First, there is additional overhead by generating a clone before each bsat() call.The second drawback, which is even worse, is that already done work in the SATsolver needs to be redone in subsequent rounds. Running the SAT solver for awhile, parts of the search tree are examined, e.g., some Boolean variables can beassigned to fixed values and conflict clauses are learned. If the SAT solver is inincremental mode in the next search call, the work can be continued. If the SATsolver is in non-incremental mode the intermediate learned results are lost andall previous work needs to be redone. However, in order to not loose the workof the rewriting engine a boolector simplify() call is done on the original instancebefore cloning. Option “-ni” enables the non-incremental mode in PBoolector.
4.4.3 Failed Literal Reduction
One method to improve the performance of PBoolector is called Failed LiteralReduction (FLR). FLR is performed before the Cube and Conquer algorithm isstarted, and tries to initially reduce the formula by applying failed literals.
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Failed literals are Boolean nodes which lead to a result in the simplificationif they are assumed to be true or false. FLR is similar to the probing methodintroduced in Sect. 4.3.2. The input formula DAG is traversed from top to bot-tom with breadth first search. If a Boolean node (literal) is found the problemis cloned 2 times. Next, the node is asserted to be true on the first clone andto be false on the second clone. Each clone is tried to be reduced by callingboolector simplify. If the simplification of one clone leads to the result UNSATthe Boolean node is a failed literal and it can be asserted in the opposite phase.After finding and asserting a failed literal the process is restarted again to findthe next failed literal. The process is continued until no failed literal can befound anymore (until fixpoint). If the simplification of one clone leads to SATa solution is found and FLR is aborted with result SAT. If the simplification ofboth clones leads to result UNSAT, FLR is aborted with the result UNSAT. Formost benchmarks FLR can only reduce the formula but does not find a result.
FLR is very cost intensive. The probing method in Sect. 4.3.2 was stated tohave quadratic time complexity in terms of the number of nodes n in the DAG.In the FLR the probing process is restarted n times (worst case) and has cubictime complexity. Therefore FLR, enabled with option “-flr”, is only calledonce before the Cube and Conquer algorithm is started. For SAT solvers an ad-vanced method called Tree Based Look-Ahead was proposed by Heule et.al. [38],where FLR was improved by using implication trees. In Treengeling the solverperformance could be improved by doing an initial Tree Based Look-Ahead be-fore Cube and Conquer is started.
Boolector has a feature to solve a problem under assumption. The API callboolector assume() adds a constraint which is removed again after the problem issolved with the assumption (boolector sat()). For FLR and also for the probingmethod a similar feature “simplification under assumption” would be desirable.Simplification under assumption would avoid the overhead of cloning the in-stance before probing is done. Unfortunately such a feature is not available incurrent SMT solvers nor SAT solvers actually.
4.4.4 Further Implementation Extensions
This section proposes some further small implementation extension to improvethe algorithm. Similar features where implemented in the parallel SAT solverTreengeling.
Memory Limitation
To avoid that PBoolector runs out of memory a memory limit was implemented.If the memory limit is exceeded the look-ahead phase is aborted to not producenew subproblems. The implemented API extension
• boolector bytes(Btor*):size treturns the number allocated bytes of a Boolector instance.
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The number of bytes for all subproblems are accumulated. If current memoryconsumption, plus the expected memory consumption which will be produced bysplitting exceeds the memory limit, the look-ahead phase is aborted. By defaultthe memory limit is set to maximal available memory on the machine/3. Theoption “-m <limit>” allows to set the memory limit in MB. Option “-m -1”runs PBoolector without memory limitations.
Dynamic Conflict Limit Extension
One extension was made to the upd(lim) function. There are two cases that zerosubproblems are added in a round. The first case is if for all subproblems nolook-ahead can be found. The second case is that the memory limit is reachedand no new subproblems are produced. In both cases, the subproblems need tobe solved in the search phase. This is done by adding a new update limit rule:If no node is added within a round force the algorithm to spend more time inthe SAT solver by increasing the conflict limit (if (!added) lim = lim * 2;)
Finishing Subproblems
If no look-ahead node can be found no further splitting is possible. By default,then subproblems are solved without a limit (subproblems are finished). Fin-ishing the subproblems sometimes leads to very high individual search times.To avoid finishing subproblems an optional flag no finish (“-nf”) can be set forPBoolector. Using the option, subproblems where no look-ahead node can befound are not directly solved. They are tried to be solved within the currentlimit again. They are kept in the list of subproblems until they can be solvedby continuously calling the limited version of the Boolector SAT call. The op-tion is only allowed if PBoolector is used in incremental mode. If subproblemsare not finished in the non-incremental mode, the main loop possibly does notterminate. This is the case if a subproblem, on which no further look-ahead canbe found, is not solved within the maximal limit.
Full Search
Another extension made is to enable a full search each n’th round. Full searchmeans that the limited search call is not only done on the maxactive smallestsubproblems, but it is called on all open subproblems in the list. The assumptionthat a subproblem’s size is proportional to its hardness does not always holdin practice. The occasionally started full search gives such “large” and “easy”subproblems, where the assumption does not hold, the chance that they can besolved and deleted from the list. The parameter n is set to be 10. Every 10thround all open subproblems are tried to be solved within the current limit.
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Chapter 5
Evaluation
This chapter provides a detailed evaluation on different sets of benchmarks. Theevaluation is separated into 3 parts. After the introductory section (Sect. 5.1)first a general evaluation is done on a big set of input formulas (Sect. 5.2).Second, the evaluation is done on a set of crafted benchmarks on which thealgorithm is expected to work well (Sect. 5.3). The third part will take a closerlook at the structure of successful SMT-LIB benchmarks to identify patternswhich are appropriate for PBoolector (Sect. 5.4). Along the evaluation somefew further optimisations are proposed to improve the performance.
5.1 Experimental Setup
All experiments are done on 2.83 GHz Intel Core 2 Quad machines with 8GB of memory running Ubuntu 14.04. If not explicitly stated otherwise thedefault parameters of the PBoolector are chosen. The default parameter arechosen on one hand in an intuitive way (also motivated by default parametersused in Treengeling) and on the other hand by informal experiments on severalbenchmarks. The parameters which seemed to be best are chosen as default:
• Initial limit (-l): 5000
• Minimal limit (-lmin): 1000 - Using a value smaller 1000, experimentsshowed to produce too many subproblems at the intitial phase of thealgorithm.
• Maximal limit (-lmax): 50000 - It was observed that if the limit is higherthan 50000, on several benchmarks solving the individual subproblems getvery slow.
• Number of workers (-t): Number of processor cores - For the 4 coremachine used during evaluation (without hyper-threading support) thenumber of workers is 4.
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• Maximal active subproblems (-a): 4 · number of workers - In Treengelingthe constant value 8 was taken. Experiments showed that for PBoolec-tor the constant value 4 achieves better performance. For the evaluationmachine the number of active nodes is thus 4 · 4 = 16.
• Memory limit (-m): Maximal memory / 3 - For the machine used in theevaluation splitting is stopped if used memory exceeds the limit 8GB / 3= 2.67GB.
• Look-ahead type (-lt): 14 - The default look-ahead is the combined orderlook-ahead type.
For the implementation Boolector version 2.0 with Lingeling version ayv asback end SAT solver is used. For all time measurements the wall-clock time isused. All experiments where run with a wall-clock time limit of 1800 secondsand a memory limit of 7GB. If a limit is reached the runtime status can beeither out of time (oot) or out of memory (oom). All used benchmarks exceptthe crafted ones are from the SMT benchmark LIB for QF BV which containsapproximately 33000 files.
5.1.1 Notations
To fully identify the denoted benchmarks they are notated by its name withthe preceding category separated with an underscore. For example benchmarkbrummayerbiere2 smulov1bw32 denotes benchmark smulov1bw32 from categorybrummayerbiere2. To denote a set of benchmarks the “* ” wildcard is used.The notation smulov1bw* denotes for example the benchmarks smulov1bw24,smulov1bw32 and so forth. Detailed information of the used table notation isfound in App. A.
5.1.2 Implementation Issues
There are two issues left in the implementation which will appear in the results.
The first issue is related to the Runlim1 tool which is used to control theexecution of the experiments. Runlim stops the execution after the specifiedtimeout is reached. The execution is aborted with the interrupt signal SIGINT.PBoolector have an installed signal handler for SIGINT where the statisticalvalues are printed before termination. For some benchmarks the signal handlerof PBoolector is not called when Runlim tries to abort the execution with SIG-INT. The reason for this problem is unclear. If the execution does not stop aftersome seconds Runlim sends the kill signal. SIGKILL can not be handled by asignal handler. On benchmarks where this issue occurs no statistical values areprinted and the according columns are empty in the detailed tables.
1http://fmv.jku.at/runlim/
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Further there is still a non-deterministic bug in the implementation whichcould not be found. Because of the non-deterministic behaviour it is assumedthat the bug is caused by some race condition. It was observed that the morethreads are used the more frequent the bug is triggered. The Valgrind memoryerror detector does not find any issue in the implementation. It was tried toidentify the problem with the Valgrind thread error detector tools Helgrind andDRD. Helgrind does not detect any error. DRD detects some errors but theissue could not be identified with the help of the error messages. It was alsotried to use the Clang compiler which includes the optional data race detec-tor ThreadSanitizer. No errors where detected in the implementation with theThreadSanitizer either. PBoolector was adapted to use different parallelizationAPIs (OpenMP, cilk) where the bug also occurred. So it could be excludedthat this issue lies somewhere in the locking mechanism implemented with thePthread API.
A minimal version of PBoolector was implemented which uses a very simplelook-ahead implementation and produces just 2 subproblems each round whichare tried to be solved in parallel. For the parallelization only 2 threads arecreated which are joined after execution without any conditional wait or lockingmechanism. The minimal implementation was made to limit the cause of theerror. The problem was also reproducible in the simplified implementation.In the simplified implementation it “almost” can be excluded that it contains arace error, therefore it is assumed that the race error might be somewhere in theBoolector implementation. The issue occurs always during the boolector clone()call where an assertion is not met. To debug the error, a set of files was selectedon which the error frequently occurred. The implementation was adapted to adda sleep call with 50000 seconds if the assertion is not met. Then the selectedbenchmark files where executed continuously until the assertion was triggeredand the execution went to sleep. With the GNU debugger it is possible to attachto the sleeping process and examine what is going wrong. However, the causeof the issue could not be found in the debugger either. The problem occurredonly on very huge benchmarks on which debugging was nearly impossible. Inthe result tables there are some few benchmarks with status sf (segmentationfault) where this issue occurred.
5.2 SMT-LIB Competition Benchmarks 2014
First, a general evaluation is done on the SMT competition benchmarks 2014(sc14 ) for QF BV. This benchmark set contains 2488 files, which is a selec-tion of the SMT benchmark LIB. The most promising PBoolector extensions(non-incremental mode, failed literal reduction and the combined look-aheadfor satisfiable instances) are compared to the default configuration.
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(a) Sequential-Boolector vs. Base-Non-Incremental reference times.
(b) Base-Incremental vs. Base-Non-Incremental reference times.
Figure 5.1: Runtime comparison of reference times.
5.2.1 Reference Runtimes
As reference times the solving times from PBoolector runs with limit -1 aretaken. If the limit option is set to “-l -1” the problem is solved immediatelyin the first initial search call without any limit. For the evaluation of sc14benchmarks 2 different reference times are used. If parallel PBoolector is runin incremental mode the runtimes are compared to incremental reference timesBase-Incremental (base-inc) with option “-l -1”. If parallel PBoolector isrun in non-incremental mode the runtimes are compared to non-incrementalreference runtimes Base-Non-Incremental (base-ninc) with option “-l -1
-ni”. The abbreviations in the parenthesis are used in the detailed tables wherethe space is sparse. The Reference-Non-Incremental runtimes should have thesame runtime behaviour to the Sequential-Boolector (sequ-btor) tool, be-cause Boolector solves QF BV formulas in non-incremental mode.
To verify that Sequential-Boolector has equal runtimes to the Base-Non-Incremental runtimes, Fig. 5.1a shows a runtime comparison. The scatter-plotshows the runtimes of both implementations for each benchmark. Below thediagonal benchmarks are listed where Base-Non-Incremental is faster and abovethe diagonal Sequential-Boolector is faster. Most of the benchmarks exhibitsimilar runtime behaviour for both tools as expected. There are some outlierswhere Base-Non-Incremental is faster than Sequential-Boolector and vice versa.One remarkable outlier is the benchmark file float test v5 r15 vr1 c1 s8236.Base-Non-Incremental solves it in 120.26s whereas Sequential-Boolector takes927.66s. Actually there is a slight difference in the implementation of Base-Non-Incremental compared to Sequential-Boolector. In the search phase of thenon-incremental BPoolector first an explicit simplification is done, next the in-
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stance is cloned and finally boolector sat() is called on the clone. In the cloneimplicitly a second simplification is done during boolector sat() (see Sect. 4.4.2).Running the simplification two times leads to a huge time difference of morethan 800s compared to Boolector (where simplification is done once in the satcall) for this benchmark. Simplification is not done until fix-point in Boolectorrather it is limited by some value. If the fix-point is not reached two simplifi-cation calls can lead to better reduction. Another reason for the more or lesssmall jitter is that the IO operations (included in the measured wall-clock time)are not constant.
The second comparison shown in Fig. 5.1b should visualize why it is impor-tant to compare incremental PBoolector with the incremental reference timesand non-incremental PBoolector with non-incremental reference times. There isa clear runtime difference for a lot of benchmarks turning incremental mode onor off. The back end SAT solver is differently configured depending on whetherit is run in incremental mode or not. In the following experiments it shouldbe compared if splitting can outperform an immediately solved instance. If thereference times would not be distinguished it would be not clear whether thespeedup comes from the incremental mode setting or from splitting. The scatterplot shows that there is a cluster of a lot of “hard” benchmarks where the non-incremental mode is much faster (below the diagonal). On the other hand thereis a cluster consisting of several “easy” benchmarks with contrary behaviour,where the incremental mode is faster (above the diagonal). Obviously sequen-tial Boolector (which uses non-incremental mode for QF BV formulas) and theSAT solver are configured to work best for hard benchmarks.
5.2.2 Results for PBoolector in Incremental Mode
For the incremental mode two different runs with different configurations aredone. In the first run PBoolector is evaluated with the default configuration andin the second run PBoolector is evaluated with failed literal reduction (FLR)enabled.
Parallel PBoolector
PBoolector with the default options (Parallel-Incremental/par-inc) is com-pared to the reference implementation Base-Incremental. Fig. 5.2a shows theresults for all sc14 benchmarks. Below the diagonal the parallel implementationis faster, and above the diagonal the reference implementation is faster. Thereis a big set of benchmarks (especially hard ones) where the parallel implementa-tion can not outperform the reference times. But there are several benchmarkswhich show an improvement. Most of the benchmarks with improvement arelocated near the diagonal where both implementations have similar runtimes.Tab. 5.1 row 1 shows this results in concrete numbers. From all 2488 bench-marks PBoolector is able to solve 47 files which are not solved by the referenceimplementation (++). 1395 benchmarks are solved faster (+), 610 are solved
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(a) All benchmarks (2488). (b) Hard Benchmarks (371), whereSequential-Boolector time is greater 100sec.
Figure 5.2: Runtime comparison of PBoolector in incremental mode (Parallel-Incremental) vs. the reference implementation (Base-Incremental).
slower (−) and 196 benchmarks are not solved by PBoolector although they aresolved by the reference implementation (−−). 240 benchmarks are not solvedby both implementations (o). To be able to give more detailed results thenumber of benchmarks is reduced. Only interesting input formulas, which arealso hard for the sequential Boolector implementation are selected (runtime ofSequential-Boolector > 100.0 sec). Using this filter 371 files are left. The re-sults for the filtered benchmark set is shown in Fig. 5.2b respectively in Tab. 5.1row 2. There are 9 benchmarks which achieve reasonable speedup whereas 93benchmarks show a clear slowdown in the parallel implementation. Most of thebenchmarks (201) are not solved in both implementations. The detailed resultsfor the selected 371 benchmarks are shown in the appendix in Tab. C.1.
Tab. C.1 is sorted primary by its success state in ascending order and sec-ondary by the time difference (to the reference implementation) in descendingorder. The ranking number consists of the success state (++ = 1, + = 2, − = 3,−− = 4 and o = 5) and an increasing ranking number within each success state.The success states are separated by an empty line.
Before discussing the results, first some general notes are made about PBoolec-tor’s execution behaviour. There exist several benchmarks which are solved inthe initial search call within the initial limit of 5000. These are all results wherethe number of rounds (rnds) is equal to zero. Another set of benchmarks is notappropriate for splitting. In all benchmarks where rnds = 1 and the number oftotal subproblems is equal to one, no look-ahead node was found for splitting.The work intensive part of the PBoolector is clearly the search phase. The look-
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compare filter number of benchmarks
ref to total ++ + − −− o
base-inc par-inc - 2488 47 1395 610 196 240base-inc par-inc sequ-btor> 100.0 371 9 30 38 93 201base-inc par-inc-flr sequ-btor> 100.0 371 5 13 42 106 205
Table 5.1: Runtime comparison for parallel PBoolector runs in incrementalmode (Detailed table notation in Sect. A.1 in the appendix).
rank benchmark sequ-btor[s] base-inc[s] par-inc[s]
1.1 brummayerbiere2 smulov1bw32 1056.25 oot 480.291.2 log-slicing bvurem 16 oot oot 1002.921.3 ...iffyInterleavedModMult-8 oot oot 1002.921.4 core ext con 056 002 1024 279.98 oot 1464.581.5 core ext con 060 002 1024 220.71 oot 1470.711.6 core ext con 052 002 1024 194.60 oot 1495.921.7 log-slicing bvudiv 18 oot oot 1554.091.8 core ext con 064 002 1024 344.29 oot 1769.721.9 log-slicing bvsmod 15 oot oot 1779.82
Table 5.2: The 9 benchmarks with success state ++ for the PBoolector (Parallel-Incremental) run.
ahead phase on almost all files have runtimes close to zero. The splitting phasehas slight higher runtimes. For almost all files more than 99% of the time isspent in the search phase. In the search phase, almost for all benchmarks 99%of the CPU time is spent in the SAT solver (tot(cpu)∼sat(cpu)). The maximaland the mean search times are indicators for parallelization capability. Thehigher the maximal search time, the higher the parallel critical path and theless speedup can be obtained by increasing the number of processor cores.
There are 9 benchmarks with success state ++. An excerpt of the detailedresult table is shown in Tab. 5.2. On 4 of these benchmarks (core ext con* ) evenno look-ahead node is found. PBoolector first makes a limited initial search andsolves the problem without any limit if no look-ahead is found in a secondsearch call. Calling boolector sat() twice solves some problems much faster thanonly calling it ones. This is probably the same behaviour as it was observedin Sect. 5.2.1. It was observed that solving times can be increased by run-ning the simplification 2 times. Benchmarks which show a clear improvement(++,+) also compared to Boolector are from the benchmark sets brummayer-biere2 smulov1bw*, log slicing bv*, . . . iffyInterleavedModMult-8, and multiply-Overflow. Benchmark 1.1 and 1.3 have a relative small maximal search time.For these benchmarks the performance might can be increased using more pro-cessor cores. The rest of the benchmarks with success state ++ have a highmaximal search time. For these formulas no great performance gain is expectedusing more than 4 processor cores. For the successful bv* benchmarks from
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par-incpar-inc-flr
++ + − −− o
++ 5 0 0 0 4+ 0 13 14 3 0− 0 0 25 13 0−− 0 0 3 90 0o 0 0 0 0 201
Table 5.3: Success state change between Parallel-Incremental and Parallel-Incremental-FLR.
category log slicing the number of produced subproblems is rather low and thenumber of finished subproblems is relatively high. There is only one satisfiableinstance (2.7) where performance is clearly improved through splitting.
On the other hand most of the benchmarks exhibit a bad performance com-pared to the reference times. It seems that for most benchmarks the splittingprocess is not able to produce simpler subproblems. There are even benchmarkswhere the splitting process seems to produce harder subproblems than the origi-nal problem. An example is benchmark with rank 4.55. It has a maximal searchtime of 833.28s for one SAT call but the reference implementation solves theformulas within 328.30s. On some benchmarks the produced subproblems cannot be closed. For instance on benchmarks 4.51, the number of total subprob-lems (184) is equal to the number of maximal subproblems (184). On most ofthe bad performing benchmarks the memory limit is reached (mem > 2.6GB)such that splitting has to be stopped. This hints towards that the high memoryconsumption is a clear bottleneck in PBoolector.
Parallel PBoolector with FLR
In the next experiment the effect of FLR is evaluated. The results of PBoolec-tor with option “-flr” enabled (Parallel-Incremental-FLR/par-inc-flr) aresummarized in Tab. 5.1 row 3. The results with FLR enabled, get slightly worseto the previous experiment. From the the 371 hard benchmarks now only 5 havesuccess state ++ and only 13 have success state +. The number of files withnegative success state have been increased. Tab. 5.3 shows the success statechange between the Parallel-Incremental run and the Parallel-Incremental-FLRrun. The table is read as follows: 5 files with state ++ in par-inc have state++ in par-inc-flr, 4 files with state ++ in par-inc have state o in par-inc-flrand so forth. Almost all benchmarks have changed to an equal or a worse suc-cess state. The only improvement was achieved on 3 files which have movedfrom −− to −. The detailed results are shown in Tab. C.2 in the appendix.The column cmp contains a cross reference to the Parallel-Incremental run. Allremaining successful benchmarks contain 0 failed literals. Benchmakrs whichcontain failed literals, the results where similar or worse. The only clear success
51
Figure 5.3: Runtime comparison of PBoolector in non-incremental mode(Parallel-Non-Incremental) vs. the reference implementation (Base-Non-Incremental).
compare filter benchmarks
ref to total ++ + − −− o
base-ninc par-ninc sequ-btor> 100.0 371 0 36 63 122 150base-ninc par-ninc-flr sequ-btor> 100.0 371 0 42 53 126 150
Table 5.4: Runtime comparison for parallel PBoolector runs in non-incrementalmode.
state improvement (from −− to −) where on benchmarks 3.38, 3.39 and 3.41,which only contain one failed literal. There is no benchmark where a resultthrough FLR can be found. FLR is not able to improve the results withoutFLR (Parallel-Incremental).
5.2.3 Results for PBoolector in Non-Incremental Mode
In this section an evaluation is done for the parallel PBoolector implementationin non-incremental mode. Here again 2 different runs are done. In the first runFLR is disabled and in the second run FLR is enabled. The evaluation is doneon the 371 hard benchmarks selected in the previous section.
Parallel PBoolector
In the first run the PBoolector (Parallel-Non-Incremental/par-ninc) is com-pared to the reference implementation Base-Non-Incremental. The results,shown in the scatter-plot in Fig. 5.3 respectively in Tab. 5.4 row 1, look verysimilar to the incremental results. Few of the benchmarks (36) achieve a per-formance gain. Tab. C.3 in the appendix shows the detailed results. Actually
52
par-nincpar-ninc-flr
++ + − −− o
++ 0 0 0 0 0+ 0 33 3 0 0− 0 9 50 4 0−− 0 0 0 122 0o 0 0 0 0 150
Table 5.5: Success state change between Parallel-Non-Incremental and Parallel-Non-Incremental-FLR.
the benchmarks from rank 2.1 to 2.6 are interesting because they have a timedifference > 100 seconds. This 6 files are mostly from the same benchmark fam-ily (category log-slicing) as the successful benchmarks in the incremental mode.For most of the benchmarks the parallel implementation is slower. Here it isinteresting that all benchmarks which where solved by the incremental PBoolec-tor with success state ++ can not be solved by the non-incremental PBoolector(4.18 smulov1bw32, 5.86 bvurem 16, 5.25 . . . iffyInterleavedModMult-8, 5.67 bvu-div 18 and 5.44 bvsmod 15 ). For these files it seems to be crucial that alreadydone work in the SAT solver is not lost (only in incremental mode).
Parallel PBoolector with FLR
In the next experiment the FLR is turned on (Parallel-Non-Incremental-FLR/par-ninc-flr). The experiment is summarized in Tab. 5.4 row 2 anddetailed results are shown in Tab. C.4 in the appendix. Tab. 5.5 shows the statechange compared to the run without FLR.
The success state of 9 core ext con* benchmarks could be improved (fileswith rank 2.2, 2.5, 2.9 . . . ). These benchmarks are solved by the FLR withresult unsat . For the rest of the benchmarks FLR leads to similar or worthruntime behaviour like it was observed in the incremental mode.
5.2.4 Results for Different Configurations
This section will show the results of PBoolector in incremental mode with dif-ferent configurations. The detailed tables in this section show only the resultsfor benchmarks with success state ++ and +.
PBoolector Look-Ahead Type for Satisfiable Instances
First an evaluation is done to run PBoolector with look-ahead type 15 (Parallel-Incremental-Sat/par-inc-sat). This type tries to reduce hard multiplicationor adder circuits by setting their parameters to be equal 0 or 1. This look-ahead type is assumed to be able to improve the results only for satisfiableinstances (see Sect. 4.3.5). The results are shown in Tab. 5.6 row 2 respectivelythe detailed results are listed in Tab. C.5 in the appendix. The results are
53
quite similar to the default Parallel-Incremental run (row 1). Actually thereare two satisfiable files (2.11, 2.20) with success state improvement compared toParallel-Incremental. Several unsatisfiable instances exhibit a more or less bigruntime slowdown. As it was expected, the runtimes could only be improvedon the satisfiable instances containing multiplication or adder circuits.
PBoolector with different conflict limits
In the following experiments an evaluation is done by running PBoolector withdifferent limit configurations. Many benchmarks produce a high number ofsubproblems. To reduce the number of produced problems it is tried to increasePBoolector’s limit configuration. Two experiments are done with different limitconfigurations.
• Parallel-Incremental-HighLimit1 (par-inc-hl1):-lmin 10000 -l 30000 -lmax 100000
• Parallel-Incremental-HighLimit2 (par-inc-hl2):-lmin 30000 -l 50000 -lmax 200000
The results are shown in Tab. 5.6 row 3 and 4 with the details in Tab. C.6and Tab. C.7 in the appendix. The success state change are shown in Tab. 5.7and 5.8. The high limit configurations are able to improve the results of the de-fault PBoolector configuration. It is remarkable that the number of benchmarkswith state −− can be decreased by increasing the conflict limits. The numberof benchmarks with success state + could be increased with higher limits. Onlythe number of files with success state ++ could not be improved. Configurationpar-inc-hl1 just solves 5 benchmarks with state ++ whereas default PBoolectorhad 9 benchmarks with ++. On the other hand 4 benchamrks moved from oto ++ with configuration par-inc-hl2. By taking a closer look at the detailedresults mostly files could be improved where sequential Boolector is quite ef-ficient anyway. Runtime improvements on files with slow sequential Boolectorruntimes would be more interesting. But finding an optimal limit configurationseems to have improvement potential for the algorithm.
compare filter number of benchmarks
ref to total ++ + − −− o
base-inc pbt sequ-btor> 100.0 371 9 30 38 93 201base-inc par-inc-sat sequ-btor> 100.0 371 8 30 37 94 202base-inc par-inc-hl1 sequ-btor> 100.0 371 5 39 65 57 205base-inc par-inc-hl2 sequ-btor> 100.0 371 9 55 70 36 201
Table 5.6: Runtime comparison for different parallel PBoolector configurationsin incremental mode.
54
par-incpar-inc-hl1
++ + − −− o
++ 5 0 0 0 4+ 0 22 7 1 0− 0 10 22 6 0−− 0 7 36 50 0o 0 0 0 0 201
Table 5.7: Success state change between Parallel-Incremental and Parallel-Incremental-HighLimit1.
par-incpar-inc-hl2
++ + − −− o
++ 5 0 0 0 4+ 0 24 6 0 0− 0 17 21 0 0−− 0 14 43 36 0o 4 0 0 0 197
Table 5.8: Success state change between Parallel-Incremental and Parallel-Incremental-HighLimit2.
PBoolector-sp - Start Parallel Instance
The final run shows the results for the competitive version of PBoolector withoption “-sp” (see Sect. 4.4.1). For a comparison, the results of all 2488 filesare shown in the scatter-plot in Fig 5.4. The Sequential-Boolector implementa-tion is compared to the Parallel-Incremental-Competitive (par-inc-comp)run. For almost all benchmarks par-inc-comp has similar or better runtimes.The cluster at the bottom is the same cluster which was already observed bycomparing incremental and non-incremental base runtimes. Below the diagonalto the right are the few hard benchmarks where PBoolector can significantlyoutperform Boolector. On benchmarks which are far above the diagonal, thementioned cache problems of the parallel instance can be observed. Heavycore utilisation of PBoolector negatively influences the parallel sequential solverinstance. Actually results are influenced more by the caching issue than byPBoolector improvements. Tab. 5.9 shows the accumulated runtimes for bothimplementations. The competitive version of PBoolector can not improve thesequential Boolector on the sc14 competition benchmarks.
5.2.5 Summary
No clear behaviour could be observed with different configurations. Each config-uration can solve some benchmarks efficiently whereas on other configurationsthe runtime for the same benchamrk get worse. For instance there are some
55
Figure 5.4: Runtime comparison of the competitive PBoolector (Parallel-Incremental-Competitive) vs. the Sequential-Boolector, for all 2488 benchmarks.
Sequential-Boolector Parallel-Incremental-Competitive
solved sum[s] solved sum[s]
2336 390587 2316 412802
Table 5.9: Number of solved instances and accumulated runtimes for Sequential-Boolector and Parallel-Incremental-Competitive (oot=oom=1800s).
benchmarks (e.g., smulov1bw32 ) which have a nice speedup in the incremen-tal mode whereas non-incremental mode leads to a clear slowdown. There aresome benchmarks from the ext con* family which can be improved only in non-incremental mode with FLR. The right configuration can achieve some goodresults for benchmarks with the appropriate structure. A set of benchmark filesfrom the category log-slicing are identified, which exhibit good performance withall configurations. The picture is mostly the same for different configurations:there are many files with a slowdown and some few files which can improve thereference times. The increase of the conflict limit configuration have shown thebest improvements.
5.3 Crafted Benchmarks
In this section more detailed experiments are done on a set of crafted formulas.Parametrized formulas are generated with well known structure which should beappropriate for PBoolector. The evaluations (especially in Sect. 5.3.1) will showthat the PBoolector implementation can be very powerful on hard benchmarksif they contain the right structure.
56
5.3.1 Lazy-If-Then-Else Example
The first experiment leads us back to the introductory example (Sect. 1.2), theLazy-If-Then-Else example (lazyitex ). The formula is taken from a publicationcomparing lazy and eager solving approaches for bit-vectors [36]. It was shownthat the lazyitex structure is applicable for the proposed lazy bit-vector solvingapproach. This benchmark is relatively hard because it contains several nested32 bit multiplication operations with conditional parameters. As shown in theexample the formula can be efficiently solved if the search space is split by as-serting the conditionals predicate. This example emphasizes the power of termrewriting rules in combination with search space splitting. After asserting apredicate the hard multiplication operations can be reduced by term rewritingrules. The subproblems get trivial and can be solved by the rewrite engine.This formula can even be solved without bit-blasting, a pure bit-vector rewrit-ing solver would lead to a solution.
The original problem F was formulated as follows:
z · (x0 = y0 ? y0 · (x1 = y1 ? y1 : 1) : 1)
6= (x0 = y0 ? x0 · (x1 = y1 ? x1 · z : z) : z) (5.1)
To recap, the search space is split into three subproblems, which are triviallyunsat after term-level rewriting:
(1) F ∧ x0 6= y0 ⇒ z · 1 6= z ⇒ z 6= z ⇒ ⊥(2) F ∧ x0 = y0 ∧ x1 6= y1 ⇒ z · y0 · 1 6= z · x0 ⇒ z · y0 6= z · y0 ⇒ ⊥(3) F ∧ x0 = y0 ∧ x1 = y1 ⇒ z · y0 · y1 6= z · x0 · x1 ⇒ z · y0 · y1 6= z · y0 · y1 ⇒ ⊥
The formula can be generalized using n nested conditions (= number of variablepairs x and y).
z · (x0 = y0 ? y0 · (x1 = y1 ? y1 · (. . . (xn−1 = yn−1 ? yn−1 : 1) . . .) : 1) : 1)
6= (x0 = y0 ? x0 · (x1 = y1 ? x1 · (. . . (xn−1 = yn−1 ? xn−1 · z : z) . . .) : z) : z)(5.2)
Crafted Lazy-If-Then-Else Benchmarks
A set of formulas was crafted using the general Equ. 5.2 and 64 benchmarkswhere generated by varying the number of conditionals n from 2 to 256 and thevariable bit-width bw from 2 to 256. The runtimes of the sequential Boolectorimplementation is compared to the PBoolector implementation. The parallelimplementation is run to split only on conditional nodes (BCONDN) usingoption “-lt 4”. In the tables listed at the end of this section for each n (rows)and each bw (columns) the wall-clock runtime is printed in seconds.
57
Run 1: Boolector - Reference Times
The sequential reference implementation (Tab. 5.10) solves all instances withlow bit-width quite fast. With increasing bit-width a clear exponential increaseof runtime is observable. This behaviour comes from the fact that bit-blastedmultiplication operations have an exponential increase of hardness with increas-ing bit-width (horizontal direction). Nearly each instance runs out of time withbit-width ≥ 8. There are 17/64 benchmarks are solved.
Run 2: PBoolector - Splitting on Conditionals
The second run (Tab. 5.11) shows solving times of PBoolector implementationwith splitting on conditionals (BCONDN). Here the parallel implementationachieves a clear improvement to the sequential run 1. The PBoolector imple-mentation solves 12 instances more within the time limit. The multiplicationscan be reduced by splitting and rewriting. Therefore the exponential behaviourwith increasing bit-width can be reduced to a kind of linear behaviour: Allbenchmarks with 2 nested conditions n are now solved quite efficient, also withvery high bit-width of 256. Splitting and term-rewriting leads to a super linearspeedup. For example on the benchmark with n = 2 and bw = 8 a speedup of745.18/1.69 = 440.93 on the 4 core machine is achieved.
In the experiment a slowdown compared to Boolector can be observed on lowbit-width benchmarks and increasing number of conditions (vertical direction).The slowdown can be explained by taking a look at the used heuristic: The orderof which conditional should be split first, is calculated by the simple heuristicpos+neg occurrences. Since each conditional occurs in the formula exactly onetime in positive phase all conditional nodes get the same score value of 1. Inthis case the ordering depends on Boolector’s internal DAG structure. If firstthe bottom most conditionals are used for splitting, subformulas can be reducedby rewriting but does not get trivial immediately. In the worst case for n nestedconditionals, 2n leaves in the search tree have to be generated until all theseformulas are trivially solved by the rewrite engine. The vertical direction in thetable shows this kind of exponential behaviour in the runtimes. The limitationof solving the instances by PBoolector is rather bounded by number of nestedconditionals instead of the bit-width.
Run 3: PBoolector - Splitting on Conditions with Improved Heuristic
For the next run an improved heuristic was implemented which chooses the con-ditional in a more sophisticated way from top to bottom. Reconsider Equ. 5.2:If the formula is split by selecting the top most conditional first, x0 6= y0 leadsto z 6= z (trivial unsat) and the branch can be closed. On the other branch withx0 = y0 the top most conditional can be removed. If the splitting is continuedon the top most conditional of the remaining formula, one branch can imme-diately closed again and on the other branch the formula can be reduced. Onthe lazyitex benchmarks this leads to a maximal number of 2 subproblems each
58
round and produces maximal n + 1 leaves in the search tree. In comparison torun 1 where in the worst case 2n leaves are produced. To select the top mostconditional first the heuristic for BCONDN is improved (BCONDN-H1). Thethe number of not shared subnodes (nssn) is added: h1 = pos + neg + nssn.Not shared subnodes are all subnodes of the conditional which are referenced(direct or indirect) only by the conditional. If the conditional’s predicate isassumed to be true all nssn of the else branch can be removed too, becausethey are not referenced elsewhere in the DAG. The new heuristic h1 refines theapproximation of the real formula reduction.
Using heuristic h1 clearly improves the results of run 2 by solving 22 instancesmore (Tab. 5.12). Benchmarks with small bit-width are able to outperform se-quential Boolector with increasing number of variables. Also performance onhigher bit-width benchmarks can significantly be improved. All remaining ex-periments in this work (also in Sect. 5.2) are done with the improved heuristicfor conditionals.
A disadvantage that maximal 2 subproblems can be produced each roundis the bad utilization of processor cores. However, the overall solving times byoccupying just 2 cores is much more better than in the previous experimentwhich had extensive use of all 4 available cores.
Tab. 5.13 contains the percentage of time, spent in the SAT solver duringall the search calls. As remainder, one search call consists of simplifying theformula and then after bit-blasting the SAT solver is called. On average 93.54%of time is spent in the SAT solver during the search phase. As we know thatthese instances can be solved on word-level only by rewriting, this ratio is quitehigh. The remaining three experiments try to improve the results by reducingthe SAT solver’s effort.
Run 4: PBoolector - Splitting on Conditionals and Non-IncrementalMode
One way to reduce the SAT solvers effort is to run the back end solver innon-incremental mode. The expectations where that less time is spent in theSAT solver (through possible optimisation in non-incremental mode) and theproblems are solved more efficient. Tab. 5.14 shows the results. Contrary to theexpectations the results get worse. Compared to run 3 where the SAT solveris used in incremental mode, 4 instances less can be solved within the timeout.Also the average SAT ratio of the search calls has been increased to 96.83% (seeTab. 5.15). It is noteable that in non-incremental mode two instance with highbit-width 256 can be solved now which were not solved in incremental mode.The non-incremental mode performs worse on medium bit-width formulas witha high number of nested conditionals.
59
Run 5: PBoolector - Splitting on Conditionals without Bit-Blasting
A more drastic way to reduce the SAT solver’s work is to turn off bit-blasting.This leads to a pure rewriting solver. Layzitex benchmarks are solvable by split-ting and rewriting. To turn off bit-blasting the implementation of the boolec-tor limited sat() method was modified. If the method is called with limit 0 onlyrewriting is done. If rewriting does not lead to a result, unknown is returnedelse the result is returned. The PBoolector implementation also leads to a resultif the formula is not solvable only by splitting and pure rewriting. If no look-ahead is found boolector sat() is called without the limit, where the subproblemis bit-blasted and eagerly solved (subproblem is finished). Option “-nf” (notfinish subproblems) is not possible in combination with limit 0 (minimal limit0 and maximal limit 0), as the algorithm possibly does not terminate.
In Tab. 5.16 now all 64 benchmark files are solved within the time limit.Remarkably solving times with a maximum of 2.41 seconds are achieved also onthe hardest instances. The pure rewriting setting is especially successful for thelazyitex benchmarks. On general benchmarks a clear loss of performance canbe observed using this setting.
Run 6: PBoolector - Failed Literal Reduction
Finally in run 6, to reduce to the SAT solver’s work, the failed literal reductionoption “-flr” is enabled. The initial failed literal reduction can immediatelysolve the problem without starting any search call (and therefore the SAT solver)in the main loop. The runtimes shown in Tab. 5.17. For all benchmark ∼ 100%of the runtime is spent in the failed literal reduction. All 64 instances are solvedwith similar performance as in run 5.
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 3.57 1.55 745.18 oot oot oot oot oot4 3.60 3.84 oot oot oot oot oot oot8 4.05 5.35 oot oot oot oot oot oot
16 4.05 19.11 oot oot oot oot oot oot32 0.49 41.03 oot oot oot oot oot oot64 0.56 105.26 oot oot oot oot oot oom
128 0.81 405.29 oot oot oot oot oot oom256 1.60 1637.27 oot oot oot oot oom oom
Table 5.10: Run 1 - Boolector - Reference times in seconds. 17/64 solved.
60
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 0.99 1.08 1.69 1.77 2.44 3.74 9.26 41.174 2.06 1.81 2.23 4.89 15.09 43.75 73.86 oot8 2.01 3.05 4.85 12.81 46.45 948.12 oot oot
16 2.06 13.33 841.23 oot oot oot oot oot32 0.33 486.17 oot oot oot oot oot oom64 0.47 oot oot oot oot oot oot oom
128 4.44 oot oot oot oot oot oom oom256 25.76 oot oot oot oot oot oom oom
Table 5.11: Run 2 - PBoolector (-lt 4) - Splitting on conditionals BCONDN.Wall-clock times in seconds. 29/64 solved.
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 3.44 0.88 3.93 4.81 5.59 7.15 17.02 55.654 3.78 4.17 5.28 6.51 11.49 33.85 83.66 160.678 3.69 4.59 6.39 9.09 22.72 68.39 150.74 oot
16 3.56 5.46 8.19 17.42 53.37 146.27 329.53 oot32 0.36 4.67 12.45 36.81 135.66 362.37 oot oom64 0.72 11.84 29.49 106.43 388.43 1376.60 oot oom
128 1.04 32.24 85.61 339.46 1254.98 oot oom oom256 2.97 101.56 289.72 1212.78 oot oot oom oom
Table 5.12: Run 3 - PBoolector (-lt 4) - Splitting on conditionals with im-proved heuristic BCONDN-H1. Wall-clock times in seconds. 51/64 solved.
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 50.00 96.70 99.04 97.65 95.68 91.94 89.06 86.404 66.66 99.34 99.01 98.35 97.74 97.17 94.66 86.828 86.20 99.18 98.63 97.73 97.09 95.78 90.06 -
16 94.44 98.79 98.12 97.43 96.29 93.14 84.41 -32 95.45 98.54 97.93 96.60 94.63 89.14 - -64 98.14 98.27 96.95 94.84 92.53 89.69 - -
128 98.10 97.84 95.33 92.78 90.36 - - -256 97.95 97.22 93.60 91.20 - - - -
Table 5.13: Run 3 - Time ratio spend in SAT solver during search in % (Average= 93.54%).
2 0.53 1.05 1.10 1.61 3.95 17.44 47.64 93.884 0.44 0.75 1.49 3.43 11.20 56.20 146.96 240.298 0.62 1.37 3.24 10.33 41.53 158.23 408.06 641.47
16 0.60 2.77 9.00 38.03 182.01 489.28 883.41 1703.4932 0.84 6.83 32.45 124.98 608.05 1558.29 oot oot64 1.12 21.30 131.89 657.28 oot oot oot oom
128 14.71 74.61 450.44 oot oot oot oot oom256 61.86 292.50 oot oot oot oot oom oom
Table 5.14: Run 4 - PBoolector (-lt 4 -ni) - Splitting on conditionals. TheSAT solver runs in non-incremental mode. Wall-clock times in seconds. 47/64solved.
61
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 80.00 97.83 99.36 98.80 98.64 98.89 98.26 96.214 91.35 98.88 98.73 98.42 98.43 98.73 97.85 94.318 91.11 98.77 98.50 98.46 98.47 98.25 97.04 91.73
16 93.63 98.03 98.43 98.46 98.66 97.73 94.54 87.8532 96.94 97.65 98.41 98.24 98.32 97.10 - -64 98.06 97.62 98.51 98.64 - - - -
128 97.12 97.52 98.19 - - - - -256 96.53 97.76 - - - - - -
Table 5.15: Run 4 - PBoolector (-lt 4 -ni) - Time ratio spend in SAT solverduring search in %(Average = 96.83%).
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 0.97 0.87 0.83 0.82 0.81 0.89 0.88 0.854 1.08 1.03 0.99 1.04 0.96 1.01 1.04 0.998 1.02 0.97 1.04 0.99 1.05 1.00 0.97 1.02
16 1.04 0.99 0.94 0.99 1.02 0.97 1.04 0.9932 1.01 1.06 0.98 1.03 1.07 1.00 1.16 1.1264 1.07 1.12 1.11 1.11 1.06 1.17 1.12 1.16
128 1.33 1.29 1.33 1.34 1.29 1.34 1.34 1.19256 2.28 2.30 2.07 2.30 2.39 2.14 2.39 2.41
Table 5.16: Run 5 - PBoolector (-lt 4 -lmin 0 -lmax 0 -l 0) - Splitting onconditionals without bit-blasting. Wall-clock times in seconds. 64/64 solved.
n bw2 bw4 bw8 bw16 bw32 bw64 bw128 bw256
2 1.07 1.22 1.14 0.99 1.04 1.10 1.05 1.104 0.93 0.98 0.98 1.03 0.99 1.03 1.04 0.998 1.04 1.09 1.04 1.09 1.06 1.01 1.06 1.10
16 1.04 1.10 1.02 1.07 1.07 1.06 1.00 1.0632 1.02 0.97 1.02 1.04 0.99 1.04 1.08 0.9564 1.08 1.08 1.02 1.08 1.18 1.04 1.10 1.10
128 1.21 1.21 1.02 1.21 1.21 1.19 1.15 1.19256 1.58 1.61 1.55 1.60 1.61 1.59 1.76 1.76
Table 5.17: Run 6 - PBoolector (-lt 4 -flr) - Doing an initial failed literalreduction. Wall-clock times in seconds. 64/64 solved.
5.3.2 Square of Sum
To analyze the impact of splitting on the least significant bit of multiplicationsa set of benchmarks were generated which checks the distribution rule of mul-tiplications for the square of a sum. The square of a sum with 2 variables isdefined as follows:
(x0 + x1)2 = x20 + 2x0x1 + x2
1 (5.3)
The general form of the square of a sum with n variables: ∑0≤i≤n
xi
2
=
∑0≤i≤n
x2i
+ 2 ·
∑0≤i<j≤n
xi · xj
(5.4)
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Crafted Square of Sum Benchmarks
A set of formulas was constructed using Equ. 5.4 and 64 benchmarks are gen-erated with varying number n of variables from 2 to 16 and variable bit-widthbw from 2 to 256. To check if the formula is valid the negation of the formula ischecked for unsat The runtimes of the sequential Boolector implementation iscompared to the parallel PBoolector. The parallel implementation is run withlook-ahead type MULNBIT0 (“-lt 11”) to split on the multiplication’s LSB.In the tables listed at the end of this section for each n (rows) and each bw(columns) the wall-clock runtime is printed in seconds.
Run 1: Boolector - Reference Times
Tab. 5.18 shows the reference times of the sequential implementation. Theresults are similar to the sequential results of the lazyitex benchmarks in theprevious section. Formulas with small bit-with are solved quite efficient, whereasincreasing the multiplication’s bit-width exhibits an exponential increase in solv-ing time.
Run 2: PBoolector - Splitting on Multiplication’s LSB
Tab. 5.19 shows the results. The PBoolector implementation is able to solve 1instance more than the sequential implementation. The improvement is not assignificant as for the lazyitex benchmarks. In contrast to the lazyitex bench-marks here “only” linear speedup can be achieved. Bit-splitting can not simplifythe formula with term rewriting rules. Therefore the speedup exhibits a linearbehaviour with respect to the used cores. For instance the formula with n = 4and bw = 6 achieves a speedup of 129.78/35.63 = 3.64 on the 4 core machine.
Run 3: Treengeling - Comparison to a Parallel SAT solver
As for bit-splitting no rewriting rules can be triggered the runtimes of PBoolec-tor are comparable to the parallel SAT solver implementation Treengeling. The64 square of sum benchmarks (QF BV) are converted to a CNF formula whichwhere run on the SAT solver. The expectation where that PBoolector performsbetter than the SAT solver, as it is able to split with term level information.In the CNF encoding the information is lost and the multiplications LSB cannot identified for splitting anymore. Results in Tab. 5.20 show the opposite,that Treengeling running on 4 cores is able to outperform PBoolector slightly.The same benchmarks are solved, but with better runtimes. This might be ex-plained by the fact, that Treengeling is already a highly optimized tool whereasPBoolector is at an initial state of implementation. Another reason could be thatthe SAT solver is able to split the CNF formula in a much more fine grainedway. Treengeling is able to select more promising Boolean variable whereasPBoolector just selects multiplication’s LSBs.
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Further Experiments
Several further enhancements where tried, but without improvements to run2. The enhancements are only mentioned in the following without comparingexplicitly the results. It was tried to split the multiplications on the mostsignificant bit. Another experiment was done to split a multiplication not onlyon one bit, rather each multiplication was split on its n LSBs with n = 2, 3, 4, . . ..
n bw2 bw4 bw6 bw8 bw10 bw12 bw14 bw16
2 0.97 1.02 1.06 2.41 36.11 1652.99 oot oot4 1.30 1.17 129.78 oot oot oot oot oot6 0.22 18.32 oot oot oot oot oot oot8 1.03 oot oot oot oot oot oot oot
10 1.33 oot oot oot oot oot oot oot12 1.46 oot oot oot oot oot oot oot14 1.51 oot oot oot oot oot oot oot16 3.01 oot oot oot oot oot oot oot
Table 5.18: Run 1 - Boolector - Reference times in seconds. 16/64 solved.
n bw2 bw4 bw6 bw8 bw10 bw12 bw14 bw16
2 1.11 1.34 1.20 2.23 13.77 380.90 oot oot4 0.88 1.00 35.63 oot oot oot oot oot6 1.16 7.51 oot oot oot oot oot oot8 1.17 205.64 oot oot oot oot oot oot
10 0.85 oot oot oot oot oot oot oot12 1.12 oot oot oot oot oot oot oot14 1.74 oot oot oot oot oot oot oot16 4.77 oot oot oot oot oot oot oot
Table 5.19: Run 2 - PBoolector (-lt 11) - Splitting on multiplications LSBMULNBIT0. Wall-clock times in seconds. 17/64 solved.
n bw2 bw4 bw6 bw8 bw10 bw12 bw14 bw16
2 0.82 0.95 0.95 2.76 11.44 161.70 oot oot4 1.07 1.35 19.73 oot oot oot oot oot6 0.55 5.01 oot oot oot oot oot oot8 1.18 146.50 oot oot oot oot oot oot
10 1.63 oot oot oot oot oot oot oot12 0.51 oot oot oot oot oot oot oot14 1.76 oot oot oot oot oot oot oot16 1.77 oot oot oot oot oot oot oot
Table 5.20: Run 3 - Treengeling - Parallel SAT solver. Wall-clock times inseconds. 17/64 solved.
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5.3.3 Carry Safe Adder
To evaluate the impact of the ADDNBIT0 look-ahead type a benchmark setwhere generated which verifies a Carry Save Adder (CSA). A CSA is a hard-ware addition circuit to fast compute the sum of 3 or more bit-vectors. Thedisadvantage of the conventional Ripple Carry Adder (RCA) summing up 3 (ormore) numbers is that the carry bit has to be propagated from one adder unit tothe other. For instance the sum of 3 bit-vector numbers a+b+c is calculated ina RCA chain by first, calculating tmp = a + b and second, doing the remainingsum s = tmp+c with the carry out bit of the first sum. Each addition is delayedby its previous additions. To give an idea of how the delay is omitted in a CSA,here a benchmark is sketched which verifies the correctness of a CSA summingup 3 numbers:
ps = a⊕ b⊕ c ∧sc = (a&b)|(a&c)|(b&c) ∧y = ps + (sc << 1) ∧x = a + b + c ∧x 6= y
(5.5)
The partial sum ps is the sum of the three numbers for each bit without a carry.For three binary numbers this is done by a simple xor operation. The shift carrysc computes a separated carry for each bit. Again for 3 binary numbers this isa simple logical operation. The CSA sum y finally is done by shifting sc andperform one arithmetic plus operation (with a RCA). The formula is unsat andtherefore the CSA is verified to be equal the simple RCA chain (x). The circuitlooks similar for adding more than three numbers, the CSA contains only fastlogical operations except one final arithmetic Ripple Carry Adder. No adderunit needs to wait for the propagated carry bit.
Crafted Benchmarks and Results
For benchmarks generation an already existing tool GenCSA (implemented atthe FMV institute at JKU) was used, which generates parametrized formulassimilar to Equ. 5.5. There are 64 benchmarks generated with varying num-ber of variables n from 2 to 16 (number of variables which are summed up bythe CSA) and varying bit-with from 2 to 16. The experiments are done in ananalogous manner as in Sect. 5.3.2. The formulas are split on the addition’sLSBs and Boolector (Tab. 5.21) is compared to the PBoolector (Tab. 5.22) andthe Treengeling (Tab. 5.23) implementation. The runtimes for PBoolector areworse than for the sequential Boolector. The sequential implementation solves34/64 benchmarks within the timeout. The PBoolector implementation solves27/64 benchmarks with a significant slowdown. The Treengeling implementa-tion clearly outperforms PBoolector (31/64 benchmarks solved), but cannot im-prove the sequential Boolector results. It could not be verified that splitting onaddition’s LSB give some improvements. Contrary to multiplications, it seemsthat setting an addition’s bit is not worth for splitting the search space. There-
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fore the combined look-ahead types (see Sect. 4.3.5) do not select ADDNBIT0for splitting.
n bw2 bw4 bw6 bw8 bw10 bw12 bw14 bw16
2 1.19 1.25 1.25 0.36 0.36 0.48 0.36 0.364 1.18 1.14 1.63 2.51 4.99 5.56 15.84 13.006 1.14 2.34 19.31 100.92 602.13 1379.47 1628.03 oot8 1.01 8.20 221.60 oot oot oot oot oot
10 0.94 39.12 oot oot oot oot oot oot12 1.10 227.14 oot oot oot oot oot oot14 0.79 253.55 oot oot oot oot oot oot16 1.26 1018.89 oot oot oot oot oot oot
Table 5.21: Run 1 - Boolector - Reference times in seconds. 34/64 solved.
n bw2 bw4 bw6 bw8 bw10 bw12 bw14 bw16
2 1.06 0.89 0.84 0.82 0.85 0.85 0.84 0.844 0.84 0.89 1.16 1.74 8.23 25.92 40.28 42.806 0.85 3.39 148.80 1360.44 oot oot oot oot8 1.10 57.98 oot oot oot oot oot oot
10 1.19 508.12 oot oot oot oot oot oot12 110.07 oot oot oot oot oot oot oot14 92.11 oot oot oot oot oot oot oot16 707.75 oot oot oot oot oot oot oot
Table 5.22: Run 2 - PBoolector (-lt 13) - Splitting on addition’s LSBADDNBIT0. Wall-clock times in seconds. 27/64 solved.
n bw2 bw4 bw6 bw8 bw10 bw12 bw14 bw16
2 1.64 1.64 0.69 1.51 1.57 0.57 1.57 1.514 1.41 1.57 2.11 4.30 12.30 56.17 40.46 68.706 1.06 2.80 73.63 328.70 1091.42 oot oot oot8 1.15 12.01 oot oot oot oot oot oot
10 1.23 78.48 oot oot oot oot oot oot12 1.22 211.84 oot oot oot oot oot oot14 1.75 527.05 oot oot oot oot oot oot16 1.78 831.22 oot oot oot oot oot oot
Table 5.23: Run 3 - Treengeling - Parallel SAT solver. Wall-clock times inseconds. 31/64 solved.
5.4 Selected Benchmarks
In the final section of the evaluation chapter a closer look is taken at the struc-ture of successful SMT-LIB benchmark files. Some more file structures will beidentified which are appropriate for PBoolector.
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1: zero (32) 2: var (32)
3: slice (1)3 1 3 1
4: concat (64)1
2
5: concat (64)1 2
6: cond (64)
12 3
7: var (32)
8: slice (1)3 1 3 1
9: concat (64)
12
10: concat (64)
12
11: cond (64)1
2 3
12: mul (64)12
13: constd (64)14: constd (64)
15: slt (1)1
2
16: slt (1)
21
17: or (1)
12
18: smulo (1)
12
19: ne (1)
1 2
SMOC
Figure 5.5: The smulov1bw32 benchmarks structure. A signed multiplicationoverflow circuit (SMOC) is checked against the naive implementation.
5.4.1 Smulov1bw*
Some of the few files which show good performance in Sect. 5.2 are the smulov1bw*files from the SMT-LIB category brummmayerbiere2. This benchmark set isverifying a signed multiplication overflow circuit (SMOC) with different bit-widths’. The circuit is checked against a naive multiplication overflow detector.The following formula checks a SMOC with bit-width 4 against the naive im-plementation like it is done in the benchmarks:
(p8 = sext(a4)8 · sext(b4)8 ∧ (p8 < −8 ∨ p8 > 7)) 6= SMOC(a4, b4)
The index denotes the bit-vector’s bit-width. If the formula is unsat theleft side (of the inequality) which is the naive implementation is equivalent tothe optimized SMOC circuit on the right side. For the naive implementationthe 4 bit-vectors a and b are sign extended to a 8 bit vector. The two 8 bitvectors are multiplied and if the product is smaller than INT MIN4 = −8or is greater than INT MAX4 = 7, then a 4 bit multiplication overflow havebeen occurred. For better reading here the variable p which saves the productis introduced. The SMOC which is stated here as an abstract function is afast overflow detector consisting of one 5 bit multiplication and several logicalconnectives. The details how the fast SMOC is implemented are found in [17].A 4 to 8 bit sign-extension is encoded in Boolector with an if conditional and 2concatenations:
sext(a4)8 ≡ a4[3]1 ? 14 ◦ a4 : 04 ◦ a4
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benchmark Boolector[s] PBoolector -lt 14
tot (wc)[s] search tot(wc)[s] search tot(cpu)[s]
smulov1bw16 9.66 6.67 6.44 21.2smulov1bw24 133.8 79.00 78.26 224.74smulov1bw32 1061.94 466.61 463.75 1421.85smulov1bw48 oot oot - -
(a) Combined look-ahead (-lt 14).
benchmark Boolector[s] PBoolector -lt 4
tot (wc)[s] search tot(wc)[s] search tot(cpu)[s]
smulov1bw16 9.66 10.47 10.46 12.79smulov1bw24 133.8 128.81 128.78 136.72smulov1bw32 1061.94 1064.22 1064.18 1090.06smulov1bw48 oot oot - -
(b) Split only on conditions (-lt 4).
Table 5.24: Comparison of Boolector vs. PBoolector implementation onsmulov1bw* benchmarks in seconds. For BPoolector the total runtime andthe runtime of the search phase is given.
If the MSB of a is one, a is concatenated with a 4 bit one-constant elsewith a 4 bit zero-constant. The bottleneck of this benchmark set is the highbit-width of the multiplications. To check a 32 bit SMOC leads to a formulaswhich contains one 64 bit multiplication (in the naive implementation) and one33 bit multiplication (in the SMOC). Fig. 5.5 shows the formula as DAG. Theoptimized SMOC is abstracted to one node. PBoolector splits the formula forexample by setting conditional 6 to be false. Node 6 is replaced with node 5which is the sign extension concatenation. A 32 bit zero-constant is concate-nated with a 32 bit variable. In the input parameter 1 of the 64 bit multiplication(id 12), now half of the bits are set to the fixed value 0. With the possibilityto fix half of the bits in the multiplication’s parameter, the SAT solver’s perfor-mance significantly can be improved. Different to the lazyitex benchmarks inSect. 5.3.1, here multiplications are not eliminated but its solving effort can bereduced. Splitting on the 2 conditionals (node 6 and 11) results into 4 subprob-lems which are relatively simpler because of the fixed bits in the multiplicationparameters.
The results for the combined look-ahead are shown in Tab. 5.24a. PBoolectoris able to outperform the sequential implementation very well. Additionally tothe total runtime the wall-clock time and cpu time of the search phase is printed.The SMOC contains several one-bit “and” and “slice” operations. With thecombined look-ahead the formula is first split on conditionals and then on theone-bit logical connectives of the SMOC. Tab. 5.24b lists the runtimes with
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splitting only on conditionals (“-lt 4”). The whole benchmark consist of 2conditionals which produce maximal 4 subproblems to solve in parallel. Thisexperiment should evaluate whether if it is sufficient to split only on the hardparts of the formula (the sign extension conditionals) or not. Actually withthe combined look-ahead the wall-clock time is clearly better. But it can beobserved that if more splitting is done some overhead is introduced. For instancethe 32 bit benchmark file occupies with pure conditional splitting only 1090.06seconds CPU time whereas the combined look-ahead takes 1421.85 seconds. Thedisadvantage of producing a low number of subproblems is that they can not besolved with reasonable speedup in parallel. For this benchmarks it is crucial tosplit on the conditional (to make the hard multiplication simpler). All furthersplitting introduces some overhead, but the splitting is important to get goodparallelization capabilities. For most other SMT-LIB benchmarks which exhibitbad performance probably the introduced overhead is too high.
5.4.2 MultiplyOverflow
The benchmarks challenge multiplyOverflow (Tab. C.1 in the appendix rank2.14) verifies the overflow detection like it is implemented in the C-API callcalloc():
1 void ∗calloc(size t a, size t b)2 {3 if( ((size t)−1) / a < b ){ errno = ENOMEM; return NULL; }4 return memset(malloc(a∗b), 0, a∗b);5 }
The verification of the 32 bit unsigned multiplication overflow is checkedagainst the naive implementation. The naive implementation is similar to thesigned multiplication overflow check which was shown in the previous section,with the difference that no conditional is needed for the unsigned value exten-sion.
p64 = (032 ◦ a32) · (032 ◦ a32) ∧ p64[63 : 32] > 032 6= ((032 − 1)/a32) < b32
The benchmark consists only of one look-ahead node, namely the 64 bit multi-plication which can be split on the LSB. Bit splitting on the hard multiplicationcan improve the result with a speedup of 1.09. With this file it could be shownthat bit splitting is also successful on a real world benchmark.
In the category challenge the benchmark integerOverflow (Tab. C.1 in theappendix rank 5.196) verifies that no signed multiplication overflow can occurin the function baz() if the variable y 6= 0:
1 int baz(int x, int y)2 { return (x / y) ∗ y; }
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benchmark Boolector[s] PBoolector[s] statusproblem 3 22.98 oot unsatproblem 7 992.74 oot satproblem 8 992.88 oot satproblem 10 oot 1.83 unsatproblem 11 oot 1.79 unsatproblem 12 oot oot unknownproblem 13 oot oot unknownproblem 14 1.09 0.96 unsatproblem 16 oot 2.99 unsatproblem 17 oot oot unknownproblem 18 oot oot unknownproblem 19 oot 22.14 unsatproblem 20 145.34 oot unsatproblem 21 oot 1.82 unsatproblem 22 191.71 oot unsatproblem 23 1458.02 oot unsat
Table 5.25: Runtime comparison of selected calypto benchmark files.
For this benchmark PBoolector should be also capable to outperform Boolectorbecause it is verified against a naive signed multiplication overflow implementa-tion like it is done in the previous section. Actually the 32 bit overflow detectionis still to hard for PBoolector that it can be solved within the time limit.
5.4.3 Calypto
The competition benchmarks of 2014 does not contain benchmarks from the cat-egory calypto. In this family several benchmarks are found which are quite fastfor PBoolector. The benchmark set contains 24 small and very hard problems.One reason for its hardness is that the problems contain hard multiplicationand/or adder operations with high bit-width. Tab. 5.25 shows the runtimecomparison between Boolector and PBoolector. Benchmarks which are solvedin less than 10 seconds by both implementations are not shown in the table(except problem 14 ). In the following the successful benchmarks are examinedin detailed.
Problem 14
In Tab. 5.25 problem 14 has fast runtime for both implementations. Actually atthe beginning of the experiments Boolector was not able to solve this file within1800 seconds whereas PBoolector could solve it within few seconds. During theinvestigation to find out why the benchmark works for PBoolector some newrewriting rules have been found such that Boolector is able to solve the file ef-ficiently too (therefore the fast runtime of 1.09 seconds). In the following, firstthe reason for fast PBoolector solving is shown and second the new rewritingrule is explained.
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This benchmark contains a similar pattern (sign-extension pattern) as thebenchmarks in Sect. 5.4.1. The structure of problem 14 is shown in Fig. B.3 inthe appendix. The DAG is relatively small but contains one hard 32 bit andone hard 33 bit multiplication . Setting for example node 8 to be false can fixthe 16 LSBs of the 32 bit multiplication’s (id 12) parameter (id 11). Furtherthe 17 MSBs of the 33 bit multiplication’s (id 21) parameter (id 16) are fixed aswell. Splitting on node 8 leads to 2 subproblems which are considerably simplerthan the original formulas, because half of the multiplication’s parameter bitscan be fixed to a constant value.
If a closer look is taken to the DAG it can be observed that 32 bits of thetwo multiplications are structural identical. The multiplication operations cannot be shared on term-level as its bit-with is different. But parts of the multi-plication can be shared on the AIG level. The problem here is that parametersof the multiplication are exchanged: multiplication 21 has the left part as firstparameter and multiplication 12 has the right part as first parameter. As multi-plication is commutative, a rewriting rule was introduced in Boolector such thatthe multiplications parameter ordering on the AIG level is normalized. This isdone for example by exchanging the parameters for multiplication 12: node 11gets parameter 1 and node 6 gets parameter 2. Then, on the AIG level 32 bitsof the two multiplications can share its structure and the problems gets verytrivial for the SAT solver. Adding this new rewriting the formula can be solvedby Boolector in 1.09s without any splitting. Actually with this rule a new classof rewriting could be introduced to Boolector. The new rewriting rule operateson the intermediate AIG layer instead on the formula DAG (term layer).
Problem 16
Benchmark problem 16 is solved by PBoolector in 2.99 seconds whereas Boolec-tor runs out of time. The benchmark file problem 16 contains 5 hard 32 bitmultiplications. The inputs of the multiplications are sign-extended variables.Similar to the smulov1bw* benchmarks presented in section 5.4.1 parts of themultiplication bit-vectors can be set to a fixed value by splitting on the condi-tional node (the encoded sign-extension operation). Fig. B.4 in the appendixshows the formula as DAG representation. As the problem is solved withoutproducing a lot of subproblems the real execution tree of a PBoolector can bevisualized (Fig. B.5 in the appendix). On the top the original formula is spliton a conditional. Two subproblems are produced in round 1, one trivial unsatsubproblem (left) which can immediately be closed and one reduced subproblem(right). The reduced formula is split until both generated subproblems can besolved within the limit (round 4). On the diagonal the formulas size (hardness)reduction can be observed. Similar to the lazyitex benchmarks with the im-proved heuristic (Sect. 5.3.1), in each round only two problems can be run inparallel because the left branch can always be closed immediately.
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Problem 19
Benchmark problem 19 is a bigger formula than the previous presented ones.It contains several logical operations and three multiplications. One 31 bitmultiplication seems to be the bottleneck for this benchmark. The partial DAGof the formula shown in Fig. B.6 in the appendix serves as input parameter forthe 31 bit multiplication. If the formula is split by setting the one bit variable(id 287) to true or false 27 bits of the input parameter can be fixed by theconcatenation operators to constant 1 or constant 0. Fixing the multiplication’sbits leads to a remarkable speed up in solving times. Actually this benchmarkis the reason why the heuristic pocc + nocc (see Sect. 4.3.4) was chosen for theVAR1 look-ahead type. The variable is referenced 27 times positively and 0times negatively. The multiplication heuristic with score 27 · 0 = 0 would notselect the variable as look-ahead node whereas the addition heuristic with score27+0 = 27 is able to select the node for splitting. The pattern shown in Fig. B.6is just an unconventional way to encode a sign extension. The one-bit variable287 is the sign-bit of variable 314.
With some new rewriting rules the encoding of the partial DAG could bereduced to a conventional sign extension like it was shown in Sect. 5.4.1 (Theindex denotes the bit-width):
x1 ◦ x1 ⇒ x1? 12 : 02
(x1? 1n : 0n) ◦ x1 ⇒ x1? 1n+1 : 0n+1
If the rules are applied recursively on the pattern, rewriting would end up in theconventional sign-extension pattern x1? 127 : 027. It is left for future work to findout if such new rewriting rules could speed up Boolector. I assume Boolectorwill not speed up because, like for other shown benchmarks, the multiplicationonly gets easy if the bits of its parameters can be fixed directly. The bits ofthe parameter can only be fixed with help of PBoolector by splitting on theconditional node of the resulting sign-extension pattern.
Other Benchmark Files
There are three more benchmarks which run into timeout with Boolector andwhich are solved by PBoolector (problem 10/11/21 ). This slow solving time isstrange, because all 3 file where solved with a previous Boolector version within2 seconds. A short investigation identified the non-incremental mode of Boolec-tor as the cause of this slowdown. PBoolector in incremental mode with limit-1 solves the three files within 5s. If PBoolector runs in non-incremental modewith limit -1 all three files run into timeout.
There are several benchmarks left where PBoolector runs into timeout whereassome of these files are solved by Boolector. It was tried to improve the results ofPBoolector by changing the limit and run it with the individual look-ahead typesto solve one of the remaining benchmarks. None of these informal experimentswas able to improve the results. The benchmarks, not solved by BPoolector,
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also contain several sign-extension operations. On some benchmarks less bitsare extended and on some benchmarks the extensions serve as input for addersor other “simple” operations. It might be that PBoolector is just able to out-perform Boolector if the sign-extensions are inputs of multiplications and a hugenumber of bits are extended, which was the case for the successful benchmarks.The satisfiable benchmarks (problem 7/8 ) are not applicable for the look-aheadtypes which are defined to improve satisfiable instances (MULNV0, MULNV1,ADDNV0), because the containing multiplication/additions have no variablesas parameters.
5.4.4 Log Slicing
In the category log-slicing several benchmarks with reasonable speedup withPBoolector are identified (see Tab. C.1 in the appendix). The benchmarks wheredeveloped along complexity evaluation of fixed-size bit-vector formulas [33]. Itwas shown that a subset of base operations (bit-wise operations, equalities andslicing) suffice to derive NEXP-TIME hardness. A set of benchmarks was gen-erated to check the translation of arithmetic operations to the base operations.Benchmark bvudiv 16 for instance checks the translation of a 16 bit unsigneddivision. Because the translation is done by introducing new variables the DAGstructure is rather flat for the bv * benchmarks and contains various constraintequality nodes. The look-ahead candidates are some few one-bit slice nodesand equalities. Several SLICE1 nodes have only one occurrence, hence it is im-portant for these benchmarks that the heuristic pocc + nocc is used. Actuallyno clear reason could be found why this kind of benchmarks work well in theparallel approach. One reason might be that the search tree here is boundedby the few number of look-ahead candidates. After some rounds no look-aheadcan be found and the problems are finished.
5.4.5 Galois - IffyInterlaevedModMul-8
Also good performance improvement could also achieved for the benchmarkiffyInterleavedModMult-8 (see Tab. C.1 in the appendix, rank 1.3). The bench-mark is explained as follows in the formula header: “The benchmarks are check-ing whether an algorithm for computing modular multiplication by interleavingthe multiplication and residue calculations, is equivalent to an algorithm thatfirst multiplies the two inputs, and then performs the residue calculation onthe result.” It contains 31 BCOND, 16 SLICE1, 1 BEQ1VN and 1 MULNBITnodes (with bit-width 16) as look-ahead candidates. As in the previous sectionno clear reason for the successful PBoolector run was found.
There are some properties which this benchmark has in common with thesuccessful log-slicing bv* benchmarks: The formula is quite hard with respectto its size (the DAG contains 247 nodes). The structure of the DAG is flat andit contains several equality constraint nodes. The look-ahead candidates aremostly direct children of the constraint nodes. There is one difference which
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can be observed that the successful log-slicing benchmarks produce few sub-problems (∼ 25) whereas the iffyInterleavedModMul-8 benchmark produces alot of subproblems (442).
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Chapter 6
Conclusion
6.1 Summary
A parallel SMT solving approach for the quantifier-free fragment of fixed-sizebit-vectors, based on the Cube and Conquer algorithm, was presented. Theapproach was implemented in the tool PBoolector based on the state-of-the-artSMT solver Boolector. No single algorithm was able to achieve overall goodperformance for all kind of benchmarks. On many benchmarks a huge slow-down was observed by the algorithm. The used look-ahead techniques are oftennot applicable for search space splitting because the solving effort of the pro-duced subproblems can not be reduced. Anyhow, it could be shown that thealgorithm is able to outperform the sequential Boolector implementation onformulas containing certain patterns, with up to super-linear speedup. The suc-cessful benchmarks are mostly related with hard multiplication circuits, whichis one of the bottlenecks in the bit-blasting algorithm. The identified patternscan be classified into 3 types.
• It was shown that nested if-then-else conditionals with interleaving multi-plications can be solved very efficient with the PBoolector approach (seeSect. 5.3.1).
• It could be shown that sign-extension encodings, in practice often usedwith multiplication overflow verification, is able to achieve a well speedupon several real-world benchmarks from the SMT-LIB (see Sect. 5.4).
• It was shown that bit-splitting on the least significant bit for hard multi-plication circuits achieves a well linear speedup (see Sect. 5.3.2). On theother hand it was shown that bit-splitting could not achieve a positivespeedup for hard adder circuits (see Sect. 5.3.3).
Some additional formula patterns were detected, different to the mentionedmultiplication patterns, which work well for PBoolector. A nice side effect wasthat a new rewriting rule on the AIG Layer was found during closer investigationof the evaluation results of PBoolector (see Sect. 5.4.3 problem 14 ).
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6.2 Future work
To conclude the thesis the final section will give an outlook of possible futureextension and research proposals to improve the introduced algorithm. Firsttwo rather small implementation extensions and next some conceptual improve-ments are proposed.
As first small extension the update function can be enhanced. The num-ber of solved subproblems by the rewrite engine and the SAT solver could becounted separately. If the number of subproblems solved by the SAT solver isgreater than the number of closed subproblems than the limit is increased as itwas proposed. But, if a lot of subproblems are solved by the rewrite engine thelimit should be decreased. In the case that the SAT solver does not contributeto solve the problems its work effort should be reduced by decreasing the limit.The second extension is to add term elimination in the rewriting engine similarto the variable elimination. This extension would not restrict several look-aheadtypes to nodes, which contain at least one variables as parameter (BEQ1VN,MULNV0, . . . ). The rewrite engine would be able to eliminate for exampleterms like a · b with a + c if it is known that a · b = a + c. The evaluation, ifthese two extensions can bring some performance gain is left for future work.
The assumption that a formula’s hardness is proportional to its word-levelDAG size might fail for many formula instances. In the future a closer investiga-tion on that topic might be beneficial. Empirical measurements, which comparethe assumed hardness and the real solving effort (runtime of the SAT solver) ofthe generated subproblems are proposed for evaluation. The hardness measuremight be improved by estimating the number of clauses and variables, whichwill be generated by bit-blasting.
Probably, there is also potential in the improvement of the heuristics for scor-ing the look-ahead candidates. The used heuristics for PBoolector are countingthe number of occurrences. A closer investigation can be done by comparing thescore of the heuristic with the score of the probing method (which is assumed toselect the perfect look-ahead candidate). The Task is to find the best heuristicfunction for each look-ahead type that maximizes the correlation between themeasured reduction and the heuristically estimated reduction. This is obviouslya hard task.
A further possible future direction is to detect hard parts in the formula. Theevaluation has shown that PBoolector is successful if the solving effort of hardarithmetic formula parts can be reduced. If it is possible to find and reducethese parts from the beginning of the search tree generation, the introducedoverhead which was observed for many benchmarks might be reduced. For ex-ample if a hard multiplication circuit is split first on an equality node which isnot related with the multiplication, two subproblems are produced (still con-taining the hard multiplication part) which the SAT solver has to solve. A
76
simple heuristic improvement would be to weight the score with the numberof arithmetic operations that are related with the look-ahead node. The morearithmetic operations the higher the weight. The already existing and in-depthevaluated (look-ahead) SAT solver heuristics might be of help to find word-levelheuristic improvements. The heuristics might be also improved by introducingand combining more statistical features which describe the formula’s structure.For a simple example the reduction of a formula is higher if a parameter of along logical AND chain is set to 0 than the parameter of a short logical ANDchain (a ∧ b ∧ . . . ∧ 0 ≡ ⊥).
In the future PBoolector should be extended to other theories supportedby Boolector, the theory of arrays, lambda expressions and their combination.An evaluation on the whole set of SMT-LIB benchmarks (for QF BV ∼ 33000files) can be done, to possibly identify further real world application patternsapplicable for PBoolector. On the other hand, additionally hard bit-vectorformulas can be crafted to further analyze PBoolectors strengths. The usedback end SAT solver Lingeling is optimized for hard standalone CNF formulas.It might be possible to find a SAT solver configuration that is more adequatefor the PBoolector application.
77
Appendices
78
Appendix A
Table Notations
A.1 Comparison Tables
In the comparison tables the improvements of the parallel implementation isshown compared to its reference implementation. The improvement is catego-rized into 5 different success states. The table contains the columns:
• compare: The left column “ref” contains the name of the reference toolto which the tool in the right column “to” is compared to.
• filter: Contains information about filtering the benchmark set.
• total: Contains the number of benchmarks which are compared.
• success state “++”: Contains the number of benchmarks which are notsolved by the left (ref) tool and which is solved by the right (to) tool.
• success state +: Contains the number of benchmarks which are solvedfaster by the right tool.
• success state −: Contains the number of benchmarks which are solvedslower by the right tool.
• success −−: Contains the number of benchmarks which are not solved byright tool although they are solver by the left tool.
• o: The number of benchmarks which are not solved by both implementa-tions.
A.2 Detailed Result Tables
The detailed result table shows statistical values to analyze a PBoolector run.Unnecessary information is not shown. For example in a run which do not useFailed Literal Reduction (FLR) the statistical values for FLR are not printed.The table can consist of following columns:
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• rank: The results are ordered by success state. Each benchmark gets aunique ranking number for each run.
• benchmark: The name of the benchmark with preceding category. If thename is to long it is abbreviated with “. . . ”. Each benchmark should beclearly identifiable by the name.
• stat: The status of the benchmark. Here to common encoding 0 (un-known), 10 (sat) and 20 (unsat) is used.
• sequ-btor: The runtime of the sequential Boolector implementation inseconds.
• ref: The runtime of the reference implementation to which the parallel runis compared (in seconds). This column contains for example the runtimeof the reference implementation Base-Incremental (base-inc).
• runtime statistics for the parallel run (e.g. PBoolector).
– tot(wc): The total wall-clock runtime in seconds.
– diff: The difference between the parallel and the reference runtimein seconds. If a timeout has occurred the time limit of 1800 secondsis used for the difference.
– su: The achieved speedup. If a timeout has occurred the time limitof 1800 seconds is used for the speedup.
– mem: The memory consumption for the parallel run in GB.
– rnd: The number of rounds in the main algorithm.
– subproblems: This section contains detailed information about theproduced subproblems.
∗ “tot”: The total number of produced subproblems.
∗ “max”: The maximal number of open subproblems.
∗ “fin”: The number of finished subproblems where no look-aheadwas found.
– flr: This section contains detailed information about the Failed Lit-eral Reduction.
∗ tot(wc): The wall-clock time spent in FLR (in seconds).
∗ fl: The number of found failed literals.
∗ res: The result of FLR (0,10 or 20).
– lkhd: The wall-clock time spent in the look-ahead phase (in seconds).
– split: The wall-clock time spent in the split phase (in seconds).
– search: This section contains detailed information about the searchphase.
∗ tot(wc): The wall-clock time spent in the search phase (in sec-onds).
80
∗ tot(cpu): The total CPU time spent in the search phase (inseconds).
∗ sat(cpu): The total CPU time spend in the SAT solver duringthe search (in seconds).
∗ min: The minimal time of all boolector sat calls in seconds.
∗ max: The maximal time of all boolector sat calls in seconds.
∗ mean: The mean time of all boolector sat calls in seconds.
• cmp: A column which lists the ranking of another parallel run for com-parison (e.g the rank of par-inc-flr).
81
Appendix B
Figures
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O0
Fig
ure
B.1
:F
orm
ula
1.1
wit
hb
it-w
idth
8en
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edas
AIG
.T
he
tria
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eis
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83
1: var (1)2: zero (32)
3: cond (32)123
4: concat (33)
21
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6: concat (64)
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8: cond (32)23 1
9: concat (33)21
10: var (31)
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14: slice (63)6 2 0
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179: slice (1)1 1
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187: mul (33)1
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189: slice (1)3 1 3 1
190: eq (1)
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192: eq (1)12
Figure B.2: Benchmark smulov1bw32. On the bottom the naive implementationwhich is checked against the Signed Multiplication Overflow Circuit (SMOC).
84
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Fig
ure
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:B
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mar
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14
conta
ins
2h
ard
mu
ltip
lica
tion
op
erati
on
sw
ith
sign
-exte
nd
edp
ara
met
ers.
85
1:
var
(1
)
2:
var
(3
2)
3:
slic
e (1
)3
1 3
1
4:
zero
(1
6)
5:
con
d (
16
)1
2 36
: sl
ice
(16
)3
1 1
6
7:
con
cat
(32
)1
2
8:
var
(3
2)
9:
slic
e (1
)3
1 3
1
10
: co
nd
(1
6)
32
1
11
: sl
ice
(16
)3
1 1
6
12
: co
nca
t (3
2)
1
2
13
: m
ul
(32
)
12
14
: v
ar (
1)
15
: v
ar (
32
)
16
: sl
ice
(1)
31
31
17
: co
nd
(1
6)
23
1
18
: sl
ice
(16
)3
1 1
6
19
: co
nca
t (3
2)
12
20
: v
ar (
32
)
21
: sl
ice
(1)
31
31
22
: co
nd
(1
6)
23
1
23
: sl
ice
(16
)3
1 1
6
24
: co
nca
t (3
2)
12
25
: m
ul
(32
)
12
26
: v
ar (
1)
27
: v
ar (
32
)
28
: sl
ice
(1)
31
31
29
: co
nd
(1
6)
23
1
30
: sl
ice
(16
)3
1 1
6
31
: co
nca
t (3
2)
12
32
: v
ar (
32
)33
: sl
ice
(1)
31
31
34
: co
nd
(1
6)
23
1
35
: sl
ice
(16
)3
1 1
6
36
: co
nca
t (3
2)
12
37
: m
ul
(32
)
12
38
: v
ar (
1)
39
: v
ar (
32
)
40
: sl
ice
(1)
31
31
41
: co
nd
(1
6)
23
1
42
: sl
ice
(16
)3
1 1
6
43
: co
nca
t (3
2)
12
44
: v
ar (
32
)
45
: sl
ice
(1)
31
31
46
: co
nd
(1
6) 23
1
47
: sl
ice
(16
)3
1 1
6
48
: co
nca
t (3
2)
12
49
: m
ul
(32
)
12
50
: ze
ro (
32
)
51
: co
nd
(3
2)
1
2
3
52
: co
nd
(3
2)
12
3
53
: co
nd
(3
2)
12
3
54
: co
nd
(3
2)
12
3
55
: an
d (
1)
12
56
: an
d (
1)
12
57
: an
d (
1)
12
58
: an
d (
1)
12
59
: an
d (
1)
12
60
: an
d (
1)
12
61
: an
d (
1)
1 2
62
: an
d (
1)
12
63
: an
d (
1)
12
64
: an
d (
1)
12
65
: an
d (
1)
12
66
: an
d (
1)
12
67
: an
d (
1)
12
68
: an
d (
1)
12
69
: ze
ro (
15
)
70
: co
nd
(1
5)
12
3
71
: co
nca
t (1
6) 2
1
72
: co
nd
(1
6)
31
2
73
: co
nd
(1
6)
123
74
: co
nd
(1
6)
12
3
75
: co
nd
(1
6)
1
2
3
76
: co
nca
t (3
2)
12
77
: an
d (
1)
12
78
: an
d (
1) 1
2
79
: an
d (
1) 1
2
80
: an
d (
1)
1 2
81
: an
d (
1)
12
82
: an
d (
1)
12
83
: an
d (
1)
12
84
: an
d (
1)
12
85
: an
d (
1) 1
2
86
: an
d (
1)
12
87
: an
d (
1) 1
2
88
: co
nd
(1
5)
23
1
89
: co
nca
t (1
6)
21
90
: co
nd
(1
6)
31
2
91
: co
nd
(1
6) 1
23
92
: co
nd
(1
6) 1
23
93
: co
nd
(1
6)
12
3
94
: co
nca
t (3
2)
12
95
: m
ul
(32
)
12
96
: eq
(1
)
12
Fig
ure
B.4
:B
ench
mar
kpro
blem
16
conta
ins
seve
ral
hard
mu
ltip
lica
tion
op
erati
ons
wit
hsi
gn
-exte
nd
edpara
met
ers.
86
1:
var
(1
)
2:
var
(3
2)
3:
slic
e (1
)3
1 3
1
4:
zero
(1
6)
5:
con
d (
16
)1
2 36
: sl
ice
(16
)3
1 1
6
7:
con
cat
(32
)1
2
8:
var
(3
2)
9:
slic
e (1
)3
1 3
1
10
: co
nd
(1
6)
32
1
11
: sl
ice
(16
)3
1 1
6
12
: co
nca
t (3
2)
1
2
13
: m
ul
(32
)
12
14
: v
ar (
1)
15
: v
ar (
32
)
16
: sl
ice
(1)
31
31
17
: co
nd
(1
6)
23
1
18
: sl
ice
(16
)3
1 1
6
19
: co
nca
t (3
2)
12
20
: v
ar (
32
)
21
: sl
ice
(1)
31
31
22
: co
nd
(1
6)
23
1
23
: sl
ice
(16
)3
1 1
6
24
: co
nca
t (3
2)
12
25
: m
ul
(32
)
12
26
: v
ar (
1)
27
: v
ar (
32
)
28
: sl
ice
(1)
31
31
29
: co
nd
(1
6)
23
1
30
: sl
ice
(16
)3
1 1
6
31
: co
nca
t (3
2)
12
32
: v
ar (
32
)33
: sl
ice
(1)
31
31
34
: co
nd
(1
6)
23
1
35
: sl
ice
(16
)3
1 1
6
36
: co
nca
t (3
2)
12
37
: m
ul
(32
)
12
38
: v
ar (
1)
39
: v
ar (
32
)
40
: sl
ice
(1)
31
31
41
: co
nd
(1
6)
23
1
42
: sl
ice
(16
)3
1 1
6
43
: co
nca
t (3
2)
12
44
: v
ar (
32
)
45
: sl
ice
(1)
31
31
46
: co
nd
(1
6) 23
1
47
: sl
ice
(16
)3
1 1
6
48
: co
nca
t (3
2)
12
49
: m
ul
(32
)
12
50
: ze
ro (
32
)
51
: co
nd
(3
2)
1
2
3
52
: co
nd
(3
2)
12
3
53
: co
nd
(3
2)
12
3
54
: co
nd
(3
2)
12
3
55
: an
d (
1)
12
56
: an
d (
1)
12
57
: an
d (
1)
12
58
: an
d (
1)
12
59
: an
d (
1)
12
60
: an
d (
1)
12
61
: an
d (
1)
1 2
62
: an
d (
1)
12
63
: an
d (
1)
12
64
: an
d (
1)
12
65
: an
d (
1)
12
66
: an
d (
1)
12
67
: an
d (
1)
12
68
: an
d (
1)
12
69
: ze
ro (
15
)
70
: co
nd
(1
5)
12
3
71
: co
nca
t (1
6) 2
1
72
: co
nd
(1
6)
31
2
73
: co
nd
(1
6)
123
74
: co
nd
(1
6)
12
3
75
: co
nd
(1
6)
1
2
3
76
: co
nca
t (3
2)
12
77
: an
d (
1)
12
78
: an
d (
1) 1
2
79
: an
d (
1) 1
2
80
: an
d (
1)
1 2
81
: an
d (
1)
12
82
: an
d (
1)
12
83
: an
d (
1)
12
84
: an
d (
1)
12
85
: an
d (
1) 1
2
86
: an
d (
1)
12
87
: an
d (
1) 1
2
88
: co
nd
(1
5)
23
1
89
: co
nca
t (1
6)
21
90
: co
nd
(1
6)
31
2
91
: co
nd
(1
6) 1
23
92
: co
nd
(1
6) 1
23
93
: co
nd
(1
6)
12
3
94
: co
nca
t (3
2)
12
95
: m
ul
(32
)
12
96
: eq
(1
)
12
1:
var
(3
2)
2:
slic
e (1
)3
1 3
13
: ze
ro (
16
)
4:
con
d (
16
)
12
3
5:
slic
e (1
6)
31
16
6:
con
cat
(32
)
12
7:
var
(3
2)
8:
slic
e (1
)3
1 3
1
9:
con
d (
16
)
32
1
10
: sl
ice
(16
)3
1 1
6
11
: co
nca
t (3
2)
12
12
: m
ul
(32
)
12
13
: ze
ro (
15
)
14
: co
nd
(1
5)
12
3
15
: co
nca
t (1
6) 2
1
16
: co
nca
t (3
2)
21
17
: co
nd
(1
5)
12
3
18
: co
nca
t (1
6)
21
19
: co
nca
t (3
2)
21
20
: m
ul
(32
) 12
21
: eq
(1
)
12
1:
var
(1
)
2:
var
(3
2)
3:
slic
e (1
)3
1 3
1
4:
zero
(1
6)
5:
con
d (
16
) 12
3
6:
slic
e (1
6)
31
16
7:
con
cat
(32
)1
2
8:
var
(3
2)
9:
slic
e (1
)3
1 3
1
10
: co
nd
(1
6)
32
1
11
: sl
ice
(16
)3
1 1
6
12
: co
nca
t (3
2)
12
13
: m
ul
(32
)1
2
14
: v
ar (
1)
15
: v
ar (
32
)
16
: sl
ice
(1)
31
31
17
: co
nd
(1
6)
23
1
18
: sl
ice
(16
)3
1 1
6
19
: co
nca
t (3
2)
12 20
: v
ar (
32
)
21
: sl
ice
(1)
31
31
22
: co
nd
(1
6)
231
23
: sl
ice
(16
)3
1 1
6
24
: co
nca
t (3
2)
12
25
: m
ul
(32
)
12
26
: v
ar (
1)
27
: v
ar (
32
)
28
: sl
ice
(1)
31
31
29
: co
nd
(1
6)
23
1
30
: sl
ice
(16
)3
1 1
6
31
: co
nca
t (3
2)
12
32
: v
ar (
32
)
33
: sl
ice
(1)
31
31
34
: co
nd
(1
6)
23
1
35
: sl
ice
(16
)3
1 1
6
36
: co
nca
t (3
2)
12
37
: m
ul
(32
)
12
38
: ze
ro (
32
)
39
: co
nd
(3
2)
12
3
40
: co
nd
(3
2)
12
3
41
: co
nd
(3
2)
12
3
42
: an
d (
1)
12
43
: an
d (
1)
12
44
: an
d (
1)
12
45
: an
d (
1)
1 2
46
: an
d (
1)
12
47
: an
d (
1)
12
48
: an
d (
1)
12
49
: an
d (
1)
12
50
: an
d (
1)
12
51
: an
d (
1)
12
52
: ze
ro (
15
)
53
: co
nd
(1
5)
12
3
54
: co
nca
t (1
6)
21
55
: co
nd
(1
6)
31
2
56
: co
nd
(1
6)
12
3
57
: co
nd
(1
6)
12
3
58
: co
nca
t (3
2)
12
59
: an
d (
1)
12
60
: an
d (
1) 1
2
61
: an
d (
1) 1
2
62
: an
d (
1) 1
2
63
: an
d (
1)
1264
: an
d (
1)
12
65
: an
d (
1)
12
66
: an
d (
1)
126
7:
con
d (
15
)
23
1
68
: co
nca
t (1
6)
21
69
: co
nd
(1
6)
31
2
70
: co
nd
(1
6) 1
23
71
: co
nd
(1
6)
12
3
72
: co
nca
t (3
2) 1
2
73
: m
ul
(32
)1
2
74
: eq
(1
)1
2
1:
var
(3
2)
2:
slic
e (1
)3
1 3
13
: ze
ro (
16
)
4:
con
d (
16
)
12
3
5:
slic
e (1
6)
31
16
6:
con
cat
(32
)
12
7:
var
(3
2)
8:
slic
e (1
)3
1 3
1
9:
con
d (
16
)
32
1
10
: sl
ice
(16
)3
1 1
6
11
: co
nca
t (3
2)
12
12
: m
ul
(32
)
12
13
: ze
ro (
15
)
14
: co
nd
(1
5)
12
3
15
: co
nca
t (1
6) 2
1
16
: co
nca
t (3
2)
21
17
: co
nd
(1
5)
12
3
18
: co
nca
t (1
6)
21
19
: co
nca
t (3
2)
21
20
: m
ul
(32
) 12
21
: eq
(1
)
12
1:
var
(1
)
2:
var
(3
2)
3:
slic
e (1
)3
1 3
14
: ze
ro (
16
)
5:
con
d (
16
)
12
3
6:
slic
e (1
6)
31
16
7:
con
cat
(32
)
12
8:
var
(3
2)
9:
slic
e (1
)3
1 3
1
10
: co
nd
(1
6)
32
1
11
: sl
ice
(16
)3
1 1
6
12
: co
nca
t (3
2)
12
13
: m
ul
(32
)
12
14
: v
ar (
1)
15
: v
ar (
32
)16
: sl
ice
(1)
31
31
17
: co
nd
(1
6)
23
1
18
: sl
ice
(16
)3
1 1
6
19
: co
nca
t (3
2)
12
20
: v
ar (
32
)21
: sl
ice
(1)
31
31
22
: co
nd
(1
6)
23
1
23
: sl
ice
(16
)3
1 1
6
24
: co
nca
t (3
2)
12
25
: m
ul
(32
)
12
26
: ze
ro (
32
)
27
: co
nd
(3
2)
12
3
28
: co
nd
(3
2) 1
23
29
: an
d (
1)
12
30
: an
d (
1)
12
31
: an
d (
1)
12
32
: an
d (
1)
12
33
: an
d (
1)
12
34
: an
d (
1)
12
35
: ze
ro (
15
)
36
: co
nd
(1
5) 1
233
7:
con
cat
(16
) 21
38
: co
nd
(1
6)
31
2
39
: co
nd
(1
6)
12
3
40
: co
nca
t (3
2) 1
2
41
: an
d (
1)
12
42
: an
d (
1)
12
43
: an
d (
1)
12
44
: an
d (
1)
1
2
45
: an
d (
1) 1
2
46
: co
nd
(1
5)
23
1
47
: co
nca
t (1
6)
21
48
: co
nd
(1
6)
31
2
49
: co
nd
(1
6)
12
3
50
: co
nca
t (3
2) 1
2
51
: m
ul
(32
) 12
52
: eq
(1
)
12
1:
var
(3
2)
2:
slic
e (1
)3
1 3
13
: ze
ro (
16
)
4:
con
d (
16
)
12
3
5:
slic
e (1
6)
31
16
6:
con
cat
(32
)
12
7:
var
(3
2)
8:
slic
e (1
)3
1 3
1
9:
con
d (
16
)
32
1
10
: sl
ice
(16
)3
1 1
6
11
: co
nca
t (3
2)
12
12
: m
ul
(32
)
12
13
: ze
ro (
15
)
14
: co
nd
(1
5)
12
3
15
: co
nca
t (1
6) 2
1
16
: co
nca
t (3
2)
21
17
: co
nd
(1
5)
12
3
18
: co
nca
t (1
6)
21
19
: co
nca
t (3
2)
21
20
: m
ul
(32
) 12
21
: eq
(1
)
12
1:
var
(1
)
2:
var
(3
2)
3:
slic
e (1
)3
1 3
14
: ze
ro (
16
)
5:
con
d (
16
)1
23
6:
slic
e (1
6)
31
16
7:
con
cat
(32
)
12
8:
var
(3
2)
9:
slic
e (1
)3
1 3
1
10
: co
nd
(1
6)
32
1
11
: sl
ice
(16
)3
1 1
6
12
: co
nca
t (3
2)
1
2
13
: m
ul
(32
)
12
14
: ze
ro (
32
)
15
: co
nd
(3
2)
12
3
16
: an
d (
1)
12
17
: ze
ro (
15
)
18
: co
nd
(1
5)
12
3
19
: co
nca
t (1
6)
21
20
: co
nca
t (1
6)
21
21
: an
d (
16
)
12
22
: co
nd
(1
6) 1
32
23
: co
nca
t (3
2)
12
24
: an
d (
1)
1
2
25
: co
nd
(1
5)
23
1
26
: co
nca
t (1
6) 2
1
27
: co
nca
t (1
6)
21
28
: an
d (
16
)1
2
29
: co
nd
(1
6)
13
2
30
: co
nca
t (3
2)
1
2
31
: m
ul
(32
) 12
32
: eq
(1
)
12
1:
var
(3
2)
2:
slic
e (1
)3
1 3
13
: ze
ro (
16
)
4:
con
d (
16
)
12
3
5:
slic
e (1
6)
31
16
6:
con
cat
(32
)
12
7:
var
(3
2)
8:
slic
e (1
)3
1 3
1
9:
con
d (
16
)
32
1
10
: sl
ice
(16
)3
1 1
6
11
: co
nca
t (3
2)
12
12
: m
ul
(32
)
12
13
: ze
ro (
15
)
14
: co
nd
(1
5)
12
31
5:
zero
(1
)
16
: co
nca
t (1
6)
12
17
: co
nca
t (1
6)
21
18
: an
d (
16
)
12
19
: co
nca
t (3
2)
21
20
: co
nd
(1
5)
12
3
21
: co
nca
t (1
6)
21
22
: co
nca
t (1
6)
21
23
: an
d (
16
)
12
24
: co
nca
t (3
2)
21
25
: m
ul
(32
) 12
26
: eq
(1
)
12
1:
zero
(1
)
inti
al
rou
nd
1
rou
nd
2
rou
nd
3
rou
nd
4
Fig
ure
B.5
:V
isu
ali
zed
exec
uti
on
tree
of
PB
oole
ctor
for
pro
blem
16.
87
1: zero (14)
2: zero (26)
3: var (4)
4: concat (30)
12
5: zero (4)
6: var (6)
7: const (6)
8: eq (1)
12
9: const (26)
10: const (6)
11: eq (1)1
2
12: zero (18)
13: const (7)
14: var (23)
15: concat (30)
12
16: slice (8)2 9 2 2
17: concat (26)
12
18: const (6)
19: eq (1)
12
20: zero (17)
21: slice (9)2 9 2 1
22: concat (26)
12
23: const (6)
24: eq (1)1
2
25: zero (16)
26: slice (10)2 9 2 0
27: concat (26)
1
2
28: const (6)
29: eq (1)
12
30: zero (15)
31: slice (11)2 9 1 9
32: concat (26)
1
2
33: const (6)
34: eq (1)
12
35: slice (12)2 9 1 8
36: concat (26)
1 2
37: const (6)
38: eq (1)1
2
39: zero (13)
40: slice (13)2 9 1 7
41: concat (26)
12
42: const (6)
43: eq (1)1
2
44: zero (12)
45: slice (14)2 9 1 6
46: concat (26)
12
47: const (6)
48: eq (1)12
49: zero (11)
50: slice (15)2 9 1 5
51: concat (26)
12
52: const (6)
53: eq (1)1
2
54: zero (10)
55: slice (16)2 9 1 4
56: concat (26)
12
57: const (6)
58: eq (1)
12
59: zero (9)
60: slice (17)2 9 1 3
61: concat (26)
12
62: const (6)
63: eq (1)1
2
64: zero (8)
65: slice (18)2 9 1 2
66: concat (26)
12
67: const (6)
68: eq (1)
12
69: zero (7)70: slice (19)
2 9 1 1
71: concat (26)
12
72: const (6)
73: eq (1)
1 2
74: zero (6)
75: slice (20)2 9 1 0
76: concat (26)
12
77: const (6)
78: eq (1)
12
79: zero (5)80: slice (21)
2 9 9
81: concat (26)12
82: const (6)
83: eq (1)
12
84: slice (22)2 9 8
85: concat (26)
12
86: const (6)
87: eq (1)
12
88: zero (3)89: slice (23)
2 9 7
90: concat (26)12
91: const (6)
92: eq (1)
1 2
93: zero (2)
94: slice (24)2 9 6
95: concat (26)
12
96: const (6)
97: eq (1)
1 2
98: zero (1)
99: slice (25)2 9 5
100: concat (26)
12
101: const (6)
102: eq (1)
1 2
103: slice (26)2 9 4
104: const (6)
105: eq (1)
12
106: slice (26)2 8 3
107: const (6)
108: eq (1)
1 2
109: slice (26)2 7 2
110: const (6)
111: eq (1)
12
112: slice (26)2 6 1
113: const (6)
114: eq (1)
12
115: const (3)
116: concat (26)21
117: const (6)
118: eq (1)
1 2
119: concat (32)1
2
120: slice (26)2 6 1
121: eq (1)1
2
122: concat (24)
2 1
123: concat (26)
21
124: cond (26)
312
125: cond (26)12
3
126: cond (26)
123
127: cond (26)1
23
128: cond (26)
12
3
129: cond (26)1
23
130: cond (26)1
2
3
131: cond (26)12
3
132: cond (26)
123
133: cond (26)1
23
134: cond (26)
123
135: cond (26)
123
136: cond (26)
12 3
137: cond (26)
123
138: cond (26)
12 3
139: cond (26)
12 3
140: cond (26)
12 3
141: cond (26)
12 3
142: cond (26)
123
143: cond (26)
12 3
144: cond (26)12 3
145: cond (26)
123
146: cond (26)
12 3
147: cond (26)
12
3
148: cond (26)
12
3
149: cond (26) 12 3
150: concat (30)
1 2
151: mul (30)
1 2
152: concat (44)1
2
153: var (23)
154: concat (26)
12
155: concat (32)2
1
156: ul t (1)2 1
157: zero (32)
158: const (6)
159: eq (1)1
2
160: zero (31)
161: const (26)
162: mul (26)1
2
163: slice (1)2 5 2 5
164: concat (32)
1 2
165: const (6)
166: eq (1)
12
167: zero (30)
168: slice (2)2 5 2 4
169: concat (32)
1 2
170: const (6)
171: eq (1)1
2
172: zero (29)173: slice (3)
2 5 2 3
174: concat (32)
1 2
175: const (6)
176: eq (1)1 2
177: zero (28)
178: slice (4)2 5 2 2
179: concat (32)
1 2
180: const (6)
181: eq (1)12
182: zero (27)
183: slice (5)2 5 2 1
184: concat (32)
1 2
185: const (6)
186: eq (1)1
2
187: slice (6)2 5 2 0
188: concat (32)12
189: zero (25)190: slice (7)
2 5 1 9
191: concat (32)
1 2
192: zero (24)193: slice (8)
2 5 1 8
194: concat (32)
1 2
195: zero (23)196: slice (9)
2 5 1 7
197: concat (32)
1 2
198: zero (22)
199: slice (10)2 5 1 6
200: concat (32)
1 2
201: zero (21)
202: slice (11)2 5 1 5
203: concat (32)
1 2
204: zero (20)205: slice (12)
2 5 1 4
206: concat (32)
1 2
207: zero (19)208: slice (13)
2 5 1 3
209: concat (32)
1 2
210: slice (14)2 5 1 2
211: concat (32)1
2
212: slice (15)2 5 1 1
213: concat (32)1 2
214: slice (16)2 5 1 0
215: concat (32)1
2
216: slice (17)2 5 9
217: concat (32)1
2
218: slice (18)2 5 8
219: concat (32)1 2
220: slice (19)2 5 7
221: concat (32)1 2
222: slice (20)2 5 6
223: concat (32)
1 2
224: slice (21)2 5 5
225: concat (32)1
2
226: slice (22)2 5 4
227: concat (32)1 2
228: slice (23)2 5 3
229: concat (32)1
2
230: slice (24)2 5 2
231: concat (32)1 2
232: slice (25)2 5 1
233: concat (32)1
2
234: concat (32)1 2
235: concat (32)
2 1
236: slice (27)3 1 5
237: concat (32)1
2
238: slice (28)3 1 4
239: concat (32)1 2
240: slice (29)3 1 3
241: concat (32)1
2
242: slice (30)3 1 2
243: concat (32)1 2
244: slice (31)3 1 1
245: concat (32)1
2
246: cond (32)1
3 2
247: cond (32)
1 23
248: cond (32)1
23
249: cond (32)1 23
250: cond (32)1
2 3
251: cond (32)
1 23
252: cond (32)1 23
253: cond (32)
1 23
254: cond (32)1
23
255: cond (32)
123
256: cond (32)
12 3
257: cond (32)
1 23
258: cond (32)
1 23
259: cond (32)
1 23
260: cond (32)1
23
261: cond (32)
1 23
262: cond (32)1
2 3
263: cond (32)
1 23
264: cond (32)12 3
265: cond (32)1 23
266: cond (32)
12
3
267: cond (32)1
23
268: cond (32)1
23
269: cond (32)
1 23
270: cond (32)1
23
271: cond (32)
123
272: cond (32)
123
273: cond (32)
123
274: cond (32)1 2
3
275: cond (32)1 2
3
276: cond (32)1 2
3
277: cond (32)1 23
278: cond (32)1
23
279: add (32)1
2
280: concat (43)
1
2
281: var (42)
282: concat (43)
1 2
283: add (43)
12
284: concat (44)
1 2
285: concat (29)1 2
286: concat (31)2
1
287: var (1)
288: concat (2)
1 2
289: concat (3)2
1
290: concat (4)
21
291: concat (5)2
1
292: concat (6)
21
293: concat (7)2
1
294: concat (8)
21
295: concat (9)2
1
296: concat (10)
21
297: concat (11)2
1
298: concat (12)
21
299: concat (13)2
1
300: concat (14)
21
301: concat (15)2
1
302: concat (16)
21
303: concat (17)2
1
304: concat (18)
21
305: concat (19)2
1
306: concat (20)
21
307: concat (21)2
1
308: concat (22)
21
309: concat (23)
21
310: concat (24)
21
311: concat (25)2
1
312: concat (26)2
1
313: concat (27)
2 1
314: var (4)
315: concat (31)
12
316: mul (31)
1
2
317: slice (1)3 0 3 0
318: concat (2)
1 2
319: concat (3)
21
320: concat (4)
2 1
321: concat (5)
2 1
322: concat (6)2
1
323: concat (7)2
1
324: concat (8)
21
325: concat (9)
21
326: concat (10)
21
327: concat (11)
2 1
328: concat (12)
21
329: concat (13)
21
330: concat (44)2
1
331: add (44)
1
2
332: add (44)
1 2
333: slice (1)4 3 4 3
Figure B.6: Partial DAG of benchmark problem 19. Node 315 serves as inputparameter for a 31 bit multiplication.
88
Appendix C
Detailed Results
89
Tab
leC
.1:
Det
aile
dre
sult
sof
PB
oole
ctor
(Para
llel
-In
crem
enta
l)co
mp
are
dto
the
refe
ren
ceti
mes
Base
-In
crem
enta
l.L
ast
colu
mn
par
-in
c-fl
rco
nta
ins
acr
oss
refe
ren
ceto
det
ail
edre
sult
sofP
ara
llel
-In
crem
enta
l-F
LR
inT
ab
.C.2
(Det
ail
edta
ble
nota
tion
sar
ein
Sec
t.A
.2).
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
1.1
brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
5oot
480.2
91319.7
13.7
50.5
54
270
49
41
0.0
32.6
4476.4
81321.3
61317.7
90.0
027.5
71.8
31.1
1.2
log-s
licin
gbvurem
16
20
oot
oot
1002.9
2797.0
81.7
90.2
824
23
22
0.0
10.3
41001.6
92526.8
42526.6
30.0
0340.7
730.4
41.2
1.3
...iffyIn
terle
avedM
odM
ult-8
20
oot
oot
1379.2
6420.7
41.3
10.4
63
442
142
00.0
71.3
41376.5
05110.8
35109.3
80.0
020.3
24.1
81.3
1.4
core
ext
con
056
002
1024
20
279.9
8oot
1464.5
8335.4
21.2
30.9
11
11
0.0
00.0
01463.7
41463.7
41463.3
4155.6
11308.1
3731.8
75.6
1.5
core
ext
con
060
002
1024
20
220.7
1oot
1470.7
1329.2
91.2
20.6
11
11
0.0
00.0
01469.9
71469.9
71469.5
6166.7
81303.1
9734.9
95.8
1.6
core
ext
con
052
002
1024
20
194.6
0oot
1495.9
2304.0
81.2
00.7
11
11
0.0
00.0
01495.1
21495.1
21494.7
5137.6
51357.4
8747.5
65.5
1.7
log-s
licin
gbvudiv
18
20
oot
oot
1554.0
9245.9
11.1
60.8
11
40
31
23
0.0
37.5
11541.4
04286.6
94285.4
60.0
0532.3
634.8
51.4
1.8
core
ext
con
064
002
1024
20
344.2
9oot
1769.7
230.2
81.0
20.9
11
11
0.0
00.0
01768.9
91768.9
91768.5
5165.4
31603.5
6884.5
05.1
11.9
log-s
licin
gbvsm
od
15
20
oot
oot
1779.8
220.1
81.0
10.4
41
184
51
118
0.1
32.8
71775.1
54042.0
44040.5
00.0
0138.0
76.0
71.5
2.1
log-s
licin
gbvudiv
17
20
1270.3
61373.3
4621.5
2751.8
22.2
10.7
10
33
23
21
0.0
25.9
4611.9
11571.4
61570.3
10.0
0187.3
915.7
12.1
2.2
log-s
licin
gbvurem
15
20
oot
1147.7
1415.1
9732.5
22.7
60.2
824
23
20
0.0
10.3
8413.9
4977.9
6977.7
70.0
0141.4
412.2
22.2
2.3
log-s
licin
gbvsm
od
14
20
1414.5
41350.5
9727.7
4622.8
51.8
60.3
39
171
46
107
0.1
22.5
1723.5
41912.0
61910.5
70.0
070.9
43.1
82.3
2.4
log-s
licin
gbvm
ul12
20
129.1
9962.3
1442.5
4519.7
72.1
70.1
22
22
0.0
00.0
2441.8
4836.4
0836.3
80.2
8441.0
5167.2
82.4
2.5
core
ext
con
048
002
1024
20
487.7
21716.6
41291.3
5425.2
91.3
30.7
11
11
0.0
00.0
01290.6
11290.6
11290.2
9124.3
01166.3
1645.3
14.6
2.6
core
ext
con
044
002
1024
20
195.8
21571.3
51263.0
8308.2
71.2
40.7
11
11
0.0
00.0
01262.3
71262.3
71262.0
8107.3
41155.0
4631.1
93.1
72.7
...test
v7
r7
vr5
c1
s3582
10
326.7
8392.5
6115.4
2277.1
43.4
04.1
21
37
16
01.2
047.9
557.1
4116.4
595.4
10.1
83.2
31.3
14.5
62.8
log-s
licin
gbvudiv
16
20
390.9
9521.1
2253.3
0267.8
22.0
60.2
824
22
20
0.0
10.5
1251.9
1738.1
1737.8
70.0
086.5
19.2
32.5
2.9
core
ext
con
036
002
1024
20
269.9
91156.1
9896.7
9259.4
01.2
90.5
11
11
0.0
00.0
0896.1
0896.1
0895.8
664.6
1831.4
9448.0
53.1
82.1
0...sm
tcom
p09
lfbv
form
ula
20
180.2
4207.7
168.5
8139.1
33.0
30.9
10
56
32
00.0
73.1
864.2
0242.2
5234.6
80.0
26.6
11.9
14.8
12.1
1lo
g-s
licin
gbvurem
14
20
757.1
6270.8
8141.2
3129.6
51.9
20.1
824
22
19
0.0
10.3
1140.0
9421.7
5421.5
80.0
055.8
45.4
12.6
2.1
2...sm
tcom
p09
nt
bv
form
ula
20
156.7
2175.2
882.1
693.1
22.1
31.8
14
80
56
00.1
47.9
271.6
4271.8
2258.4
70.0
520.6
41.3
13.4
02.1
3lo
g-s
licin
gbvsm
od
13
20
360.2
6438.2
6350.8
887.3
81.2
50.3
36
156
50
100
0.1
12.1
9346.4
6891.5
2890.2
70.0
029.1
51.5
82.7
2.1
4challenge
multip
lyO
verflow
20
1035.6
11002.1
1917.7
084.4
11.0
90.1
22
22
0.0
00.0
2917.0
01499.0
61499.0
30.5
6915.2
2299.8
12.8
2.1
5lo
g-s
licin
gbvudiv
15
20
162.7
0192.9
5116.9
775.9
81.6
50.2
823
21
18
0.0
10.4
9115.6
7339.0
4338.8
30.0
040.5
24.5
22.9
2.1
6brum
mayerbie
re
nlz
be256
20
117.5
1157.9
797.0
960.8
81.6
30.7
35
40
0.0
14.1
792.0
2126.4
3122.4
70.1
533.2
114.0
53.2
02.1
7lo
g-s
licin
gbvurem
13
20
147.9
7116.6
757.3
259.3
52.0
40.1
721
17
17
0.0
10.2
656.2
9180.4
0180.2
30.0
022.8
62.6
92.1
02.1
8brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3132.0
173.3
258.6
91.8
00.3
25
132
42
15
0.0
10.6
471.7
8220.1
9219.1
20.0
015.6
00.6
72.1
12.1
9lo
g-s
licin
gbvsm
od
12
20
138.5
8174.8
0166.8
27.9
81.0
50.3
31
150
52
79
0.0
81.9
9163.3
7456.7
0455.6
20.0
013.6
50.9
22.1
22.2
0...y
erbie
re3
isqrtin
validvc
10
181.6
615.8
99.5
06.3
91.6
70.1
416
16
00.0
00.0
78.7
029.1
728.2
50.2
32.2
80.9
42.1
32.2
1bm
c-b
vm
agic
10
443.8
6453.4
4452.9
90.4
51.0
00.1
01
10
0.0
00.0
00.4
60.4
60.3
90.4
60.4
60.4
63.2
2.2
2...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
710.9
110.6
30.2
81.0
30.5
01
10
0.0
00.0
09.3
79.3
70.7
19.3
79.3
79.3
73.2
42.2
3...m
ayerbie
re2
sm
ulo
v2bw384
20
122.0
615.6
415.3
70.2
71.0
20.4
01
10
0.0
00.0
014.3
314.3
38.7
914.3
314.3
314.3
33.9
2.2
4...m
ayerbie
re2
sm
ulo
v2bw448
20
150.6
343.6
943.5
00.1
91.0
00.5
01
10
0.0
00.0
042.3
242.3
234.5
642.3
242.3
242.3
23.1
12.2
5...m
ayerbie
re2
sm
ulo
v2bw512
20
198.3
836.6
836.5
30.1
51.0
00.6
01
10
0.0
00.0
035.1
135.1
122.7
635.1
135.1
135.1
13.1
32.2
6...a
yerbie
re2
sm
ulo
v4bw0512
10
128.7
82.5
72.4
20.1
51.0
60.1
01
10
0.0
00.0
01.5
81.5
80.1
71.5
81.5
81.5
83.1
02.2
7...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
33.9
63.8
10.1
51.0
40.2
01
10
0.0
00.0
02.8
92.8
90.2
72.8
92.8
92.8
93.1
42.2
8...a
yerbie
re2
sm
ulo
v4bw0768
10
563.4
25.8
05.6
60.1
41.0
20.4
01
10
0.0
00.0
04.6
94.6
90.4
04.6
94.6
94.6
93.1
52.2
9...a
yerbie
re2
sm
ulo
v4bw0896
10
1274.5
28.2
68.1
60.1
01.0
10.5
01
10
0.0
00.0
06.9
66.9
60.5
46.9
66.9
66.9
63.2
12.3
0brum
mayerbie
re
bitrev8192
20
101.9
7101.4
9101.4
20.0
71.0
00.0
01
10
0.0
00.0
00.0
40.0
40.0
00.0
40.0
40.0
43.1
3.1
bm
c-b
vgraycode
10
442.5
4449.0
3449.1
5-0
.12
1.0
00.1
01
10
0.0
00.0
00.0
40.0
40.0
10.0
40.0
40.0
43.3
3.2
...e
rbie
re4
unconstrain
ed02
10
oot
44.2
344.4
9-0
.26
0.9
94.2
01
10
0.0
00.0
039.8
539.8
57.7
739.8
539.8
539.8
53.4
3.3
RW
SExam
ple
18.txt
10
160.6
1193.3
5193.8
0-0
.45
1.0
00.1
01
10
0.0
00.0
0192.9
7192.9
7192.8
1192.9
7192.9
7192.9
73.5
3.4
...m
ayerbie
re2
um
ulo
v1bw224
20
108.4
290.1
694.6
1-4
.45
0.9
50.3
12
20
0.0
02.7
390.9
190.9
189.4
20.0
088.3
530.3
03.6
Contin
ued
on
next
page
90
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
3.5
...m
ayerbie
re2
um
ulo
v1bw256
20
143.9
4119.0
5134.0
7-1
5.0
20.8
90.4
12
20
0.0
03.1
3129.9
4129.9
5127.8
80.0
0101.9
443.3
23.8
3.6
log-s
licin
gbvsub
04000
20
545.8
4842.7
7878.9
0-3
6.1
30.9
60.1
11
11
0.0
00.0
0877.7
5877.7
5876.8
35.8
4871.9
1438.8
83.7
3.7
...test
v5
r15
vr1
c1
s8236
20
927.6
6285.1
7327.3
2-4
2.1
50.8
73.1
34
57
80
2.5
496.2
8206.8
6373.9
4329.0
60.2
513.3
63.3
14.7
03.8
...st
v5
r15
vr10
c1
s11127
10
346.0
31009.9
91054.6
3-4
4.6
40.9
63.7
82
95
90
5.4
7201.0
4814.5
91495.2
11434.2
10.0
245.2
96.4
24.1
93.9
...test
v5
r5
vr5
c1
s24018
10
182.8
1186.5
3321.8
4-1
35.3
10.5
83.4
69
159
29
01.6
776.3
8226.1
1576.0
0537.1
30.0
07.1
61.7
34.8
63.1
0...e
st
v7
r7
vr10
c1
s24535
10
237.8
8246.3
0397.0
0-1
50.7
00.6
23.6
62
116
14
03.1
0127.3
3241.3
8445.7
1392.6
40.0
06.8
91.9
04.7
63.1
1float
mult2.c
.50
10
109.3
940.0
2198.8
3-1
58.8
10.2
04.5
11
30
12
00.4
125.3
6167.8
2411.6
0373.7
80.0
034.2
14.2
44.1
06
3.1
2...b
iere3
isqrtaddin
validvc
10
185.6
016.0
0190.3
4-1
74.3
40.0
81.9
50
302
219
82
0.0
51.7
6187.5
8674.7
1660.5
40.0
05.8
20.4
73.1
23.1
3...sm
tcom
p09
std
bv
form
ula
20
108.8
6169.6
1345.9
0-1
76.2
90.4
93.1
30
169
96
00.2
09.5
7333.9
21296.6
41270.4
00.0
126.0
22.1
63.3
43.1
4lo
g-s
licin
gbvsrem
15
20
791.1
0927.6
21138.0
9-2
10.4
70.8
20.3
24
92
46
76
0.0
61.4
41135.4
42967.7
52966.8
90.0
0142.6
78.2
93.1
93.1
5lo
g-s
licin
gbvsrem
14
20
256.9
0269.5
8490.0
2-2
20.4
40.5
50.3
22
92
46
68
0.0
61.3
4487.4
81351.2
91350.5
30.0
048.9
84.0
33.1
63.1
6...m
tcom
p09
full
bv
form
ula
20
172.4
6167.3
1411.6
3-2
44.3
20.4
13.5
30
186
93
00.1
79.6
0399.7
41494.8
31473.4
50.0
027.2
32.6
63.3
53.1
7pspace
shift1add.2
2961
20
110.7
1110.8
6365.9
5-2
55.0
90.3
00.3
11
11
0.0
00.0
0365.1
6365.1
6364.6
712.6
4352.5
2182.5
83.2
23.1
8pspace
shift1add.2
3958
20
114.0
8115.5
0402.4
3-2
86.9
30.2
90.4
11
11
0.0
00.0
0401.5
8401.5
8401.0
613.8
5387.7
3200.7
93.2
33.1
9pspace
shift1add.2
4955
20
128.4
7121.7
9436.5
9-3
14.8
00.2
80.4
11
11
0.0
00.0
0435.7
7435.7
7435.2
314.8
4420.9
4217.8
93.2
53.2
0pspace
shift1add.2
5952
20
143.2
8129.3
8467.5
1-3
38.1
30.2
80.4
11
11
0.0
00.0
0466.7
5466.7
5466.1
915.5
0451.2
6233.3
83.2
63.2
1pspace
shift1add.2
6949
20
152.3
3135.5
8495.2
9-3
59.7
10.2
70.4
11
11
0.0
00.0
0494.5
4494.5
4493.9
616.3
7478.1
6247.2
73.2
73.2
2brum
mayerbie
re3
isqrt
20
145.8
1148.1
1527.6
7-3
79.5
60.2
81.4
52
324
164
62
0.0
51.8
4524.8
51809.4
31792.9
80.0
335.0
61.4
93.3
33.2
3pspace
shift1add.2
7946
20
167.8
1143.3
4530.4
0-3
87.0
60.2
70.6
11
11
0.0
00.0
0529.6
3529.6
3529.0
417.5
9512.0
5264.8
23.2
83.2
4...test
v5
r5
vr1
c1
s14623
10
143.2
0147.5
7546.7
6-3
99.1
90.2
73.2
55
163
24
01.6
470.9
9458.8
91350.7
11308.1
90.0
084.9
43.9
54.9
23.2
5pspace
shift1add.2
8943
20
186.8
4148.3
3563.4
2-4
15.0
90.2
60.6
11
11
0.0
00.0
0562.3
8562.3
8561.7
718.0
3544.3
5281.1
93.2
93.2
6pspace
shift1add.2
9940
20
195.3
8157.2
3595.7
5-4
38.5
20.2
60.6
11
11
0.0
00.0
0595.0
1595.0
1594.3
819.1
8575.8
3297.5
03.3
13.2
7tacas07
Y86
std
20
129.6
5193.9
7659.1
9-4
65.2
20.2
93.7
22
108
33
00.2
730.6
0625.8
12233.4
62187.1
50.0
166.1
47.2
04.8
43.2
8brum
mayerbie
re3
isqrtadd
20
144.7
3148.8
8639.4
6-4
90.5
80.2
31.6
41
289
186
11
0.0
41.6
4636.9
22364.6
22349.8
50.0
137.2
42.2
83.3
23.2
9lo
g-s
licin
gbvsdiv
16
20
182.3
5223.3
9735.0
7-5
11.6
80.3
00.4
23
94
51
73
0.0
71.5
7731.0
61932.9
01932.0
20.0
066.8
15.3
43.3
03.3
0...test
v7
r12
vr5
c1
s8938
10
615.1
6403.9
41013.2
3-6
09.2
90.4
03.7
53
94
10
05.3
9191.1
8778.2
31125.5
01056.2
50.2
6110.8
15.7
14.5
33.3
1...test
v5
r5
vr10
c1
s7194
10
141.8
4299.2
6982.3
9-6
83.1
30.3
04.1
70
164
30
01.8
681.9
9881.9
43005.0
82960.4
80.0
097.6
87.0
94.6
83.3
2lfsr
lfsr
004
111
112
20
178.0
5181.6
1873.1
9-6
91.5
80.2
13.6
18
97
52
00.3
031.5
3834.7
63049.4
83031.6
40.0
6114.5
710.5
53.3
73.3
3brum
mayerbie
re3
maxand256
20
102.0
5252.0
31015.4
7-7
63.4
40.2
53.8
22
79
31
00.3
524.6
6986.6
22318.2
62263.9
20.1
2199.1
58.9
54.1
33.3
4...e
st
v5
r15
vr1
c1
s32559
20
163.7
5217.6
01055.8
7-8
38.2
70.2
14.9
47
80
10
03.8
0134.1
2890.5
01904.8
51830.5
50.2
4150.0
77.5
94.7
93.3
5lfsr
lfsr
008
031
064
20
oot
42.9
7915.9
7-8
73.0
00.0
53.4
36
202
133
00.7
019.8
2890.3
63443.0
23413.9
40.0
733.4
14.3
93.4
23.3
6lo
g-s
licin
gbvsdiv
17
20
502.6
4442.9
21495.5
8-1
052.6
60.3
01.2
24
92
53
80
0.0
811.4
41481.2
04162.0
34159.8
70.0
1128.2
611.1
33.3
63.3
7float
newton.5
.2.i
20
249.5
3196.3
71671.2
4-1
474.8
70.1
24.1
34
136
27
01.3
554.6
71604.1
23147.7
33088.5
30.0
1208.4
68.1
14.8
33.3
8lfsr
lfsr
008
063
096
20
oot
144.8
31780.4
3-1
635.6
00.0
83.6
34
145
82
00.9
658.2
21709.7
36504.9
96471.5
60.1
383.2
310.7
34.9
4
4.1
pspace
power2sum
.6097
20
388.1
21779.4
3oot
-20.5
70.9
92.2
11
10
0.0
00.0
01621.0
51621.0
51620.6
31621.0
51621.0
51621.0
54.1
4.2
log-s
licin
gbvslt
15402
20
1048.3
41765.7
2oot
-34.2
80.9
82.8
836
36
00.0
122.8
31353.6
66172.8
56168.7
49.5
5258.4
669.3
64.2
4.3
...e
st
v7
r12
vr1
c1
s10576
20
oot
1756.9
8oot
-43.0
20.9
84.8
0.0
00.0
00.0
00.0
00.0
04.3
4.4
pspace
power2sum
.5699
20
347.9
21748.1
2oot
-51.8
80.9
72.0
11
10
0.0
00.0
01333.3
41333.3
41332.9
61333.3
41333.3
41333.3
44.4
4.5
log-s
licin
gbvsub
05994
20
414.3
11726.2
5oot
-73.7
50.9
60.1
11
10
0.0
00.0
06.5
96.5
94.6
06.5
96.5
96.5
94.5
4.6
...test
v5
r15
vr5
c1
s8246
20
1106.6
41711.4
6oot
-88.5
40.9
54.5
39
83
10
03.4
1132.6
3143.0
9325.0
0262.5
70.2
34.3
21.9
04.7
4.7
log-s
licin
gbvult
14973
20
1019.5
01697.3
0oot
-102.7
00.9
43.1
10
38
38
00.0
225.4
41649.8
65963.8
45959.4
02.0
8261.8
753.7
34.8
4.8
log-s
licin
gbvsdiv
18
20
1485.5
21599.0
3oot
-200.9
70.8
91.2
16
76
52
35
0.0
59.1
91710.0
14589.4
74587.7
90.0
0300.8
417.8
64.9
4.9
...test
v7
r7
vr5
c1
s19694
10
1370.1
11539.9
4oot
-260.0
60.8
64.1
0.0
00.0
00.0
00.0
00.0
04.1
04.1
0...e
st
v7
r7
vr10
c1
s10625
10
852.7
61379.9
4oot
-420.0
60.7
73.9
0.0
00.0
00.0
00.0
00.0
04.1
24.1
1lo
g-s
licin
gbvslt
12430
20
oot
1349.7
6oot
-450.2
40.7
52.9
739
39
00.0
121.7
51548.1
76500.9
86497.3
213.9
3235.2
292.8
74.1
44.1
2...e
st
v5
r15
vr5
c1
s26657
20
oot
1282.8
4oot
-517.1
60.7
14.5
91
129
10
07.0
0250.4
9725.0
21473.2
91382.1
30.0
114.5
14.5
94.1
54.1
3lo
g-s
licin
gbvslt
10944
20
505.6
81114.2
5oot
-685.7
50.6
22.9
841
41
00.0
221.6
31538.5
16500.0
86496.3
70.7
4168.7
573.8
64.1
64.1
4...test
v7
r7
vr1
c1
s4574
10
609.3
31077.2
4oot
-722.7
60.6
04.1
57
104
17
02.7
8115.1
2419.9
6980.1
2919.6
70.1
688.4
42.8
84.1
74.1
5...st
v5
r10
vr10
c1
s21502
20
697.8
21056.4
7oot
-743.5
30.5
94.2
82
151
16
03.9
7180.9
7873.4
82399.9
42326.4
60.0
0139.6
26.0
04.1
84.1
6VS3
VS3-b
enchm
ark-A
910
268.4
51009.1
3oot
-790.8
70.5
63.3
45
272
171
00.4
321.6
21774.3
16839.2
86832.5
00.0
032.6
86.1
04.2
0
Contin
ued
on
next
page
91
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.1
7...e
st
v5
r10
vr1
c1
s32538
20
208.1
6996.7
5oot
-803.2
50.5
53.9
56
126
16
03.2
1128.8
31471.1
33913.7
93842.6
10.0
0114.4
112.1
24.2
14.1
8...e
st
v5
r10
vr5
c1
s13679
10
899.6
4937.6
8oot
-862.3
20.5
24.4
88
214
15
05.5
9218.8
7956.2
63164.7
83053.2
80.0
019.4
36.1
34.2
24.1
9VS3
VS3-b
enchm
ark-A
710
985.7
9934.5
9oot
-865.4
10.5
23.3
62
390
148
00.6
728.9
71730.1
86717.9
26709.4
90.0
022.8
14.4
94.2
34.2
0lo
g-s
licin
gbvslt
7972
20
305.2
5931.6
7oot
-868.3
30.5
23.1
12
67
67
00.0
218.7
01729.7
96696.4
46692.1
13.0
4154.2
534.3
44.2
44.2
1...servers
sla
pd
avc149923
10
325.6
4886.7
7oot
-913.2
30.4
93.3
30
67
67
00.0
614.2
61775.1
16952.0
26949.1
30.2
850.8
713.1
94.2
54.2
2...e
st
v7
r7
vr10
c1
s32506
10
1754.9
5864.5
0oot
-935.5
00.4
84.0
0.0
00.0
00.0
00.0
00.0
04.2
64.2
3lo
g-s
licin
gbvslt
9458
20
539.1
6833.9
7oot
-966.0
30.4
62.9
844
44
00.0
222.6
81461.7
56272.5
66268.7
115.0
5137.5
967.4
54.2
74.2
4...test
v7
r7
vr1
c1
s24449
20
679.7
0826.2
8oot
-973.7
20.4
64.1
0.0
00.0
00.0
00.0
00.0
04.2
84.2
5...servers
sla
pd
avc149922
10
325.6
8788.4
7oot
-1011.5
30.4
43.3
29
67
67
00.0
614.4
71711.1
96905.6
26902.7
60.3
050.8
213.2
04.2
94.2
6...e
st
v5
r10
vr1
c1
s19145
10
316.4
9780.8
2oot
-1019.1
80.4
33.8
67
155
16
03.7
6161.7
31104.3
23075.5
73002.1
80.0
1125.8
98.3
84.3
04.2
7...y
erbie
re
countbitssrl0
32
20
387.0
6713.5
6oot
-1086.4
40.4
00.5
27
148
18
23
0.0
52.0
51676.3
63311.1
33299.9
00.0
0801.5
49.8
54.3
14.2
8lo
g-s
licin
gbvult
8991
20
542.6
1709.9
0oot
-1090.1
00.3
92.9
846
44
00.0
223.2
71714.5
76010.1
16006.3
70.1
9171.0
473.2
94.3
24.2
9lo
g-s
licin
gbvult
7994
20
230.1
0691.4
3oot
-1108.5
70.3
83.2
10
64
64
00.0
218.2
01383.4
36682.0
06678.0
92.4
3154.8
742.8
34.3
34.3
0lo
g-s
licin
gbvslt
6486
20
191.6
5687.4
2oot
-1112.5
80.3
83.1
16
95
69
00.0
323.5
81682.1
86433.3
56427.7
00.0
4130.0
225.0
34.3
44.3
1lo
g-s
licin
gbvult
6997
20
201.1
9669.2
9oot
-1130.7
10.3
73.7
10
53
53
00.0
214.4
31046.4
06668.1
96664.5
88.4
9108.6
641.6
84.3
54.3
2...e
st
v5
r10
vr5
c1
s13195
10
337.5
5662.6
0oot
-1137.4
00.3
74.0
0.0
00.0
00.0
00.0
00.0
04.3
64.3
3...1
alt
true.B
V.c
.cil.c
.17
20
oot
621.7
5oot
-1178.2
50.3
53.9
22
53
26
00.1
710.6
51646.4
16136.2
16104.0
70.0
2123.4
519.7
94.3
74.3
4lo
g-s
licin
gbvult
6000
20
406.3
8608.3
3oot
-1191.6
70.3
43.0
16
98
71
00.0
422.9
21763.6
26632.0
16627.4
30.4
0123.5
926.0
14.3
84.3
5lo
g-s
licin
gbvslt
8715
20
342.5
6598.5
0oot
-1201.5
00.3
32.8
844
40
00.0
220.1
61777.5
06604.2
66600.7
714.6
8156.8
983.6
04.3
94.3
6...test
v7
r7
vr1
c1
s22845
10
316.6
3593.4
8oot
-1206.5
20.3
33.8
57
195
18
04.3
8183.3
6266.7
0584.2
3503.3
30.0
247.3
51.3
44.4
04.3
7lfsr
lfsr
008
127
112
20
176.0
0591.6
3oot
-1208.3
70.3
33.6
18
37
32
00.3
338.0
11656.8
96131.3
86109.5
32.0
3286.5
225.1
34.4
14.3
8lo
g-s
licin
gbvslt
7229
20
217.0
1590.3
8oot
-1209.6
20.3
33.0
16
94
70
00.0
324.9
41564.8
06495.0
66489.3
10.0
0149.9
724.9
84.4
24.3
9...e
st
v5
r15
vr1
c1
s26845
20
619.0
9589.8
4oot
-1210.1
60.3
34.6
39
95
10
03.5
4132.6
1205.5
2590.1
4514.1
00.2
75.6
32.6
04.4
34.4
0lfsr
lfsr
008
143
112
20
205.3
9587.5
9oot
-1212.4
10.3
34.1
19
28
28
00.4
533.5
91628.6
46390.5
96345.6
81.9
595.5
922.8
24.4
44.4
1...e
st
v5
r10
vr1
c1
s13516
10
231.3
6574.5
3oot
-1225.4
70.3
23.7
0.0
00.0
00.0
00.0
00.0
04.4
54.4
2...tcom
p09
stall
bv
form
ula
20
609.5
3493.1
7oot
-1306.8
30.2
74.3
12
37
31
00.3
517.1
91405.7
26359.9
56321.9
51.0
3250.6
339.7
54.4
64.4
3float
mul03
30
420
514.8
0488.9
7oot
-1311.0
30.2
73.3
0.0
00.0
00.0
00.0
00.0
04.4
74.4
4float
newton.5
.3.i
20
415.2
8486.8
0oot
-1313.2
00.2
74.1
15
59
23
00.9
336.3
4260.5
8784.7
4744.5
50.0
146.4
74.3
44.4
84.4
5...test
v5
r10
vr5
c1
s8690
20
353.9
3485.9
1oot
-1314.0
90.2
74.4
58
133
17
03.4
6137.3
4587.2
81576.5
61495.3
50.0
271.2
73.8
64.4
94.4
6float
newton.2
.3.i
20
289.3
8445.9
2oot
-1354.0
80.2
54.2
19
89
23
01.3
851.9
1230.2
4627.6
4580.5
40.0
030.0
82.5
64.5
04.4
7float
newton.6
.3.i
10
564.0
4437.0
6oot
-1362.9
40.2
44.3
20
106
19
01.6
661.5
4586.2
91751.0
61694.5
90.0
066.4
66.7
34.5
14.4
8lfsr
lfsr
008
143
128
20
261.7
9428.5
3oot
-1371.4
70.2
44.0
16
31
31
00.6
352.2
61554.0
46170.1
06122.5
33.2
8251.5
427.5
54.5
24.4
9VS3
VS3-b
enchm
ark-A
610
111.9
3397.0
9oot
-1402.9
10.2
22.1
82
612
131
00.9
940.4
71723.2
76617.3
36606.8
40.0
018.9
43.6
04.5
44.5
0...test
v7
r7
vr5
c1
s14675
10
364.5
5393.9
7oot
-1406.0
30.2
24.1
0.0
00.0
00.0
00.0
00.0
04.5
54.5
1lfsr
lfsr
008
015
128
20
277.3
6380.8
7oot
-1419.1
30.2
14.1
59
184
184
01.7
621.9
51766.8
46900.9
36808.7
60.1
444.0
34.7
44.5
74.5
2lo
g-s
licin
gbvslt
5743
20
444.5
8367.8
2oot
-1432.1
80.2
03.0
20
125
73
00.0
428.7
7911.8
36155.2
56147.7
70.0
2293.3
817.9
53.3
94.5
3lfsr
lfsr
008
095
128
20
148.8
9366.3
4oot
-1433.6
60.2
03.6
20
49
37
00.5
142.3
31418.3
66305.3
66277.9
21.3
4284.1
322.1
24.5
84.5
4float
newton.1
.3.i
20
385.3
4360.0
3oot
-1439.9
70.2
04.3
24
124
22
01.9
771.9
2806.0
22455.9
52387.5
50.0
075.7
17.5
34.5
94.5
5brum
mayerbie
re3
min
and256
20
289.6
6328.3
0oot
-1471.7
00.1
81.1
56
33
0.0
34.6
21661.9
01865.8
31860.8
90.7
6833.2
8133.2
74.6
14.5
6lfsr
lfsr
008
111
112
20
148.6
0324.7
8oot
-1475.2
20.1
83.6
18
44
37
00.4
139.2
91568.7
36017.5
25994.7
53.0
9190.6
524.3
64.6
24.5
7float
newton.4
.3.i
20
334.8
4321.3
7oot
-1478.6
30.1
83.9
17
88
22
01.3
250.4
5328.2
4877.6
7832.5
30.0
048.1
53.8
74.6
34.5
8brum
mayerbie
re3
min
or256
20
116.1
3321.3
3oot
-1478.6
70.1
85.2
14
30
18
00.1
514.9
7743.1
55040.3
94982.5
60.1
2473.1
527.1
04.1
14.5
9lo
g-s
licin
gbvslt
5000
20
296.4
0315.4
5oot
-1484.5
50.1
82.5
20
140
59
00.0
528.9
11452.9
66429.9
46422.7
70.0
2297.7
420.0
93.3
84.6
0float
newton.3
.3.i
20
355.4
9314.2
5oot
-1485.7
50.1
74.5
0.0
00.0
00.0
00.0
00.0
04.6
44.6
1lfsr
lfsr
008
159
080
20
194.3
3310.3
5oot
-1489.6
50.1
73.5
17
50
36
00.3
047.0
41657.1
96106.3
36088.1
81.0
6116.9
326.5
54.6
54.6
2...y
erbie
re3
min
andm
axor064
20
207.7
9308.5
2oot
-1491.4
80.1
74.0
62
111
14
00.2
431.1
01763.8
22525.1
82479.2
40.1
1111.3
39.0
24.6
64.6
3lfsr
lfsr
008
159
112
20
275.9
4300.0
3oot
-1499.9
70.1
73.5
917
17
00.1
425.6
71159.8
16265.1
86252.0
67.6
7223.5
856.9
64.6
74.6
4lfsr
lfsr
004
159
128
20
112.1
0299.7
4oot
-1500.2
60.1
73.8
15
54
33
00.2
241.7
41451.3
25980.8
25961.9
90.1
1297.7
527.8
23.4
14.6
5lfsr
lfsr
008
143
096
20
173.9
4291.3
7oot
-1508.6
30.1
63.5
23
47
39
00.5
836.5
61755.2
26673.0
36631.5
41.4
4247.6
819.2
34.6
94.6
6lfsr
lfsr
008
015
112
20
388.4
4270.5
7oot
-1529.4
30.1
53.9
60
190
190
01.5
018.6
41560.6
66969.7
66885.2
10.1
335.6
74.4
44.7
14.6
7...test
v5
r5
vr5
c1
s2800
10
228.1
4268.6
0oot
-1531.4
00.1
54.2
69
191
31
02.1
697.0
6393.0
31288.9
41233.0
10.0
023.3
52.2
64.7
2
Contin
ued
on
next
page
92
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.6
8lfsr
lfsr
004
143
128
20
116.2
1259.8
5oot
-1540.1
50.1
43.9
24
75
47
00.6
957.8
01686.3
26477.8
96438.6
00.3
6174.0
616.9
64.7
34.6
9lfsr
lfsr
008
079
128
20
107.7
6256.8
8oot
-1543.1
20.1
43.6
23
79
48
00.7
560.6
91705.5
26600.4
36568.8
20.2
4136.8
918.3
34.7
44.7
0lfsr
lfsr
008
095
096
20
105.5
2255.9
1oot
-1544.0
90.1
43.7
20
67
45
00.4
744.1
21124.2
96661.5
06637.6
70.0
9280.3
121.5
64.7
54.7
1lfsr
lfsr
008
111
096
20
133.3
2238.7
2oot
-1561.2
80.1
33.6
19
59
40
00.4
351.8
11371.2
56375.9
76352.7
21.4
7192.7
023.1
94.7
74.7
2brum
mayerbie
re3
maxor064
20
209.1
4224.2
7oot
-1575.7
30.1
21.9
68
69
343
0.1
315.7
91553.2
61701.6
61682.4
80.1
1367.1
09.4
54.7
84.7
3...sm
tcom
p09
sw
bv
form
ula
20
202.4
2210.0
8oot
-1589.9
20.1
24.7
14
34
34
00.2
010.0
11158.3
36215.7
76181.8
60.3
4287.1
730.3
24.8
04.7
4lfsr
lfsr
008
127
128
20
212.8
0198.4
4oot
-1601.5
60.1
13.5
17
33
30
00.3
542.0
01519.8
85760.9
65737.6
83.4
2212.0
225.3
84.8
24.7
5lfsr
lfsr
008
143
064
20
103.4
9192.9
4oot
-1607.0
60.1
14.0
22
60
60
00.4
747.5
21735.5
46733.7
96697.7
50.4
2217.1
718.8
64.8
54.7
6tacas07
Y86
btnft
20
135.7
7186.2
1oot
-1613.7
90.1
05.0
11
37
24
00.1
519.4
31324.7
16496.2
16452.0
40.1
2173.4
445.4
34.6
04.7
7float
newton.2
.2.i
20
143.1
8178.6
2oot
-1621.3
80.1
04.0
0.0
00.0
00.0
00.0
00.0
04.8
74.7
8lfsr
lfsr
008
063
128
20
oot
178.5
0oot
-1621.5
00.1
03.7
34
136
65
01.3
770.9
71716.1
06544.4
86500.4
30.2
9244.8
411.5
04.8
84.7
9float
newton.1
.2.i
20
139.9
8173.6
3oot
-1626.3
70.1
04.2
0.0
00.0
00.0
00.0
00.0
04.8
94.8
0float
newton.4
.2.i
20
143.2
5155.9
4oot
-1644.0
60.0
94.0
30
150
26
01.3
460.1
5547.7
31765.0
21703.4
60.0
055.3
24.6
24.9
04.8
1float
newton.3
.2.i
20
186.5
6147.7
7oot
-1652.2
30.0
84.4
0.0
00.0
00.0
00.0
00.0
04.9
14.8
2lfsr
lfsr
008
159
096
20
275.8
6145.8
5oot
-1654.1
50.0
83.2
16
46
39
00.2
753.9
01710.5
16573.7
06558.6
51.0
9134.0
931.1
64.9
34.8
3lfsr
lfsr
008
143
080
20
112.1
3144.7
5oot
-1655.2
50.0
83.9
21
44
44
00.4
538.9
81752.2
06775.1
56738.5
10.7
5306.0
121.3
14.9
54.8
4lfsr
lfsr
008
095
112
20
129.0
8142.4
7oot
-1657.5
30.0
83.6
19
50
39
00.4
741.6
31484.5
36354.4
56330.7
71.5
8245.5
923.4
54.9
64.8
5lfsr
lfsr
008
127
096
20
113.0
4135.2
3oot
-1664.7
70.0
83.6
20
49
37
00.3
541.3
61582.9
06137.5
86115.4
52.9
2157.0
621.7
64.9
74.8
6brum
mayerbie
re3
maxxor032
20
143.9
5127.0
4oot
-1672.9
60.0
71.9
74
543
179
00.2
57.1
31784.5
66343.1
76324.4
90.0
150.0
44.1
94.9
84.8
7lfsr
lfsr
008
015
096
20
272.7
5123.5
9oot
-1676.4
10.0
73.9
60
242
242
01.5
121.2
11442.8
96973.6
26892.3
60.1
134.0
73.8
44.9
94.8
8brum
mayerbie
re
countbits064
20
129.3
4121.6
1oot
-1678.3
90.0
73.3
40
217
206
00.1
04.1
9618.6
66995.7
26984.3
00.0
149.0
06.3
14.1
00
4.8
9lfsr
lfsr
008
111
128
20
160.6
8112.3
2oot
-1687.6
80.0
63.2
19
54
37
00.5
050.9
91674.8
46446.7
06424.4
52.9
1133.9
524.0
54.1
01
4.9
0lfsr
lfsr
008
047
096
20
160.4
099.3
6oot
-1700.6
40.0
63.7
45
235
77
01.7
069.4
41704.8
86581.8
96531.3
70.1
1136.1
87.7
34.1
02
4.9
1brum
mayerbie
re3
min
xor064
20
111.8
493.3
5oot
-1706.6
50.0
51.6
80
537
78
00.4
59.4
31770.5
25906.3
25857.8
70.0
146.6
44.7
04.1
03
4.9
2lfsr
lfsr
008
159
064
20
149.5
283.4
8oot
-1716.5
20.0
53.3
14
51
36
00.1
738.2
61582.5
76494.8
26484.7
73.7
7138.9
833.6
54.1
04
4.9
3lfsr
lfsr
008
159
048
20
100.5
644.3
2oot
-1755.6
80.0
23.5
15
61
41
00.1
328.0
41610.1
76647.9
56638.3
52.2
4113.6
230.0
84.1
05
5.1
...test
v7
r12
vr1
c1
s703
0oot
oot
sf
1445.7
85.0
84.0
13
20
90
1.2
746.5
550.8
7106.9
987.1
10.2
66.4
52.3
35.2
35.2
...e
st
v5
r10
vr10
c1
s7608
20
1199.1
4oot
sf
171.1
71.1
14.3
75
165
14
04.8
1192.5
6351.1
9899.9
2817.5
40.0
19.7
52.2
75.2
05.3
...e
rbie
re4
unconstrain
ed01
0oom
oom
oom
1.1
51.0
27.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.3
5.4
...e
rbie
re4
unconstrain
ed07
0oom
oom
oom
0.3
11.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
5.5
...e
rbie
re4
unconstrain
ed08
0oom
oom
oom
0.1
41.0
07.1
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
02
5.6
...e
rbie
re4
unconstrain
ed05
0oom
oom
oom
0.0
91.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
04
5.7
core
ext
con
060
001
1024
20
110.6
0oot
oot
0.0
01.0
00.1
11
10
0.0
00.0
0141.1
9141.1
9140.8
0141.1
9141.1
9141.1
95.7
5.8
core
ext
con
060
256
1024
20
101.7
3oot
oot
0.0
01.0
03.6
617
17
021.8
516.4
01373.7
05130.9
54990.9
116.4
0273.8
795.0
25.9
5.9
core
ext
con
064
001
1024
20
128.2
5oot
oot
0.0
01.0
00.2
11
10
0.0
00.0
0151.5
4151.5
4151.1
3151.5
4151.5
4151.5
45.1
05.1
0fft
Sz1024
34824
0oot
oot
oot
0.0
01.0
03.3
40
216
216
00.8
08.7
4483.3
17055.6
47022.5
70.2
738.5
76.3
05.1
25.1
1fft
Sz512
15128
00
oot
oot
oot
0.0
01.0
03.2
70
413
413
00.6
24.3
9471.8
27114.8
07078.3
10.1
517.2
62.5
15.1
35.1
2fft
Sz512
15128
10
oot
oot
oot
0.0
01.0
03.2
70
411
411
00.6
24.2
0566.5
17110.7
57074.7
40.1
617.2
62.5
35.1
45.1
3fft
Sz512
15128
20
oot
oot
oot
0.0
01.0
03.2
70
428
428
00.5
44.1
5560.2
37133.5
67102.5
90.1
616.5
82.5
15.1
55.1
4fft
Sz512
15128
40
oot
oot
oot
0.0
01.0
03.1
70
416
416
00.6
24.3
0477.4
67125.1
27088.6
80.1
417.5
32.5
25.1
65.1
5fft
Sz512
15128
50
oot
oot
oot
0.0
01.0
03.2
70
413
413
00.6
34.3
7577.7
37116.6
07080.4
20.1
617.5
52.5
35.1
75.1
6fft
Sz512
15128
60
oot
oot
oot
0.0
01.0
03.2
70
434
434
00.5
54.1
8454.0
07112.6
67081.8
00.1
516.6
42.4
85.1
85.1
7...st
v5
r10
vr10
c1
s15708
0oot
oot
oot
0.0
01.0
04.3
0.0
00.0
00.0
00.0
00.0
05.1
95.1
8...e
st
v5
r15
vr5
c1
s23844
0oot
oot
oot
0.0
01.0
04.2
0.0
00.0
00.0
00.0
00.0
05.2
15.1
9...st
v7
r12
vr10
c1
s15994
10
1367.7
1oot
oot
0.0
01.0
04.4
0.0
00.0
00.0
00.0
00.0
05.2
25.2
0...e
st
v7
r12
vr5
c1
s14336
10
933.1
7oot
oot
0.0
01.0
03.8
0.0
00.0
00.0
00.0
00.0
05.2
45.2
1...e
st
v7
r12
vr5
c1
s29826
10
896.5
0oot
oot
0.0
01.0
04.4
0.0
00.0
00.0
00.0
00.0
05.2
55.2
2...e
st
v7
r17
vr1
c1
s23882
0oot
oot
oot
0.0
01.0
04.0
33
59
60
4.6
6151.1
8164.2
4341.4
9275.3
50.4
18.0
32.8
75.2
65.2
3...e
st
v7
r17
vr1
c1
s30331
0oot
oot
oot
0.0
01.0
04.3
0.0
00.0
00.0
00.0
00.0
05.2
75.2
4...fyIn
terle
avedM
odM
ult-1
28
0oot
oot
oot
0.0
01.0
04.3
12
88
00.0
228.4
41677.1
15683.7
25677.6
01.4
5302.3
778.9
45.2
8
Contin
ued
on
next
page
93
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.2
5...ffy
Interle
avedM
odM
ult-3
20
oot
oot
oot
0.0
01.0
03.3
29
116
116
00.0
511.1
61556.4
16475.3
06471.6
40.2
7110.5
911.2
45.2
95.2
6lfsr
lfsr
008
159
128
20
285.7
4oot
oot
0.0
01.0
03.6
11
20
20
00.1
935.0
91523.0
06241.7
26221.5
74.3
9228.7
944.9
05.3
05.2
7lo
g-s
licin
gbvadd
18209
20
101.2
6oot
oot
0.0
01.0
01.2
11
10
0.0
00.0
068.0
568.0
562.3
968.0
568.0
568.0
55.3
15.2
8lo
g-s
licin
gbvadd
19096
20
109.7
8oot
oot
0.0
01.0
01.2
11
10
0.0
00.0
076.1
376.1
370.6
076.1
376.1
376.1
35.3
25.2
9lo
g-s
licin
gbvadd
19983
20
121.5
9oot
oot
0.0
01.0
01.2
11
10
0.0
00.0
082.3
282.3
275.3
782.3
282.3
282.3
25.3
35.3
0lo
g-s
licin
gbvadd
20870
20
132.0
0oot
oot
0.0
01.0
01.2
11
10
0.0
00.0
089.8
189.8
182.0
289.8
189.8
189.8
15.3
45.3
1lo
g-s
licin
gbvadd
21757
20
165.0
7oot
oot
0.0
01.0
01.4
11
10
0.0
00.0
098.6
098.6
088.9
098.6
098.6
098.6
05.3
55.3
2lo
g-s
licin
gbvadd
22644
20
156.0
3oot
oot
0.0
01.0
01.4
11
10
0.0
00.0
0102.2
9102.2
994.2
2102.2
9102.2
9102.2
95.3
65.3
3lo
g-s
licin
gbvadd
23531
20
167.4
5oot
oot
0.0
01.0
01.4
11
10
0.0
00.0
0110.7
6110.7
6100.3
0110.7
6110.7
6110.7
65.3
75.3
4lo
g-s
licin
gbvadd
24418
20
198.0
9oot
oot
0.0
01.0
01.4
11
10
0.0
00.0
0118.5
3118.5
3106.4
7118.5
3118.5
3118.5
35.3
85.3
5lo
g-s
licin
gbvadd
25305
20
186.3
4oot
oot
0.0
01.0
01.4
11
10
0.0
00.0
0125.4
8125.4
8111.8
6125.4
8125.4
8125.4
85.3
95.3
6lo
g-s
licin
gbvadd
26192
20
202.5
2oot
oot
0.0
01.0
01.6
11
10
0.0
00.0
0131.6
8131.6
8117.3
3131.6
8131.6
8131.6
85.4
05.3
7lo
g-s
licin
gbvadd
27079
20
216.4
1oot
oot
0.0
01.0
01.6
11
10
0.0
00.0
0139.5
2139.5
2121.9
7139.5
2139.5
2139.5
25.4
15.3
8lo
g-s
licin
gbvadd
27966
20
232.0
1oot
oot
0.0
01.0
01.7
11
10
0.0
00.0
0147.6
1147.6
1127.8
1147.6
1147.6
1147.6
15.4
25.3
9lo
g-s
licin
gbvadd
28853
20
245.3
3oot
oot
0.0
01.0
01.7
11
10
0.0
00.0
0154.3
3154.3
3132.0
1154.3
3154.3
3154.3
35.4
35.4
0lo
g-s
licin
gbvadd
29740
20
264.9
3oot
oot
0.0
01.0
01.7
11
10
0.0
00.0
0162.2
3162.2
3136.9
1162.2
3162.2
3162.2
35.4
45.4
1lo
g-s
licin
gbvm
ul13
20
428.6
3oot
oot
0.0
01.0
00.1
22
21
0.0
00.0
20.9
01744.9
21744.9
00.2
71743.7
5436.2
35.4
55.4
2lo
g-s
licin
gbvm
ul14
20
1477.5
0oot
oot
0.0
01.0
00.2
22
20
0.0
00.0
20.8
91.1
81.1
60.2
90.5
60.3
95.4
65.4
3lo
g-s
licin
gbvm
ul15
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00.0
30.9
31.2
51.2
30.3
20.5
80.4
25.4
75.4
4lo
g-s
licin
gbvm
ul16
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00.0
20.8
51.1
51.1
30.3
00.5
40.3
85.4
85.4
5lo
g-s
licin
gbvm
ul17
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00.2
20.9
21.2
31.1
20.3
10.5
70.4
15.4
95.4
6lo
g-s
licin
gbvm
ul18
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00.2
30.9
41.2
11.1
00.2
70.5
70.4
05.5
05.4
7lo
g-s
licin
gbvsdiv
19
0oot
oot
oot
0.0
01.0
01.3
13
66
57
21
0.0
48.6
01383.7
33818.2
13816.7
80.0
1718.0
617.7
65.5
15.4
8lo
g-s
licin
gbvsdiv
20
0oot
oot
oot
0.0
01.0
01.4
11
61
57
10
0.0
38.0
6830.7
83206.8
73205.4
90.0
1901.5
317.5
25.5
25.4
9lo
g-s
licin
gbvsdiv
21
0oot
oot
oot
0.0
01.0
01.4
10
59
54
00.0
37.6
110.7
248.6
447.2
50.0
01.0
00.3
25.5
35.5
0lo
g-s
licin
gbvsdiv
22
0oot
oot
oot
0.0
01.0
01.5
10
60
55
10.0
38.0
011.2
3125.3
1123.8
40.0
070.4
10.7
75.5
45.5
1lo
g-s
licin
gbvsdiv
23
0oot
oot
oot
0.0
01.0
01.5
10
58
56
30.0
37.9
611.8
3473.8
6472.4
40.0
0382.9
82.8
95.5
55.5
2lo
g-s
licin
gbvsdiv
24
0oot
oot
oot
0.0
01.0
01.7
11
64
64
00.0
39.0
818.9
468.2
466.6
60.0
11.1
60.3
95.5
65.5
3lo
g-s
licin
gbvsdiv
25
0oot
oot
oot
0.0
01.0
01.6
10
56
56
30.0
38.3
712.4
7565.8
2564.3
00.0
6424.8
63.6
75.5
75.5
4lo
g-s
licin
gbvslt
10201
20
552.9
2oot
oot
0.0
01.0
02.9
843
43
00.0
221.9
01617.9
36398.0
26394.2
58.2
6161.9
376.1
75.5
85.5
5lo
g-s
licin
gbvslt
11687
20
665.8
6oot
oot
0.0
01.0
03.0
942
42
00.0
222.6
21749.7
06469.9
96466.1
93.7
3229.0
868.1
15.5
95.5
6lo
g-s
licin
gbvslt
13173
20
801.4
6oot
oot
0.0
01.0
02.9
839
39
00.0
123.8
21646.3
76426.0
66422.1
89.2
7144.4
375.6
05.6
05.5
7lo
g-s
licin
gbvslt
13916
20
908.9
9oot
oot
0.0
01.0
02.9
838
38
00.0
224.1
51679.8
36424.9
46421.0
52.6
8222.8
376.4
95.6
15.5
8lo
g-s
licin
gbvslt
14659
20
838.9
9oot
oot
0.0
01.0
02.9
837
37
00.0
223.1
31612.1
16192.5
46188.5
911.0
5193.4
875.5
25.6
25.5
9lo
g-s
licin
gbvslt
16145
20
1188.5
6oot
oot
0.0
01.0
02.9
936
36
00.0
224.4
51674.9
16298.2
36293.9
10.1
1212.9
363.6
25.6
35.6
0lo
g-s
licin
gbvslt
16888
20
1502.5
6oot
oot
0.0
01.0
02.9
521
21
00.0
124.2
11529.6
25643.9
55637.5
530.8
4307.3
9166.0
05.6
45.6
1lo
g-s
licin
gbvslt
17631
20
1750.5
6oot
oot
0.0
01.0
02.6
416
16
00.0
117.3
2922.4
45067.3
95061.7
637.0
5338.1
0194.9
05.6
55.6
2lo
g-s
licin
gbvslt
18374
0oot
oot
oot
0.0
01.0
02.2
416
16
00.0
117.4
61763.2
85922.1
05917.7
8123.5
9664.6
1394.8
15.6
65.6
3lo
g-s
licin
gbvslt
19117
0oot
oot
oot
0.0
01.0
03.1
722
22
00.0
126.9
21681.6
06164.5
46158.1
615.0
9293.9
294.8
45.6
75.6
4lo
g-s
licin
gbvslt
19860
0oot
oot
oot
0.0
01.0
03.0
722
22
00.0
127.3
21558.4
05753.2
95746.9
816.1
4262.0
988.5
15.6
85.6
5lo
g-s
licin
gbvsm
od
16
0oot
oot
oot
0.0
01.0
00.5
27
120
57
75
0.0
82.1
91706.5
23794.9
93793.8
30.0
0244.8
48.2
75.6
95.6
6lo
g-s
licin
gbvsm
od
17
0oot
oot
oot
0.0
01.0
01.3
17
82
55
34
0.0
610.5
01663.8
43763.6
43761.7
80.0
0594.5
713.9
45.7
05.6
7lo
g-s
licin
gbvsm
od
18
0oot
oot
oot
0.0
01.0
01.4
13
70
60
90.0
48.5
61551.0
92248.6
92247.1
30.0
01049.2
110.9
75.7
15.6
8lo
g-s
licin
gbvsm
od
19
0oot
oot
oot
0.0
01.0
01.4
12
68
62
11
0.0
38.1
8143.2
12493.0
32491.4
60.0
0870.1
812.2
25.7
25.6
9lo
g-s
licin
gbvsm
od
20
0oot
oot
oot
0.0
01.0
01.4
10
58
56
40.0
37.3
8853.9
71382.8
91381.3
70.0
0844.5
18.3
85.7
35.7
0lo
g-s
licin
gbvsrem
16
0oot
oot
oot
0.0
01.0
00.4
19
81
48
52
0.0
51.3
81683.8
83729.3
13728.5
20.0
0358.4
612.6
05.7
45.7
1lo
g-s
licin
gbvsrem
17
0oot
oot
oot
0.0
01.0
01.0
16
69
46
39
0.0
58.2
11691.7
74551.8
24550.2
50.0
0545.0
219.0
55.7
55.7
2lo
g-s
licin
gbvsrem
18
0oot
oot
oot
0.0
01.0
01.2
12
59
52
15
0.0
46.8
21463.9
73856.0
33854.7
00.0
1843.8
020.9
65.7
65.7
3lo
g-s
licin
gbvsrem
19
0oot
oot
oot
0.0
01.0
01.3
10
56
51
40.0
37.0
1739.9
91616.5
01615.2
30.0
1731.5
110.7
15.7
75.7
4lo
g-s
licin
gbvsrem
20
0oot
oot
oot
0.0
01.0
01.1
954
50
10.0
26.7
08.3
4687.9
1686.8
50.0
0656.1
16.2
55.7
85.7
5lo
g-s
licin
gbvsub
06991
20
307.6
8oot
oot
0.0
01.0
00.2
11
10
0.0
00.0
07.3
67.3
64.6
97.3
67.3
67.3
65.7
9
Contin
ued
on
next
page
94
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.7
6lo
g-s
licin
gbvsub
07988
20
321.6
9oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
08.2
98.2
94.8
58.2
98.2
98.2
95.8
05.7
7lo
g-s
licin
gbvsub
08985
20
378.5
6oot
oot
0.0
01.0
00.1
11
10
0.0
00.0
09.1
99.1
94.9
29.1
99.1
99.1
95.8
15.7
8lo
g-s
licin
gbvsub
09982
20
328.0
6oot
oot
0.0
01.0
00.2
11
10
0.0
00.0
010.1
910.1
95.1
410.1
910.1
910.1
95.8
25.7
9lo
g-s
licin
gbvsub
10979
20
161.2
1oot
oot
0.0
01.0
00.1
11
10
0.0
00.0
011.1
811.1
85.1
511.1
811.1
811.1
85.8
35.8
0lo
g-s
licin
gbvsub
11976
20
664.6
3oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
012.7
812.7
85.3
612.7
812.7
812.7
85.8
45.8
1lo
g-s
licin
gbvsub
12973
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
013.9
813.9
85.3
913.9
813.9
813.9
85.8
55.8
2lo
g-s
licin
gbvsub
13970
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
015.0
915.0
95.6
115.0
915.0
915.0
95.8
65.8
3lo
g-s
licin
gbvsub
14967
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
016.9
016.9
05.5
616.9
016.9
016.9
05.8
75.8
4lo
g-s
licin
gbvsub
15964
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
018.9
018.9
05.7
618.9
018.9
018.9
05.8
85.8
5lo
g-s
licin
gbvsub
16961
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
020.5
620.5
65.8
220.5
620.5
620.5
65.8
95.8
6lo
g-s
licin
gbvsub
17958
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
015.8
215.8
20.2
015.8
215.8
215.8
25.9
05.8
7lo
g-s
licin
gbvsub
18955
0oot
oot
oot
0.0
01.0
00.6
11
10
0.0
00.0
018.0
418.0
40.2
118.0
418.0
418.0
45.9
15.8
8lo
g-s
licin
gbvsub
19952
0oot
oot
oot
0.0
01.0
00.6
11
10
0.0
00.0
019.5
519.5
50.2
119.5
519.5
519.5
55.9
25.8
9lo
g-s
licin
gbvsub
20949
0oot
oot
oot
0.0
01.0
00.6
11
10
0.0
00.0
022.7
822.7
80.2
222.7
822.7
822.7
85.9
35.9
0lo
g-s
licin
gbvsub
21946
0oot
oot
oot
0.0
01.0
00.7
11
10
0.0
00.0
024.9
524.9
50.2
324.9
524.9
524.9
55.9
45.9
1lo
g-s
licin
gbvsub
22943
0oot
oot
oot
0.0
01.0
00.7
11
10
0.0
00.0
027.3
927.3
90.2
427.3
927.3
927.3
95.9
55.9
2lo
g-s
licin
gbvsub
23940
0oot
oot
oot
0.0
01.0
00.8
11
10
0.0
00.0
032.5
632.5
60.2
532.5
632.5
632.5
65.9
65.9
3lo
g-s
licin
gbvudiv
19
0oot
oot
oot
0.0
01.0
00.8
829
28
11
0.0
25.6
11783.5
64623.6
44622.5
10.0
01349.4
353.1
55.9
75.9
4lo
g-s
licin
gbvudiv
20
0oot
oot
oot
0.0
01.0
00.9
832
30
90.0
25.8
41144.5
83234.6
03233.4
50.0
01123.7
236.3
45.9
85.9
5lo
g-s
licin
gbvudiv
21
0oot
oot
oot
0.0
01.0
01.0
732
30
50.0
16.4
05.9
2501.2
4500.1
70.0
0413.0
86.6
05.9
95.9
6lo
g-s
licin
gbvudiv
22
0oot
oot
oot
0.0
01.0
01.0
733
32
40.0
26.5
86.2
11882.5
11881.3
60.0
01648.6
924.7
75.1
00
5.9
7lo
g-s
licin
gbvudiv
23
0oot
oot
oot
0.0
01.0
01.0
733
32
40.0
26.4
96.3
7771.8
2770.6
90.0
1663.6
310.1
65.1
01
5.9
8lo
g-s
licin
gbvudiv
24
0oot
oot
oot
0.0
01.0
01.0
628
26
00.0
15.4
55.5
417.7
516.6
20.0
61.3
20.3
35.1
02
5.9
9lo
g-s
licin
gbvudiv
25
0oot
oot
oot
0.0
01.0
01.2
736
32
10.0
16.9
76.7
51555.7
81554.4
50.0
21531.6
620.4
75.1
03
5.1
00
log-s
licin
gbvult
10985
20
556.6
2oot
oot
0.0
01.0
02.7
740
40
00.0
221.2
71550.5
16576.1
76572.2
43.5
3229.7
290.0
85.1
04
5.1
01
log-s
licin
gbvult
11982
20
518.1
6oot
oot
0.0
01.0
02.9
734
34
00.0
119.1
91516.4
26232.6
16228.8
31.2
1173.0
383.1
05.1
05
5.1
02
log-s
licin
gbvult
12979
20
893.4
9oot
oot
0.0
01.0
02.9
940
40
00.0
123.6
91748.2
46385.2
06381.0
62.2
1195.3
267.2
15.1
06
5.1
03
log-s
licin
gbvult
13976
20
893.7
3oot
oot
0.0
01.0
03.0
839
39
00.0
225.1
71712.6
16534.5
46530.3
30.1
5183.2
977.7
95.1
07
5.1
04
log-s
licin
gbvult
15970
20
1070.6
3oot
oot
0.0
01.0
03.1
10
36
36
00.0
124.7
71351.2
34874.4
64869.7
70.2
0190.9
543.1
45.1
08
5.1
05
log-s
licin
gbvult
16967
0oot
oot
oot
0.0
01.0
02.9
521
21
00.0
123.3
11372.3
45776.0
45770.0
551.6
5256.2
0144.4
05.1
09
5.1
06
log-s
licin
gbvult
17964
0oot
oot
oot
0.0
01.0
02.9
521
21
00.0
125.9
31268.4
75503.7
25497.7
839.3
3252.8
5127.9
95.1
10
5.1
07
log-s
licin
gbvult
18961
0oot
oot
oot
0.0
01.0
02.3
416
16
00.0
118.2
21566.8
95066.3
95062.2
573.3
4508.6
1298.0
25.1
11
5.1
08
log-s
licin
gbvult
19958
0oot
oot
oot
0.0
01.0
03.2
822
22
00.0
127.3
61738.2
15681.4
45675.4
90.1
9481.6
571.9
25.1
12
5.1
09
log-s
licin
gbvult
20955
0oot
oot
oot
0.0
01.0
03.0
721
21
00.0
126.5
71156.9
85151.5
65145.7
310.8
5529.5
175.7
65.1
13
5.1
10
log-s
licin
gbvult
21952
0oot
oot
oot
0.0
01.0
03.0
821
21
00.0
126.0
21525.4
75716.0
55710.3
33.4
2465.0
771.4
55.1
14
5.1
11
log-s
licin
gbvult
22949
0oot
oot
oot
0.0
01.0
03.0
821
21
00.0
126.8
81005.1
45915.0
85909.2
212.8
3757.1
870.4
25.1
15
5.1
12
log-s
licin
gbvult
23946
0oot
oot
oot
0.0
01.0
02.9
720
20
00.0
123.7
1655.3
16183.7
96177.9
80.2
8674.8
879.2
85.1
16
5.1
13
log-s
licin
gbvult
24943
0oot
oot
oot
0.0
01.0
03.0
819
19
00.0
127.2
61112.2
24499.9
84494.4
20.2
7414.3
755.5
65.1
17
5.1
14
log-s
licin
gbvult
25940
0oot
oot
oot
0.0
01.0
03.0
819
19
00.0
124.3
5708.3
13575.9
63570.2
80.2
1690.9
543.0
85.1
18
5.1
15
log-s
licin
gbvult
9988
20
555.0
3oot
oot
0.0
01.0
02.9
844
44
00.0
222.9
51618.9
06318.0
66314.0
94.8
1194.9
671.8
05.1
19
5.1
16
log-s
licin
gbvurem
17
0oot
oot
oot
0.0
01.0
00.5
830
24
16
0.0
23.4
3894.0
33730.5
13729.8
40.0
0774.5
040.9
95.1
20
5.1
17
log-s
licin
gbvurem
18
0oot
oot
oot
0.0
01.0
00.6
729
29
70.0
13.5
41115.8
52549.9
22549.3
50.0
01111.6
935.9
15.1
21
5.1
18
log-s
licin
gbvurem
19
0oot
oot
oot
0.0
01.0
00.6
629
27
20.0
13.4
74.0
4443.3
7442.7
60.0
0393.3
67.1
55.1
22
5.1
19
log-s
licin
gbvurem
20
0oot
oot
oot
0.0
01.0
00.8
732
30
10.0
23.8
84.9
2428.8
3428.2
10.0
0413.9
16.6
05.1
23
5.1
20
log-s
licin
gbvurem
21
0oot
oot
oot
0.0
01.0
00.7
630
30
00.0
13.8
24.2
115.2
614.6
20.1
50.9
20.2
55.1
24
5.1
21
log-s
licin
gbvurem
22
0oot
oot
oot
0.0
01.0
00.9
732
32
00.0
24.4
35.4
516.7
116.0
60.0
00.9
60.2
65.1
25
5.1
22
pip
epip
e-n
oabs.a
tla
s.q
fbv
20
1290.1
0oot
oot
0.0
01.0
00.7
0.0
00.0
00.0
00.0
00.0
05.1
26
5.1
23
pspace
power2sum
.5500
20
496.1
6oot
oot
0.0
01.0
02.0
11
10
0.0
00.0
01208.9
31208.9
31208.5
71208.9
31208.9
31208.9
35.1
27
5.1
24
pspace
power2sum
.5898
20
343.5
7oot
oot
0.0
01.0
02.2
11
10
0.0
00.0
01472.5
21472.5
21472.1
21472.5
21472.5
21472.5
25.1
28
5.1
25
pspace
power2sum
.6296
20
434.7
6oot
oot
0.0
01.0
02.2
11
10
0.0
00.0
01780.5
21780.5
21780.0
81780.5
21780.5
21780.5
25.1
29
5.1
26
pspace
power2sum
.6495
20
504.5
1oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
30
Contin
ued
on
next
page
95
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.1
27
pspace
power2sum
.6694
20
1005.8
1oot
oot
0.0
01.0
02.0
0.0
00.0
00.0
00.0
00.0
05.1
31
5.1
28
pspace
power2sum
.6893
20
1102.0
2oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
32
5.1
29
pspace
power2sum
.7092
20
452.5
0oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
33
5.1
30
pspace
power2sum
.7291
20
971.6
8oot
oot
0.0
01.0
02.0
0.0
00.0
00.0
00.0
00.0
05.1
34
5.1
31
pspace
power2sum
.7490
20
505.6
9oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
35
5.1
32
pspace
power2sum
.7689
20
515.9
3oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
36
5.1
33
pspace
power2sum
.7888
20
1145.4
9oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
37
5.1
34
pspace
power2sum
.8087
0oot
oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
38
5.1
35
pspace
power2sum
.8286
20
898.4
1oot
oot
0.0
01.0
02.0
0.0
00.0
00.0
00.0
00.0
05.1
39
5.1
36
pspace
power2sum
.8485
20
872.3
0oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
40
5.1
37
pspace
power2sum
.8684
20
1028.3
0oot
oot
0.0
01.0
02.0
0.0
00.0
00.0
00.0
00.0
05.1
41
5.1
38
pspace
power2sum
.8883
20
1035.8
8oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
42
5.1
39
pspace
power2sum
.9082
20
1301.9
1oot
oot
0.0
01.0
02.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
43
5.1
40
pspace
power2sum
.9281
20
1338.5
3oot
oot
0.0
01.0
02.0
0.0
00.0
00.0
00.0
00.0
05.1
44
5.1
41
pspace
power2sum
.9480
20
751.8
3oot
oot
0.0
01.0
02.1
0.0
00.0
00.0
00.0
00.0
05.1
45
5.1
42
pspace
power2sum
.9679
0oot
oot
oot
0.0
01.0
02.0
0.0
00.0
00.0
00.0
00.0
05.1
46
5.1
43
pspace
power2sum
.9878
20
573.3
5oot
oot
0.0
01.0
02.1
0.0
00.0
00.0
00.0
00.0
05.1
47
5.1
44
RW
SExam
ple
15.txt
0oot
oot
oot
0.0
01.0
00.3
0.0
00.0
00.0
00.0
00.0
05.1
48
5.1
45
RW
SExam
ple
19.txt
0oot
oot
oot
0.0
01.0
00.3
0.0
00.0
00.0
00.0
00.0
05.1
49
5.1
46
RW
SExam
ple
20.txt
0oot
oot
oot
0.0
01.0
00.4
0.0
00.0
00.0
00.0
00.0
05.1
50
5.1
47
RW
SExam
ple
7.txt
0oot
oot
oot
0.0
01.0
00.1
0.0
00.0
00.0
00.0
00.0
05.1
51
5.1
48
uum
uum
12
0oot
oot
oot
0.0
01.0
00.6
0.0
00.0
00.0
00.0
00.0
05.1
52
5.1
49
uum
uum
16
0oot
oot
oot
0.0
01.0
01.9
60
452
301
00.3
53.0
2637.6
26943.6
76935.1
40.0
042.0
13.5
45.1
53
5.1
50
uum
uum
20
0oot
oot
oot
0.0
01.0
03.1
0.0
00.0
00.0
00.0
00.0
05.1
54
5.1
51
uum
uum
24
0oot
oot
oot
0.0
01.0
03.2
0.0
00.0
00.0
00.0
00.0
05.1
55
5.1
52
uum
uum
28
0oot
oot
oot
0.0
01.0
03.4
0.0
00.0
00.0
00.0
00.0
05.1
56
5.1
53
uum
uum
32
0oot
oot
oot
0.0
01.0
03.2
0.0
00.0
00.0
00.0
00.0
05.1
57
5.1
54
VS3
VS3-b
enchm
ark-A
10
0oot
oot
oot
0.0
01.0
03.4
45
207
207
00.2
917.3
31766.6
57005.4
96999.8
00.1
744.4
55.9
95.1
58
5.1
55
VS3
VS3-b
enchm
ark-A
11
0oot
oot
oot
0.0
01.0
03.4
47
213
211
00.3
116.7
41763.3
86973.0
16967.4
60.1
544.3
95.7
85.1
59
5.1
56
VS3
VS3-b
enchm
ark-A
12
0oot
oot
oot
0.0
01.0
03.4
32
160
160
00.2
212.8
71775.7
67071.2
27067.5
00.3
744.3
79.0
85.1
60
5.1
57
VS3
VS3-b
enchm
ark-A
13
0oot
oot
oot
0.0
01.0
03.2
30
111
111
00.1
711.7
0639.7
47067.9
37063.9
40.3
753.6
310.5
35.1
61
5.1
58
VS3
VS3-b
enchm
ark-A
80
oot
oot
oot
0.0
01.0
03.2
34
153
153
00.2
313.9
21784.0
77033.2
07029.4
30.3
744.8
39.2
95.1
62
5.1
59
VS3
VS3-b
enchm
ark-S
20
oot
oot
oot
0.0
01.0
03.5
20
42
42
00.0
715.6
41745.3
06779.8
96777.3
61.7
9108.5
122.9
15.1
63
5.1
60
...1
alt
true.B
V.c
.cil.c
.21
0oot
oot
oot
0.0
01.0
04.7
11
15
14
00.0
65.9
61597.0
35953.7
35920.1
00.5
3321.5
958.9
55.1
64
5.1
61
brum
mayerbie
re2
sm
ulo
v1bw48
0oot
oot
oot
0.0
01.0
01.4
0.0
00.0
00.0
00.0
00.0
05.1
65
5.1
62
...e
rbie
re3
isqrtaddeqcheck
0oot
oot
oot
0.0
01.0
03.3
60
413
248
13
0.1
15.1
1723.7
86504.2
66484.7
30.0
093.8
13.7
25.1
66
5.1
63
...m
ayerbie
re3
isqrtaddnoif
0oot
oot
oot
0.0
01.0
01.9
30
217
110
00.0
73.8
61358.8
66672.7
66663.8
50.0
2165.2
211.1
25.1
67
5.1
64
...m
ayerbie
re3
isqrteqcheck
0oot
oot
oot
0.0
01.0
03.2
50
336
228
00.0
94.7
61103.3
96636.5
36615.9
40.0
099.2
54.7
15.1
68
5.1
65
brum
mayerbie
re3
isqrtnoif
0oot
oot
oot
0.0
01.0
02.7
29
200
122
00.0
43.9
81689.6
46590.9
86580.9
30.0
0186.2
211.9
65.1
69
5.1
66
brum
mayerbie
re3
maxor128
0oot
oot
oot
0.0
01.0
00.9
10
11
38
0.0
56.9
71564.5
11697.8
31692.1
91.1
9266.0
658.5
55.1
70
5.1
67
brum
mayerbie
re3
maxor256
0oot
oot
oot
0.0
01.0
04.4
13
83
10.0
526.7
61305.7
32666.0
12643.6
20.7
1423.9
480.7
95.1
71
5.1
68
brum
mayerbie
re3
maxxor064
0oot
oot
oot
0.0
01.0
03.9
28
144
95
00.1
37.4
81623.1
66507.3
06490.5
90.0
1130.8
212.1
65.1
72
5.1
69
brum
mayerbie
re3
maxxor128
0oot
oot
oot
0.0
01.0
04.1
16
38
38
00.0
612.0
11779.0
16561.6
46545.4
20.5
2147.0
928.6
55.1
73
5.1
70
brum
mayerbie
re3
maxxor256
0oot
oot
oot
0.0
01.0
03.6
910
10
00.0
316.4
71208.7
65502.2
05486.9
61.2
6466.2
976.4
25.1
74
5.1
71
...b
iere3
maxxorm
axorand032
0oot
oot
oot
0.0
01.0
03.4
130
898
304
00.9
714.7
71737.7
15979.5
15943.0
90.0
046.3
01.9
75.1
75
5.1
72
...b
iere3
maxxorm
axorand064
0oot
oot
oot
0.0
01.0
03.9
67
424
82
01.1
231.4
71721.5
85382.3
35332.6
80.0
0109.0
24.7
05.1
76
5.1
73
...b
iere3
maxxorm
axorand128
0oot
oot
oot
0.0
01.0
04.1
16
31
25
00.1
917.8
41634.6
45628.7
15594.8
50.0
8181.5
026.1
85.1
77
5.1
74
...b
iere3
maxxorm
axorand256
0oot
oot
oot
0.0
01.0
04.0
0.0
00.0
00.0
00.0
00.0
05.1
78
5.1
75
...y
erbie
re3
min
andm
axor128
0oot
oot
oot
0.0
01.0
04.3
17
67
17
10.3
028.0
91557.5
15156.8
15105.1
20.1
3376.4
728.8
15.1
79
5.1
76
...y
erbie
re3
min
andm
axor256
0oot
oot
oot
0.0
01.0
04.9
910
71
0.0
921.6
11174.3
44471.8
94445.5
28.4
7548.1
086.0
05.1
80
5.1
77
brum
mayerbie
re3
min
xor128
0oot
oot
oot
0.0
01.0
04.8
33
179
77
00.3
411.1
81782.2
86607.9
46575.1
40.0
8103.2
110.8
15.1
81
Contin
ued
on
next
page
96
Table
C.1
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.1
78
brum
mayerbie
re3
min
xor256
0oot
oot
oot
0.0
01.0
04.8
14
32
18
00.1
413.3
11410.1
25711.4
05672.8
80.1
4342.0
530.3
85.1
82
5.1
79
...e
rbie
re3
min
xorm
inand064
0oot
oot
oot
0.0
01.0
04.8
55
316
127
00.8
015.5
41756.8
26361.5
36298.9
60.0
158.2
85.4
75.1
83
5.1
80
...e
rbie
re3
min
xorm
inand128
0oot
oot
oot
0.0
01.0
04.8
17
47
43
00.2
712.2
51629.4
06366.9
06337.2
70.0
7201.3
124.7
75.1
84
5.1
81
...e
rbie
re3
min
xorm
inand256
0oot
oot
oot
0.0
01.0
04.6
912
11
00.1
717.8
31070.3
94184.5
14155.4
80.6
4614.6
056.5
55.1
85
5.1
82
brum
mayerbie
re3
mulh
s16
0oot
oot
oot
0.0
01.0
00.3
14
89
53
20.0
00.2
412.0
41744.2
71744.0
80.0
01222.1
27.8
25.1
86
5.1
83
brum
mayerbie
re3
mulh
s32
0oot
oot
oot
0.0
01.0
01.1
21
142
92
10.0
11.5
427.1
498.6
097.9
30.0
013.0
80.2
55.1
87
5.1
84
brum
mayerbie
re3
mulh
s64
0oot
oot
oot
0.0
01.0
03.9
35
142
78
00.0
110.8
91722.1
56349.7
96347.0
40.0
0320.3
19.5
55.1
88
5.1
85
...m
ayerbie
re
countbits1024
0oot
oot
oot
0.0
01.0
01.8
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
89
5.1
86
brum
mayerbie
re
countbits128
0oot
oot
oot
0.0
01.0
03.5
25
72
70
00.0
79.6
91752.7
56623.5
26608.3
00.0
5104.1
815.3
05.1
90
5.1
87
brum
mayerbie
re
countbits256
0oot
oot
oot
0.0
01.0
03.8
11
22
20
00.0
514.6
61738.5
66026.4
76007.4
80.3
0253.9
550.6
45.1
91
5.1
88
brum
mayerbie
re
countbits512
0oot
oot
oot
0.0
01.0
03.7
56
40
0.0
523.2
81493.6
92955.0
02937.2
01.1
4485.6
9184.6
95.1
92
5.1
89
...b
iere
countbitsrotate032
0oot
oot
oot
0.0
01.0
00.2
516
16
00.0
10.0
817.8
850.3
950.1
40.6
93.9
31.6
35.1
93
5.1
90
...b
iere
countbitsrotate064
0oot
oot
oot
0.0
01.0
00.7
628
28
00.0
20.9
2113.6
4479.6
2477.4
70.0
814.8
47.7
45.1
94
5.1
91
...b
iere
countbitsrotate128
0oot
oot
oot
0.0
01.0
01.6
516
16
00.0
32.9
6386.3
41037.0
71031.8
30.2
580.7
533.4
55.1
95
5.1
92
...b
iere
countbitsrotate256
0oot
oot
oot
0.0
01.0
04.3
58
80
0.0
37.4
41091.9
83909.7
43893.0
07.5
1521.4
7156.3
95.1
96
5.1
93
...y
erbie
re
countbitssrl0
64
0oot
oot
oot
0.0
01.0
01.3
13
68
20
00.0
54.2
677.7
6218.2
9207.0
90.0
018.7
71.5
85.1
97
5.1
94
...y
erbie
re
countbitssrl1
28
0oot
oot
oot
0.0
01.0
03.8
15
66
23
00.1
420.2
4691.1
72940.1
62910.1
80.0
0120.6
615.8
15.1
98
5.1
95
...y
erbie
re
countbitssrl2
56
0oot
oot
oot
0.0
01.0
04.4
13
10
50
0.0
630.3
91564.8
54214.2
14168.6
80.1
2489.0
078.0
45.1
99
5.1
96
challenge
integerO
verflow
0oot
oot
oot
0.0
01.0
00.4
512
80
0.0
00.1
713.5
239.9
839.4
70.0
09.7
21.7
45.2
00
5.1
97
...e
rbie
re4
unconstrain
ed04
0oom
oom
oom
-0.2
11.0
07.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.4
5.1
98
...e
rbie
re4
unconstrain
ed03
0oom
oom
oom
-0.3
60.9
96.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
01
5.1
99
...e
rbie
re4
unconstrain
ed06
0oom
oom
oom
-0.3
80.9
96.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
5.2
00
...e
rbie
re4
unconstrain
ed09
0oom
oom
oom
-0.6
50.9
96.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
03
5.2
01
...e
rbie
re4
unconstrain
ed10
0oom
oom
oom
-1.2
00.9
96.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
05
97
Tab
leC
.2:
Det
aile
dre
sult
sof
PB
oole
ctor
wit
hF
LR
(Para
llel
-In
crem
enta
l-F
LR
)co
mp
are
dto
the
refe
ren
ceti
me
Base
-In
crem
enta
l.L
ast
colu
mn
par-
inc
conta
ins
cross
refe
ren
ceto
det
ail
edre
sult
sof
Para
llel
-Incr
emen
tal
inT
ab
.C.1
.
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crem
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
1.1
brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
5oot
492.8
51307.1
53.6
50.6
55
276
52
43
0.4
00
0488.5
01373.2
41369.5
60.0
028.2
91.8
71.1
1.2
log-s
licin
gbvurem
16
20
oot
oot
1009.3
2790.6
81.7
80.2
824
23
22
0.0
40
01008.1
42527.6
12527.4
30.0
0340.6
230.4
51.2
1.3
...iffyIn
terle
avedM
odM
ult-8
20
oot
oot
1406.2
6393.7
41.2
80.4
67
484
132
00.1
02
01403.1
85227.0
35225.5
70.0
021.5
03.9
21.3
1.4
log-s
licin
gbvudiv
18
20
oot
oot
1544.4
8255.5
21.1
70.8
11
40
31
23
0.0
80
01532.7
34254.4
44253.2
20.0
0535.6
134.5
91.7
1.5
log-s
licin
gbvsm
od
15
20
oot
oot
1752.2
547.7
51.0
30.4
40
182
51
113
0.1
00
01747.4
74090.6
14089.0
60.0
0137.7
76.2
21.9
2.1
log-s
licin
gbvudiv
17
20
1270.3
61373.3
4623.3
9749.9
52.2
00.7
10
33
23
21
0.0
80
0613.8
41576.8
41575.6
80.0
0186.4
415.7
72.1
2.2
log-s
licin
gbvurem
15
20
oot
1147.7
1416.2
0731.5
12.7
60.2
824
23
20
0.0
50
0415.0
2963.4
8963.2
80.0
0136.8
912.0
42.2
2.3
log-s
licin
gbvsm
od
14
20
1414.5
41350.5
9745.4
8605.1
11.8
10.4
41
180
46
118
0.1
00
0740.9
61779.4
21777.9
00.0
063.2
92.8
12.3
2.4
log-s
licin
gbvm
ul12
20
129.1
9962.3
1423.7
7538.5
42.2
70.1
22
22
0.0
00
0423.0
7817.8
3817.8
10.2
8422.2
8163.5
72.4
2.5
log-s
licin
gbvudiv
16
20
390.9
9521.1
2251.4
1269.7
12.0
70.2
824
22
20
0.0
40
0250.0
3739.9
3739.6
70.0
086.7
89.2
52.8
2.6
log-s
licin
gbvurem
14
20
757.1
6270.8
8133.1
4137.7
42.0
30.1
824
22
19
0.0
50
0132.0
4445.3
9445.2
20.0
060.8
75.7
12.1
12.7
log-s
licin
gbvsm
od
13
20
360.2
6438.2
6359.2
878.9
81.2
20.3
35
158
50
98
0.1
00
0355.4
2925.8
2924.5
60.0
028.0
91.6
42.1
32.8
challenge
multip
lyO
verflow
20
1035.6
11002.1
1925.2
276.8
91.0
80.1
22
22
0.0
00
0924.4
61514.0
31514.0
00.5
6922.6
8302.8
12.1
42.9
log-s
licin
gbvudiv
15
20
162.7
0192.9
5117.6
975.2
61.6
40.2
823
21
18
0.0
40
0116.3
8340.9
7340.7
50.0
041.7
54.5
52.1
52.1
0lo
g-s
licin
gbvurem
13
20
147.9
7116.6
757.1
359.5
42.0
40.1
721
17
17
0.0
50
056.0
2180.8
8180.7
50.0
023.1
92.7
02.1
72.1
1brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3132.0
173.3
558.6
61.8
00.3
25
133
42
15
0.2
40
071.6
6220.8
9219.7
10.0
015.9
90.6
72.1
82.1
2lo
g-s
licin
gbvsm
od
12
20
138.5
8174.8
0161.4
913.3
11.0
80.3
32
150
51
77
0.1
00
0157.8
7459.8
4458.7
60.0
013.9
30.9
32.1
92.1
3...y
erbie
re3
isqrtin
validvc
10
181.6
615.8
99.4
96.4
01.6
70.1
416
16
00.1
70
08.6
029.1
728.2
70.2
32.2
60.9
42.2
0
3.1
brum
mayerbie
re
bitrev8192
20
101.9
7101.4
9101.5
0-0
.01
1.0
00.0
01
10
0.0
00
00.0
40.0
40.0
00.0
40.0
40.0
42.3
03.2
bm
c-b
vm
agic
10
443.8
6453.4
4453.4
7-0
.03
1.0
00.2
01
10
0.2
11
00.4
50.4
50.3
90.4
50.4
50.4
52.2
13.3
bm
c-b
vgraycode
10
442.5
4449.0
3449.3
7-0
.34
1.0
00.2
01
10
0.2
11
00.0
40.0
40.0
00.0
40.0
40.0
43.1
3.4
...e
rbie
re4
unconstrain
ed02
10
oot
44.2
344.7
5-0
.52
0.9
94.2
01
10
0.0
00
039.9
139.9
17.7
939.9
139.9
139.9
13.2
3.5
RW
SExam
ple
18.txt
10
160.6
1193.3
5198.6
3-5
.28
0.9
70.1
01
10
5.0
50
0192.7
6192.7
6192.6
0192.7
6192.7
6192.7
63.3
3.6
...m
ayerbie
re2
um
ulo
v1bw224
20
108.4
290.1
6112.0
7-2
1.9
10.8
00.3
12
20
17.4
80
090.9
090.9
089.4
20.0
088.3
530.3
03.4
3.7
log-s
licin
gbvsub
04000
20
545.8
4842.7
7880.5
3-3
7.7
60.9
60.1
11
11
0.0
00
0879.3
7879.3
7878.4
55.8
4873.5
3439.6
83.6
3.8
...m
ayerbie
re2
um
ulo
v1bw256
20
143.9
4119.0
5157.2
5-3
8.2
00.7
60.4
12
20
22.6
00
0130.5
1130.5
1128.4
40.0
0102.5
643.5
03.5
3.9
...m
ayerbie
re2
sm
ulo
v2bw384
20
122.0
615.6
489.0
7-7
3.4
30.1
80.4
01
10
74.0
40
014.0
714.0
78.8
514.0
714.0
714.0
72.2
33.1
0...a
yerbie
re2
sm
ulo
v4bw0512
10
128.7
82.5
787.7
9-8
5.2
20.0
30.1
01
10
85.3
80
01.5
81.5
80.1
71.5
81.5
81.5
82.2
63.1
1...m
ayerbie
re2
sm
ulo
v2bw448
20
150.6
343.6
9139.1
5-9
5.4
60.3
10.5
01
10
95.5
90
042.5
042.5
034.7
442.5
042.5
042.5
02.2
43.1
2...b
iere3
isqrtaddin
validvc
10
185.6
016.0
0126.9
1-1
10.9
10.1
31.9
40
270
222
30
0.1
60
0124.2
0472.2
5458.5
10.0
05.5
30.4
23.1
23.1
3...m
ayerbie
re2
sm
ulo
v2bw512
20
198.3
836.6
8161.1
7-1
24.4
90.2
30.6
01
10
124.5
10
035.2
735.2
722.9
635.2
735.2
735.2
72.2
53.1
4...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
33.9
6135.3
1-1
31.3
50.0
30.2
01
10
131.5
90
02.8
72.8
70.2
72.8
72.8
72.8
72.2
73.1
5...a
yerbie
re2
sm
ulo
v4bw0768
10
563.4
25.8
0181.9
8-1
76.1
80.0
30.4
01
10
176.2
60
04.7
24.7
20.4
14.7
24.7
24.7
22.2
83.1
6lo
g-s
licin
gbvsrem
14
20
256.9
0269.5
8477.6
4-2
08.0
60.5
60.3
23
91
46
71
0.0
80
0475.0
91359.3
31358.5
70.0
055.6
34.0
83.1
53.1
7core
ext
con
044
002
1024
20
195.8
21571.3
51786.9
1-2
15.5
60.8
80.3
11
11
7.5
24
01778.6
91778.6
91778.4
188.8
51689.8
4889.3
52.6
3.1
8core
ext
con
036
002
1024
20
269.9
91156.1
91377.9
6-2
21.7
70.8
40.2
11
11
5.0
94
01372.2
31372.2
31372.0
068.2
11304.0
2686.1
12.9
3.1
9lo
g-s
licin
gbvsrem
15
20
791.1
0927.6
21154.0
9-2
26.4
70.8
00.3
23
92
46
76
0.0
80
01151.3
42926.2
42925.4
50.0
0133.0
68.3
13.1
43.2
0brum
mayerbie
re
nlz
be256
20
117.5
1157.9
7389.2
6-2
31.2
90.4
11.3
715
60
48.8
51
0332.1
3674.7
9663.6
00.1
0137.6
523.2
72.1
63.2
1...a
yerbie
re2
sm
ulo
v4bw0896
10
1274.5
28.2
6246.9
1-2
38.6
50.0
30.5
01
10
238.9
00
06.9
46.9
40.5
46.9
46.9
46.9
42.2
93.2
2pspace
shift1add.2
2961
20
110.7
1110.8
6369.9
4-2
59.0
80.3
00.3
11
11
0.0
00
0369.2
2369.2
2368.7
312.6
8356.5
3184.6
13.1
73.2
3pspace
shift1add.2
3958
20
114.0
8115.5
0403.7
1-2
88.2
10.2
90.4
11
11
0.0
00
0402.9
4402.9
4402.4
213.8
7389.0
7201.4
73.1
83.2
4...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
710.9
1317.4
2-3
06.5
10.0
30.5
01
10
306.9
20
09.3
29.3
20.7
19.3
29.3
29.3
22.2
23.2
5pspace
shift1add.2
4955
20
128.4
7121.7
9436.1
2-3
14.3
30.2
80.4
11
11
0.0
00
0435.4
2435.4
2434.8
714.6
1420.8
1217.7
13.1
93.2
6pspace
shift1add.2
5952
20
143.2
8129.3
8468.0
0-3
38.6
20.2
80.4
11
11
0.0
00
0467.1
9467.1
9466.6
415.5
1451.6
8233.6
03.2
03.2
7pspace
shift1add.2
6949
20
152.3
3135.5
8494.5
3-3
58.9
50.2
70.4
11
11
0.0
00
0493.7
6493.7
6493.1
816.3
4477.4
2246.8
83.2
1
Contin
ued
on
next
page
98
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
3.2
8pspace
shift1add.2
7946
20
167.8
1143.3
4530.9
1-3
87.5
70.2
70.6
11
11
0.0
00
0530.1
7530.1
7529.5
717.6
5512.5
2265.0
83.2
33.2
9pspace
shift1add.2
8943
20
186.8
4148.3
3563.8
7-4
15.5
40.2
60.6
11
11
0.0
00
0563.1
1563.1
1562.4
918.4
2544.6
9281.5
63.2
53.3
0lo
g-s
licin
gbvsdiv
16
20
182.3
5223.3
9661.9
7-4
38.5
80.3
40.4
24
94
50
75
0.0
80
0659.1
31901.0
41900.1
70.0
066.8
85.2
13.2
93.3
1pspace
shift1add.2
9940
20
195.3
8157.2
3596.6
4-4
39.4
10.2
60.6
11
11
0.0
00
0595.8
9595.8
9595.2
619.3
8576.5
1297.9
53.2
63.3
2brum
mayerbie
re3
isqrtadd
20
144.7
3148.8
8638.1
1-4
89.2
30.2
31.6
41
289
186
11
0.1
60
0635.6
82364.0
52349.2
70.0
137.9
62.2
83.2
83.3
3brum
mayerbie
re3
isqrt
20
145.8
1148.1
1912.7
5-7
64.6
40.1
61.5
100
538
170
243
0.1
60
0908.4
23248.2
73229.0
60.0
033.8
11.3
13.2
23.3
4...sm
tcom
p09
std
bv
form
ula
20
108.8
6169.6
11024.8
1-8
55.2
00.1
73.5
41
258
86
0696.4
521
0311.9
51212.3
01175.7
90.0
120.7
11.5
23.1
33.3
5...m
tcom
p09
full
bv
form
ula
20
172.4
6167.3
11060.7
6-8
93.4
50.1
63.6
30
186
93
0627.7
218
0419.7
51649.4
71621.9
20.0
119.5
62.9
73.1
63.3
6lo
g-s
licin
gbvsdiv
17
20
502.6
4442.9
21390.9
6-9
48.0
40.3
21.2
24
92
53
80
0.1
20
01376.7
64150.7
94148.5
70.0
1130.4
611.1
03.3
63.3
7lfsr
lfsr
004
111
112
20
178.0
5181.6
11141.9
5-9
60.3
40.1
63.6
17
92
53
0377.4
61
0727.2
92630.1
62612.8
40.1
996.6
59.5
33.3
23.3
8lo
g-s
licin
gbvslt
5000
20
296.4
0315.4
51354.5
3-1
039.0
80.2
32.6
17
107
60
00.4
91
01319.5
54768.2
84762.5
80.0
0273.0
617.7
34.5
93.3
9lo
g-s
licin
gbvslt
5743
20
444.5
8367.8
21435.7
3-1
067.9
10.2
61.8
14
80
47
00.4
91
01407.1
85223.8
75219.1
00.0
6241.1
226.3
84.5
23.4
0...sm
tcom
p09
nt
bv
form
ula
20
156.7
2175.2
81326.8
4-1
151.5
60.1
33.6
40
248
104
0684.2
622
0628.4
52352.3
32323.0
40.0
061.8
82.6
82.1
23.4
1lfsr
lfsr
004
159
128
20
112.1
0299.7
41502.3
7-1
202.6
30.2
03.8
16
75
34
0508.0
81
0930.7
43256.0
33233.2
90.1
2117.8
914.8
04.6
43.4
2lfsr
lfsr
008
031
064
20
oot
42.9
71403.1
5-1
360.1
80.0
33.2
36
200
120
0489.4
61
0888.0
43432.9
23406.8
90.1
537.8
84.6
33.3
5
4.1
pspace
power2sum
.6097
20
388.1
21779.4
3oot
-20.5
70.9
92.2
11
10
0.0
00
01620.8
21620.8
21620.4
01620.8
21620.8
21620.8
24.1
4.2
log-s
licin
gbvslt
15402
20
1048.3
41765.7
2oot
-34.2
80.9
82.9
937
37
00.6
31
01384.2
76328.1
96324.0
28.3
4218.9
661.4
44.2
4.3
...e
st
v7
r12
vr1
c1
s10576
20
oot
1756.9
8oot
-43.0
20.9
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
4.4
pspace
power2sum
.5699
20
347.9
21748.1
2oot
-51.8
80.9
72.0
11
10
0.0
00
01333.5
81333.5
81333.1
91333.5
81333.5
81333.5
84.4
4.5
log-s
licin
gbvsub
05994
20
414.3
11726.2
5oot
-73.7
50.9
60.1
11
10
0.0
00
06.6
06.6
04.6
16.6
06.6
06.6
04.5
4.6
core
ext
con
048
002
1024
20
487.7
21716.6
4oot
-83.3
60.9
50.1
11
10
8.9
24
0101.9
1101.9
1101.6
2101.9
1101.9
1101.9
12.5
4.7
...test
v5
r15
vr5
c1
s8246
20
1106.6
41711.4
6oot
-88.5
40.9
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
4.8
log-s
licin
gbvult
14973
20
1019.5
01697.3
0oot
-102.7
00.9
43.1
10
38
38
00.3
50
01655.9
45997.0
65992.6
22.0
1263.7
854.0
34.7
4.9
log-s
licin
gbvsdiv
18
20
1485.5
21599.0
3oot
-200.9
70.8
91.2
15
70
52
33
0.1
20
01539.5
24133.0
04131.4
00.0
0283.9
417.1
54.8
4.1
0...test
v7
r7
vr5
c1
s19694
10
1370.1
11539.9
4oot
-260.0
60.8
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
4.1
1brum
mayerbie
re3
min
or256
20
116.1
3321.3
3sf
-305.5
20.5
13.7
930
18
066.7
20
0235.3
1535.9
9498.4
10.1
182.5
05.8
94.5
84.1
2...e
st
v7
r7
vr10
c1
s10625
10
852.7
61379.9
4oot
-420.0
60.7
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
04.1
3brum
mayerbie
re3
maxand256
20
102.0
5252.0
3sf
-443.7
70.3
64.0
12
52
31
064.2
90
0307.2
3784.9
4754.4
30.1
188.9
85.0
03.3
34.1
4lo
g-s
licin
gbvslt
12430
20
oot
1349.7
6oot
-450.2
40.7
52.5
632
32
00.6
31
01495.0
86284.5
16281.1
313.7
2226.9
5108.3
54.1
14.1
5...e
st
v5
r15
vr5
c1
s26657
20
oot
1282.8
4oot
-517.1
60.7
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
24.1
6lo
g-s
licin
gbvslt
10944
20
505.6
81114.2
5oot
-685.7
50.6
22.9
943
43
00.6
31
01518.5
06438.8
16434.9
00.1
2208.5
365.0
44.1
34.1
7...test
v7
r7
vr1
c1
s4574
10
609.3
31077.2
4oot
-722.7
60.6
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
44.1
8...st
v5
r10
vr10
c1
s21502
20
697.8
21056.4
7oot
-743.5
30.5
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
54.1
9...st
v5
r15
vr10
c1
s11127
10
346.0
31009.9
9oot
-790.0
10.5
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.8
4.2
0VS3
VS3-b
enchm
ark-A
910
268.4
51009.1
3oot
-790.8
70.5
63.3
45
290
144
01.1
21
01771.4
36798.1
66791.3
80.0
031.4
16.3
54.1
64.2
1...e
st
v5
r10
vr1
c1
s32538
20
208.1
6996.7
5oot
-803.2
50.5
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
74.2
2...e
st
v5
r10
vr5
c1
s13679
10
899.6
4937.6
8oot
-862.3
20.5
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
84.2
3VS3
VS3-b
enchm
ark-A
710
985.7
9934.5
9oot
-865.4
10.5
23.2
60
348
148
00.9
91
01631.6
86789.3
76781.3
30.0
023.7
94.6
84.1
94.2
4lo
g-s
licin
gbvslt
7972
20
305.2
5931.6
7oot
-868.3
30.5
23.2
13
73
64
00.4
91
01755.2
16705.6
16700.6
32.3
5158.4
731.7
84.2
04.2
5...servers
sla
pd
avc149923
10
325.6
4886.7
7oot
-913.2
30.4
93.3
30
67
67
00.1
61
01764.7
36928.0
36925.1
50.3
049.4
313.0
54.2
14.2
6...e
st
v7
r7
vr10
c1
s32506
10
1754.9
5864.5
0oot
-935.5
00.4
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.2
24.2
7lo
g-s
licin
gbvslt
9458
20
539.1
6833.9
7oot
-966.0
30.4
62.9
944
44
00.6
31
01656.3
46188.6
56184.7
014.9
4155.4
963.1
54.2
34.2
8...test
v7
r7
vr1
c1
s24449
20
679.7
0826.2
8oot
-973.7
20.4
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.2
44.2
9...servers
sla
pd
avc149922
10
325.6
8788.4
7oot
-1011.5
30.4
43.3
29
67
67
00.2
11
01706.4
76916.3
86913.5
40.3
051.0
213.1
74.2
54.3
0...e
st
v5
r10
vr1
c1
s19145
10
316.4
9780.8
2oot
-1019.1
80.4
30.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.2
64.3
1...y
erbie
re
countbitssrl0
32
20
387.0
6713.5
6oot
-1086.4
40.4
00.5
10
0.0
00.0
00.0
00.0
00.0
04.2
74.3
2lo
g-s
licin
gbvult
8991
20
542.6
1709.9
0oot
-1090.1
00.3
92.9
846
44
00.3
50
01711.1
96016.2
86012.5
30.1
8171.6
573.3
74.2
84.3
3lo
g-s
licin
gbvult
7994
20
230.1
0691.4
3oot
-1108.5
70.3
83.3
10
64
64
00.2
70
01382.7
76679.8
66675.9
12.4
0155.6
642.8
24.2
94.3
4lo
g-s
licin
gbvslt
6486
20
191.6
5687.4
2oot
-1112.5
80.3
82.9
16
112
60
00.4
91
01609.9
66031.9
56025.7
90.0
0398.6
824.0
34.3
04.3
5lo
g-s
licin
gbvult
6997
20
201.1
9669.2
9oot
-1130.7
10.3
73.7
10
53
53
00.2
70
01051.2
86652.7
86649.3
07.8
0108.6
541.8
44.3
1
Contin
ued
on
next
page
99
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.3
6...e
st
v5
r10
vr5
c1
s13195
10
337.5
5662.6
0oot
-1137.4
00.3
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
24.3
7...1
alt
true.B
V.c
.cil.c
.17
20
oot
621.7
5oot
-1178.2
50.3
53.8
11
24
23
01126.5
426
0468.5
12354.5
02334.7
10.1
2105.9
718.5
44.3
34.3
8lo
g-s
licin
gbvult
6000
20
406.3
8608.3
3oot
-1191.6
70.3
43.0
16
98
71
00.2
70
01761.6
06646.6
86642.1
30.3
9125.0
026.0
74.3
44.3
9lo
g-s
licin
gbvslt
8715
20
342.5
6598.5
0oot
-1201.5
00.3
32.9
945
45
00.6
31
01756.7
76395.2
56391.3
912.2
6155.4
467.3
24.3
54.4
0...test
v7
r7
vr1
c1
s22845
10
316.6
3593.4
8oot
-1206.5
20.3
30.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
64.4
1lfsr
lfsr
008
127
112
20
176.0
0591.6
3oot
-1208.3
70.3
33.4
819
19
01435.9
71
0277.7
91145.7
91133.8
11.5
081.2
212.5
94.3
74.4
2lo
g-s
licin
gbvslt
7229
20
217.0
1590.3
8oot
-1209.6
20.3
33.3
20
122
65
00.4
91
01538.1
95682.9
85675.7
20.0
095.9
417.8
24.3
84.4
3...e
st
v5
r15
vr1
c1
s26845
20
619.0
9589.8
4oot
-1210.1
60.3
30.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
94.4
4lfsr
lfsr
008
143
112
20
205.3
9587.5
9oot
-1212.4
10.3
30.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
04.4
5...e
st
v5
r10
vr1
c1
s13516
10
231.3
6574.5
3oot
-1225.4
70.3
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
14.4
6...tcom
p09
stall
bv
form
ula
20
609.5
3493.1
7oot
-1306.8
30.2
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
24.4
7float
mul03
30
420
514.8
0488.9
7oot
-1311.0
30.2
73.6
70
431
146
093.9
928
01163.5
16328.6
46308.4
10.0
061.7
94.1
44.4
34.4
8float
newton.5
.3.i
20
415.2
8486.8
0oot
-1313.2
00.2
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
44.4
9...test
v5
r10
vr5
c1
s8690
20
353.9
3485.9
1oot
-1314.0
90.2
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
54.5
0float
newton.2
.3.i
20
289.3
8445.9
2oot
-1354.0
80.2
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
64.5
1float
newton.6
.3.i
10
564.0
4437.0
6oot
-1362.9
40.2
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
74.5
2lfsr
lfsr
008
143
128
20
261.7
9428.5
3oot
-1371.4
70.2
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
84.5
3...test
v7
r12
vr5
c1
s8938
10
615.1
6403.9
4oot
-1396.0
60.2
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.3
04.5
4VS3
VS3-b
enchm
ark-A
610
111.9
3397.0
9oot
-1402.9
10.2
22.0
82
613
131
00.9
01
01737.4
66516.7
46506.0
90.0
021.9
33.5
54.4
94.5
5...test
v7
r7
vr5
c1
s14675
10
364.5
5393.9
7oot
-1406.0
30.2
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
04.5
6...test
v7
r7
vr5
c1
s3582
10
326.7
8392.5
6oot
-1407.4
40.2
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
02.7
4.5
7lfsr
lfsr
008
015
128
20
277.3
6380.8
7oot
-1419.1
30.2
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
14.5
8lfsr
lfsr
008
095
128
20
148.8
9366.3
4oot
-1433.6
60.2
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
34.5
9float
newton.1
.3.i
20
385.3
4360.0
3oot
-1439.9
70.2
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
44.6
0tacas07
Y86
btnft
20
135.7
7186.2
1oom
-1446.6
60.1
16.8
0.0
00.0
00.0
00.0
00.0
04.7
64.6
1brum
mayerbie
re3
min
and256
20
289.6
6328.3
0oot
-1471.7
00.1
81.1
56
33
93.0
90
01660.2
31864.2
91859.1
90.8
3833.0
2133.1
64.5
54.6
2lfsr
lfsr
008
111
112
20
148.6
0324.7
8oot
-1475.2
20.1
82.9
924
24
01424.6
11
0320.8
81070.1
41061.5
75.5
155.5
411.2
64.5
64.6
3float
newton.4
.3.i
20
334.8
4321.3
7oot
-1478.6
30.1
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
74.6
4float
newton.3
.3.i
20
355.4
9314.2
5oot
-1485.7
50.1
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
04.6
5lfsr
lfsr
008
159
080
20
194.3
3310.3
5oot
-1489.6
50.1
73.3
10
23
23
0777.7
91
0675.9
33518.5
43507.2
04.5
2157.4
329.5
74.6
14.6
6...y
erbie
re3
min
andm
axor064
20
207.7
9308.5
2oot
-1491.4
80.1
74.0
61
110
14
025.3
90
01695.8
52457.5
42412.0
10.1
2111.7
68.8
14.6
24.6
7lfsr
lfsr
008
159
112
20
275.9
4300.0
3oot
-1499.9
70.1
73.6
716
16
01437.0
81
0286.0
71066.8
61054.2
01.4
578.8
915.4
64.6
34.6
8...test
v5
r5
vr10
c1
s7194
10
141.8
4299.2
6oot
-1500.7
40.1
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.3
14.6
9lfsr
lfsr
008
143
096
20
173.9
4291.3
7oot
-1508.6
30.1
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
54.7
0...test
v5
r15
vr1
c1
s8236
20
927.6
6285.1
7oot
-1514.8
30.1
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.7
4.7
1lfsr
lfsr
008
015
112
20
388.4
4270.5
7oot
-1529.4
30.1
53.3
41
223
223
01406.5
71
0366.1
91426.8
91364.8
10.1
129.1
71.3
14.6
64.7
2...test
v5
r5
vr5
c1
s2800
10
228.1
4268.6
0oot
-1531.4
00.1
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
74.7
3lfsr
lfsr
004
143
128
20
116.2
1259.8
5oot
-1540.1
50.1
43.8
12
34
34
01496.5
71
0215.8
01013.4
6994.8
30.3
142.9
65.9
64.6
84.7
4lfsr
lfsr
008
079
128
20
107.7
6256.8
8oot
-1543.1
20.1
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
94.7
5lfsr
lfsr
008
095
096
20
105.5
2255.9
1oot
-1544.0
90.1
43.4
13
44
44
01035.3
01
0702.7
22728.3
22716.4
73.6
766.1
314.2
84.7
04.7
6...e
st
v7
r7
vr10
c1
s24535
10
237.8
8246.3
0oot
-1553.7
00.1
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.1
04.7
7lfsr
lfsr
008
111
096
20
133.3
2238.7
2oot
-1561.2
80.1
33.3
11
28
28
01005.3
61
0654.1
02568.2
42555.0
74.4
868.2
417.9
64.7
14.7
8brum
mayerbie
re3
maxor064
20
209.1
4224.2
7oot
-1575.7
30.1
21.9
68
69
343
5.9
80
01556.4
21701.6
51682.6
80.1
7370.9
99.4
54.7
24.7
9...e
st
v5
r15
vr1
c1
s32559
20
163.7
5217.6
0oot
-1582.4
00.1
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.3
44.8
0...sm
tcom
p09
sw
bv
form
ula
20
202.4
2210.0
8oot
-1589.9
20.1
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
34.8
1...sm
tcom
p09
lfbv
form
ula
20
180.2
4207.7
1oot
-1592.2
90.1
23.6
33
203
108
01076.2
853
0700.6
12723.9
92701.1
60.0
058.7
14.0
22.1
04.8
2lfsr
lfsr
008
127
128
20
212.8
0198.4
4oot
-1601.5
60.1
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
44.8
3float
newton.5
.2.i
20
249.5
3196.3
7oot
-1603.6
30.1
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.3
74.8
4tacas07
Y86
std
20
129.6
5193.9
7oot
-1606.0
30.1
10.9
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.2
74.8
5lfsr
lfsr
008
143
064
20
103.4
9192.9
4oot
-1607.0
60.1
13.7
11
35
35
01474.9
81
0291.2
21002.2
8986.6
71.9
427.0
66.8
64.7
54.8
6...test
v5
r5
vr5
c1
s24018
10
182.8
1186.5
3oot
-1613.4
70.1
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.9
Contin
ued
on
next
page
100
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.8
7float
newton.2
.2.i
20
143.1
8178.6
2oot
-1621.3
80.1
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
74.8
8lfsr
lfsr
008
063
128
20
oot
178.5
0oot
-1621.5
00.1
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
84.8
9float
newton.1
.2.i
20
139.9
8173.6
3oot
-1626.3
70.1
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
94.9
0float
newton.4
.2.i
20
143.2
5155.9
4oot
-1644.0
60.0
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
04.9
1float
newton.3
.2.i
20
186.5
6147.7
7oot
-1652.2
30.0
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
14.9
2...test
v5
r5
vr1
c1
s14623
10
143.2
0147.5
7oot
-1652.4
30.0
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.2
44.9
3lfsr
lfsr
008
159
096
20
275.8
6145.8
5oot
-1654.1
50.0
83.1
920
20
01014.9
61
0489.1
22713.3
72705.9
09.7
281.0
225.3
64.8
24.9
4lfsr
lfsr
008
063
096
20
oot
144.8
3oot
-1655.1
70.0
83.4
20
74
58
01041.1
51
0630.0
12716.1
62697.5
50.3
845.2
77.9
73.3
84.9
5lfsr
lfsr
008
143
080
20
112.1
3144.7
5oot
-1655.2
50.0
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
34.9
6lfsr
lfsr
008
095
112
20
129.0
8142.4
7oot
-1657.5
30.0
83.3
10
28
28
01429.9
41
0307.4
51090.8
21078.8
13.5
529.6
29.6
54.8
44.9
7lfsr
lfsr
008
127
096
20
113.0
4135.2
3oot
-1664.7
70.0
83.1
16
48
48
01037.7
21
0675.9
32675.9
62663.1
41.9
547.9
011.8
94.8
54.9
8brum
mayerbie
re3
maxxor032
20
143.9
5127.0
4oot
-1672.9
60.0
71.7
70
481
156
00.7
90
01782.8
66443.2
06426.3
20.0
139.1
74.8
74.8
64.9
9lfsr
lfsr
008
015
096
20
272.7
5123.5
9oot
-1676.4
10.0
73.4
30
131
131
01020.6
21
0249.6
82996.4
02957.9
00.2
434.1
64.1
64.8
74.1
00
brum
mayerbie
re
countbits064
20
129.3
4121.6
1oot
-1678.3
90.0
73.2
10
0.0
00.0
00.0
00.0
00.0
04.8
84.1
01
lfsr
lfsr
008
111
128
20
160.6
8112.3
2oot
-1687.6
80.0
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
94.1
02
lfsr
lfsr
008
047
096
20
160.4
099.3
6oot
-1700.6
40.0
63.6
24
107
85
01033.5
71
0725.1
42818.7
92792.7
40.1
381.8
96.5
14.9
04.1
03
brum
mayerbie
re3
min
xor064
20
111.8
493.3
5oot
-1706.6
50.0
51.2
76
438
78
04.4
60
01769.3
45288.7
95249.0
50.0
145.8
65.2
54.9
14.1
04
lfsr
lfsr
008
159
064
20
149.5
283.4
8oot
-1716.5
20.0
53.3
12
37
29
0489.0
61
01099.6
34780.8
24772.7
92.1
9130.2
328.9
74.9
24.1
05
lfsr
lfsr
008
159
048
20
100.5
644.3
2oot
-1755.6
80.0
23.5
14
57
41
0267.6
91
01282.5
55763.9
05754.6
94.3
1113.7
127.7
14.9
34.1
06
float
mult2.c
.50
10
109.3
940.0
2oot
-1759.9
80.0
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.1
1
5.1
...e
rbie
re4
unconstrain
ed06
0oom
oom
oom
0.8
81.0
26.8
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
99
5.2
...e
rbie
re4
unconstrain
ed07
0oom
oom
oom
0.7
01.0
16.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.4
5.3
...e
rbie
re4
unconstrain
ed01
0oom
oom
oom
0.4
81.0
16.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.3
5.4
...e
rbie
re4
unconstrain
ed04
0oom
oom
oom
0.1
81.0
06.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
97
5.5
core
ext
con
052
002
1024
20
194.6
0oot
oot
0.0
01.0
00.1
11
10
10.4
44
0112.8
3112.8
3112.4
8112.8
3112.8
3112.8
31.6
5.6
core
ext
con
056
002
1024
20
279.9
8oot
oot
0.0
01.0
00.1
11
10
11.9
94
0122.1
5122.1
5121.7
8122.1
5122.1
5122.1
51.4
5.7
core
ext
con
060
001
1024
20
110.6
0oot
oot
0.0
01.0
00.1
11
10
1.0
22
0140.8
8140.8
8140.4
9140.8
8140.8
8140.8
85.7
5.8
core
ext
con
060
002
1024
20
220.7
1oot
oot
0.0
01.0
00.1
11
10
13.7
54
0130.0
3130.0
3129.6
4130.0
3130.0
3130.0
31.5
5.9
core
ext
con
060
256
1024
20
101.7
3oot
oot
0.0
01.0
00.2
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.8
5.1
0core
ext
con
064
001
1024
20
128.2
5oot
oot
0.0
01.0
00.1
11
10
1.1
62
0150.1
5150.1
5149.7
3150.1
5150.1
5150.1
55.9
5.1
1core
ext
con
064
002
1024
20
344.2
9oot
oot
0.0
01.0
00.1
11
10
15.6
54
0145.7
3145.7
3145.3
1145.7
3145.7
3145.7
31.8
5.1
2fft
Sz1024
34824
0oot
oot
oot
0.0
01.0
03.3
40
216
216
0118.2
10
0483.4
36561.0
56528.1
90.2
638.2
15.9
35.1
05.1
3fft
Sz512
15128
00
oot
oot
oot
0.0
01.0
03.1
70
413
413
022.6
10
0471.5
57022.9
26986.3
80.1
416.9
72.4
85.1
15.1
4fft
Sz512
15128
10
oot
oot
oot
0.0
01.0
03.2
70
411
411
022.6
30
0565.8
37025.4
86989.4
40.1
517.1
72.5
05.1
25.1
5fft
Sz512
15128
20
oot
oot
oot
0.0
01.0
03.3
70
428
428
021.3
11
0561.1
37049.6
77018.9
20.1
716.6
72.4
85.1
35.1
6fft
Sz512
15128
40
oot
oot
oot
0.0
01.0
03.1
70
416
416
022.6
20
0477.8
77040.1
17003.7
20.1
517.6
22.4
95.1
45.1
7fft
Sz512
15128
50
oot
oot
oot
0.0
01.0
03.2
70
413
413
022.6
10
0577.6
47017.3
06981.0
40.1
517.1
92.5
05.1
55.1
8fft
Sz512
15128
60
oot
oot
oot
0.0
01.0
03.1
70
434
434
021.4
31
0454.0
87050.9
67020.2
30.1
416.6
52.4
65.1
65.1
9...st
v5
r10
vr10
c1
s15708
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
75.2
0...e
st
v5
r10
vr10
c1
s7608
20
1199.1
4oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
5.2
1...e
st
v5
r15
vr5
c1
s23844
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
85.2
2...st
v7
r12
vr10
c1
s15994
10
1367.7
1oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
95.2
3...test
v7
r12
vr1
c1
s703
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
5.2
4...e
st
v7
r12
vr5
c1
s14336
10
933.1
7oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
05.2
5...e
st
v7
r12
vr5
c1
s29826
10
896.5
0oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
15.2
6...e
st
v7
r17
vr1
c1
s23882
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
25.2
7...e
st
v7
r17
vr1
c1
s30331
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
35.2
8...fyIn
terle
avedM
odM
ult-1
28
0oot
oot
oot
0.0
01.0
04.1
12
88
018.5
52
01592.9
15654.8
45650.2
20.8
4388.5
274.4
15.2
45.2
9...ffy
Interle
avedM
odM
ult-3
20
oot
oot
oot
0.0
01.0
03.3
29
116
116
01.1
42
01752.9
56483.8
16480.3
60.2
2131.0
911.5
25.2
55.3
0lfsr
lfsr
008
159
128
20
285.7
4oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
6
Contin
ued
on
next
page
101
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.3
1lo
g-s
licin
gbvadd
18209
20
101.2
6oot
oot
0.0
01.0
01.2
11
10
0.0
00
068.3
468.3
463.3
968.3
468.3
468.3
45.2
75.3
2lo
g-s
licin
gbvadd
19096
20
109.7
8oot
oot
0.0
01.0
01.2
11
10
0.0
00
075.3
775.3
769.1
375.3
775.3
775.3
75.2
85.3
3lo
g-s
licin
gbvadd
19983
20
121.5
9oot
oot
0.0
01.0
01.2
11
10
0.0
00
081.3
281.3
275.2
181.3
281.3
281.3
25.2
95.3
4lo
g-s
licin
gbvadd
20870
20
132.0
0oot
oot
0.0
01.0
01.2
11
10
0.0
00
089.1
489.1
482.3
189.1
489.1
489.1
45.3
05.3
5lo
g-s
licin
gbvadd
21757
20
165.0
7oot
oot
0.0
01.0
01.4
11
10
0.0
00
096.6
496.6
488.5
896.6
496.6
496.6
45.3
15.3
6lo
g-s
licin
gbvadd
22644
20
156.0
3oot
oot
0.0
01.0
01.4
11
10
0.0
00
0104.1
9104.1
994.4
2104.1
9104.1
9104.1
95.3
25.3
7lo
g-s
licin
gbvadd
23531
20
167.4
5oot
oot
0.0
01.0
01.4
11
10
0.0
00
0109.5
2109.5
299.0
6109.5
2109.5
2109.5
25.3
35.3
8lo
g-s
licin
gbvadd
24418
20
198.0
9oot
oot
0.0
01.0
01.4
11
10
0.0
00
0120.7
9120.7
9107.5
3120.7
9120.7
9120.7
95.3
45.3
9lo
g-s
licin
gbvadd
25305
20
186.3
4oot
oot
0.0
01.0
01.4
11
10
0.0
00
0125.1
2125.1
2111.1
7125.1
2125.1
2125.1
25.3
55.4
0lo
g-s
licin
gbvadd
26192
20
202.5
2oot
oot
0.0
01.0
01.6
11
10
0.0
00
0131.6
3131.6
3117.8
8131.6
3131.6
3131.6
35.3
65.4
1lo
g-s
licin
gbvadd
27079
20
216.4
1oot
oot
0.0
01.0
01.6
11
10
0.0
00
0140.7
3140.7
3122.6
0140.7
3140.7
3140.7
35.3
75.4
2lo
g-s
licin
gbvadd
27966
20
232.0
1oot
oot
0.0
01.0
01.7
11
10
0.0
00
0147.9
9147.9
9128.6
1147.9
9147.9
9147.9
95.3
85.4
3lo
g-s
licin
gbvadd
28853
20
245.3
3oot
oot
0.0
01.0
01.7
11
10
0.0
00
0154.6
7154.6
7132.2
5154.6
7154.6
7154.6
75.3
95.4
4lo
g-s
licin
gbvadd
29740
20
264.9
3oot
oot
0.0
01.0
01.7
11
10
0.0
00
0160.6
6160.6
6136.6
8160.6
6160.6
6160.6
65.4
05.4
5lo
g-s
licin
gbvm
ul13
20
428.6
3oot
oot
0.0
01.0
00.1
22
21
0.0
00
00.9
01742.2
11742.1
80.2
71741.0
4435.5
55.4
15.4
6lo
g-s
licin
gbvm
ul14
20
1477.5
0oot
oot
0.0
01.0
00.2
22
20
0.0
00
00.8
91.1
91.1
60.2
90.5
60.4
05.4
25.4
7lo
g-s
licin
gbvm
ul15
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00
00.9
31.2
51.2
30.3
20.5
80.4
25.4
35.4
8lo
g-s
licin
gbvm
ul16
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00
00.8
51.1
61.1
30.3
00.5
40.3
95.4
45.4
9lo
g-s
licin
gbvm
ul17
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00
00.9
21.2
31.1
10.3
10.5
70.4
15.4
55.5
0lo
g-s
licin
gbvm
ul18
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00
00.9
41.2
11.1
00.2
70.5
70.4
05.4
65.5
1lo
g-s
licin
gbvsdiv
19
0oot
oot
oot
0.0
01.0
01.3
14
70
53
19
0.1
20
01496.5
63763.1
43761.6
10.0
0742.8
817.6
75.4
75.5
2lo
g-s
licin
gbvsdiv
20
0oot
oot
oot
0.0
01.0
01.3
13
68
54
14
0.1
20
01428.9
52759.0
92757.4
70.0
1776.4
313.7
35.4
85.5
3lo
g-s
licin
gbvsdiv
21
0oot
oot
oot
0.0
01.0
01.4
10
61
57
10.1
20
010.9
078.7
277.3
10.0
025.2
70.4
75.4
95.5
4lo
g-s
licin
gbvsdiv
22
0oot
oot
oot
0.0
01.0
01.4
10
55
53
20.1
20
018.0
5135.4
9134.0
70.0
066.1
60.9
15.5
05.5
5lo
g-s
licin
gbvsdiv
23
0oot
oot
oot
0.0
01.0
01.5
10
58
56
30.1
20
011.7
3473.6
7472.2
50.0
0382.9
42.8
95.5
15.5
6lo
g-s
licin
gbvsdiv
24
0oot
oot
oot
0.0
01.0
01.7
11
64
64
00.1
20
018.9
368.8
567.3
00.0
11.1
60.3
95.5
25.5
7lo
g-s
licin
gbvsdiv
25
0oot
oot
oot
0.0
01.0
01.5
10
58
57
30.1
20
012.3
01572.3
71570.7
80.0
0553.9
29.5
35.5
35.5
8lo
g-s
licin
gbvslt
10201
20
552.9
2oot
oot
0.0
01.0
02.9
944
44
00.6
31
01528.0
36488.6
56484.7
20.1
4224.6
165.5
45.5
45.5
9lo
g-s
licin
gbvslt
11687
20
665.8
6oot
oot
0.0
01.0
02.9
942
42
00.6
31
01548.5
06025.5
66021.5
910.1
6217.2
062.1
25.5
55.6
0lo
g-s
licin
gbvslt
13173
20
801.4
6oot
oot
0.0
01.0
02.9
940
40
00.6
31
01500.0
66249.7
26245.8
10.5
7206.7
363.1
35.5
65.6
1lo
g-s
licin
gbvslt
13916
20
908.9
9oot
oot
0.0
01.0
02.9
939
39
00.6
31
01573.7
06495.9
96492.0
04.3
9159.7
463.6
95.5
75.6
2lo
g-s
licin
gbvslt
14659
20
838.9
9oot
oot
0.0
01.0
03.0
938
38
00.6
31
01646.8
86673.9
26669.8
911.2
3174.6
266.0
85.5
85.6
3lo
g-s
licin
gbvslt
16145
20
1188.5
6oot
oot
0.0
01.0
03.1
10
36
36
00.6
31
01612.5
85341.8
15337.4
20.1
1512.0
248.1
25.5
95.6
4lo
g-s
licin
gbvslt
16888
20
1502.5
6oot
oot
0.0
01.0
02.9
521
21
00.8
21
01508.0
45333.2
85326.5
630.8
7339.7
1156.8
65.6
05.6
5lo
g-s
licin
gbvslt
17631
20
1750.5
6oot
oot
0.0
01.0
02.6
416
16
00.8
11
0927.8
85521.8
15516.0
049.9
7382.5
1197.2
15.6
15.6
6lo
g-s
licin
gbvslt
18374
0oot
oot
oot
0.0
01.0
01.4
38
80
0.8
11
01027.7
75530.1
15525.7
0123.1
5658.7
9425.3
95.6
25.6
7lo
g-s
licin
gbvslt
19117
0oot
oot
oot
0.0
01.0
03.0
622
22
00.8
11
01627.8
26481.3
56474.7
848.0
6226.5
1127.0
95.6
35.6
8lo
g-s
licin
gbvslt
19860
0oot
oot
oot
0.0
01.0
03.0
621
21
00.8
21
01740.4
96593.6
46587.3
140.8
6293.5
4140.2
95.6
45.6
9lo
g-s
licin
gbvsm
od
16
0oot
oot
oot
0.0
01.0
00.5
26
122
57
66
0.1
00
01485.7
43634.6
73633.4
60.0
0230.1
98.1
75.6
55.7
0lo
g-s
licin
gbvsm
od
17
0oot
oot
oot
0.0
01.0
01.3
17
82
55
34
0.1
50
01656.7
83763.7
03761.8
20.0
0594.8
713.9
45.6
65.7
1lo
g-s
licin
gbvsm
od
18
0oot
oot
oot
0.0
01.0
01.4
13
70
60
90.1
50
01533.4
62236.5
42235.0
10.0
01032.9
510.9
15.6
75.7
2lo
g-s
licin
gbvsm
od
19
0oot
oot
oot
0.0
01.0
01.5
12
68
62
11
0.1
50
0143.3
22494.4
42492.8
60.0
0878.4
012.2
35.6
85.7
3lo
g-s
licin
gbvsm
od
20
0oot
oot
oot
0.0
01.0
01.4
10
58
56
40.1
50
0850.3
81380.7
41379.2
50.0
0840.9
98.3
75.6
95.7
4lo
g-s
licin
gbvsrem
16
0oot
oot
oot
0.0
01.0
00.4
20
83
48
55
0.0
80
01744.9
84140.7
54139.9
50.0
0349.1
713.4
05.7
05.7
5lo
g-s
licin
gbvsrem
17
0oot
oot
oot
0.0
01.0
01.0
16
69
46
39
0.1
10
01694.7
74564.3
74562.7
80.0
0547.4
619.1
05.7
15.7
6lo
g-s
licin
gbvsrem
18
0oot
oot
oot
0.0
01.0
01.2
12
59
52
15
0.1
10
01418.8
33814.7
63813.4
50.0
0804.1
920.7
35.7
25.7
7lo
g-s
licin
gbvsrem
19
0oot
oot
oot
0.0
01.0
01.4
10
56
51
70.1
10
011.9
74839.0
44837.7
80.0
11188.5
930.0
65.7
35.7
8lo
g-s
licin
gbvsrem
20
0oot
oot
oot
0.0
01.0
01.1
954
50
10.1
10
08.3
7689.3
1688.2
40.0
0657.5
56.2
75.7
45.7
9lo
g-s
licin
gbvsub
06991
20
307.6
8oot
oot
0.0
01.0
00.2
11
10
0.0
00
07.3
57.3
54.6
97.3
57.3
57.3
55.7
55.8
0lo
g-s
licin
gbvsub
07988
20
321.6
9oot
oot
0.0
01.0
00.4
11
10
0.0
00
08.3
18.3
14.8
68.3
18.3
18.3
15.7
65.8
1lo
g-s
licin
gbvsub
08985
20
378.5
6oot
oot
0.0
01.0
00.1
11
10
0.0
00
09.1
99.1
94.9
29.1
99.1
99.1
95.7
7
Contin
ued
on
next
page
102
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.8
2lo
g-s
licin
gbvsub
09982
20
328.0
6oot
oot
0.0
01.0
00.2
11
10
0.0
00
010.1
810.1
85.1
310.1
810.1
810.1
85.7
85.8
3lo
g-s
licin
gbvsub
10979
20
161.2
1oot
oot
0.0
01.0
00.1
11
10
0.0
00
011.2
211.2
25.1
711.2
211.2
211.2
25.7
95.8
4lo
g-s
licin
gbvsub
11976
20
664.6
3oot
oot
0.0
01.0
00.3
11
10
0.0
00
012.7
712.7
75.3
512.7
712.7
712.7
75.8
05.8
5lo
g-s
licin
gbvsub
12973
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
013.9
413.9
45.3
613.9
413.9
413.9
45.8
15.8
6lo
g-s
licin
gbvsub
13970
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
015.0
615.0
65.6
115.0
615.0
615.0
65.8
25.8
7lo
g-s
licin
gbvsub
14967
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
016.7
616.7
65.5
716.7
616.7
616.7
65.8
35.8
8lo
g-s
licin
gbvsub
15964
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
018.7
718.7
75.7
418.7
718.7
718.7
75.8
45.8
9lo
g-s
licin
gbvsub
16961
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
020.4
320.4
35.8
420.4
320.4
320.4
35.8
55.9
0lo
g-s
licin
gbvsub
17958
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00
015.9
415.9
40.1
915.9
415.9
415.9
45.8
65.9
1lo
g-s
licin
gbvsub
18955
0oot
oot
oot
0.0
01.0
00.6
11
10
0.0
00
017.9
117.9
10.2
017.9
117.9
117.9
15.8
75.9
2lo
g-s
licin
gbvsub
19952
0oot
oot
oot
0.0
01.0
00.6
11
10
0.0
00
019.6
119.6
10.2
119.6
119.6
119.6
15.8
85.9
3lo
g-s
licin
gbvsub
20949
0oot
oot
oot
0.0
01.0
00.6
11
10
0.0
00
023.0
623.0
60.2
223.0
623.0
623.0
65.8
95.9
4lo
g-s
licin
gbvsub
21946
0oot
oot
oot
0.0
01.0
00.7
11
10
0.0
00
024.4
824.4
80.2
324.4
824.4
824.4
85.9
05.9
5lo
g-s
licin
gbvsub
22943
0oot
oot
oot
0.0
01.0
00.7
11
10
0.0
00
027.9
827.9
80.2
427.9
827.9
827.9
85.9
15.9
6lo
g-s
licin
gbvsub
23940
0oot
oot
oot
0.0
01.0
00.8
11
10
0.0
00
032.7
732.7
70.2
532.7
732.7
732.7
75.9
25.9
7lo
g-s
licin
gbvudiv
19
0oot
oot
oot
0.0
01.0
00.8
729
28
10
0.0
80
0376.1
43630.2
63629.2
30.0
01346.7
846.5
45.9
35.9
8lo
g-s
licin
gbvudiv
20
0oot
oot
oot
0.0
01.0
00.9
832
30
90.0
80
01150.8
13231.6
43230.5
20.0
01130.1
836.3
15.9
45.9
9lo
g-s
licin
gbvudiv
21
0oot
oot
oot
0.0
01.0
01.0
732
30
50.0
80
05.9
4497.9
1496.8
10.0
0410.6
56.5
55.9
55.1
00
log-s
licin
gbvudiv
22
0oot
oot
oot
0.0
01.0
01.0
733
32
40.0
80
06.2
31876.5
61875.4
00.0
01642.6
424.6
95.9
65.1
01
log-s
licin
gbvudiv
23
0oot
oot
oot
0.0
01.0
01.0
733
32
40.0
80
06.3
5786.1
6785.0
50.0
1677.0
810.3
45.9
75.1
02
log-s
licin
gbvudiv
24
0oot
oot
oot
0.0
01.0
01.0
628
26
00.0
80
05.5
517.7
516.6
30.0
51.3
20.3
35.9
85.1
03
log-s
licin
gbvudiv
25
0oot
oot
oot
0.0
01.0
01.2
736
32
10.0
80
06.7
21553.8
91552.5
70.0
21529.8
220.4
55.9
95.1
04
log-s
licin
gbvult
10985
20
556.6
2oot
oot
0.0
01.0
02.7
740
40
00.3
50
01554.5
46317.6
86313.7
73.6
7222.3
988.9
85.1
00
5.1
05
log-s
licin
gbvult
11982
20
518.1
6oot
oot
0.0
01.0
02.9
734
34
00.3
50
01521.6
46261.3
36257.5
61.1
9172.8
383.4
85.1
01
5.1
06
log-s
licin
gbvult
12979
20
893.4
9oot
oot
0.0
01.0
02.9
940
40
00.3
50
01753.0
36397.0
36392.8
92.2
0197.4
467.3
45.1
02
5.1
07
log-s
licin
gbvult
13976
20
893.7
3oot
oot
0.0
01.0
02.9
839
39
00.3
50
01718.6
46555.1
26550.9
02.9
1178.3
378.0
45.1
03
5.1
08
log-s
licin
gbvult
15970
20
1070.6
3oot
oot
0.0
01.0
03.1
10
36
36
00.3
50
01350.6
24872.2
44867.5
40.2
4194.4
043.1
25.1
04
5.1
09
log-s
licin
gbvult
16967
0oot
oot
oot
0.0
01.0
02.9
521
21
00.4
60
01378.1
25835.0
65829.1
452.9
5259.3
6145.8
85.1
05
5.1
10
log-s
licin
gbvult
17964
0oot
oot
oot
0.0
01.0
02.9
521
21
00.4
50
01261.3
45638.1
95632.2
539.7
7253.3
7131.1
25.1
06
5.1
11
log-s
licin
gbvult
18961
0oot
oot
oot
0.0
01.0
02.4
416
16
00.4
50
01537.1
75454.0
25449.5
273.4
3508.2
8287.0
55.1
07
5.1
12
log-s
licin
gbvult
19958
0oot
oot
oot
0.0
01.0
03.0
822
22
00.4
40
01730.8
85664.3
95658.4
80.1
8481.2
871.7
05.1
08
5.1
13
log-s
licin
gbvult
20955
0oot
oot
oot
0.0
01.0
03.0
721
21
00.4
50
01187.2
15162.1
85156.4
110.7
9520.6
875.9
15.1
09
5.1
14
log-s
licin
gbvult
21952
0oot
oot
oot
0.0
01.0
03.0
821
21
00.4
50
01502.0
05927.4
65921.7
93.2
6470.0
273.1
85.1
10
5.1
15
log-s
licin
gbvult
22949
0oot
oot
oot
0.0
01.0
03.0
821
21
00.4
50
01009.1
25921.7
65915.9
112.7
1757.1
070.5
05.1
11
5.1
16
log-s
licin
gbvult
23946
0oot
oot
oot
0.0
01.0
02.9
720
20
00.4
40
0662.1
36185.7
56179.9
80.2
6666.1
279.3
05.1
12
5.1
17
log-s
licin
gbvult
24943
0oot
oot
oot
0.0
01.0
03.0
819
19
00.4
40
01105.4
54451.9
04446.2
70.1
9413.6
554.9
65.1
13
5.1
18
log-s
licin
gbvult
25940
0oot
oot
oot
0.0
01.0
03.0
819
19
00.4
50
0705.0
73550.8
33545.2
70.2
1700.4
642.7
85.1
14
5.1
19
log-s
licin
gbvult
9988
20
555.0
3oot
oot
0.0
01.0
02.9
844
44
00.3
50
01623.9
16332.1
76328.2
04.7
6205.9
671.1
55.1
15
5.1
20
log-s
licin
gbvurem
17
0oot
oot
oot
0.0
01.0
00.5
830
24
16
0.0
80
0892.3
13711.9
63711.3
10.0
0773.3
540.7
95.1
16
5.1
21
log-s
licin
gbvurem
18
0oot
oot
oot
0.0
01.0
00.7
729
29
70.0
80
01109.7
52544.0
62543.4
70.0
01105.6
435.8
35.1
17
5.1
22
log-s
licin
gbvurem
19
0oot
oot
oot
0.0
01.0
00.6
629
27
20.0
80
04.0
2443.6
9443.1
10.0
0393.7
67.1
65.1
18
5.1
23
log-s
licin
gbvurem
20
0oot
oot
oot
0.0
01.0
00.8
732
30
10.0
80
04.9
0426.8
1426.2
10.0
0411.8
86.5
75.1
19
5.1
24
log-s
licin
gbvurem
21
0oot
oot
oot
0.0
01.0
00.8
630
30
00.0
80
04.2
515.3
414.7
00.1
60.9
20.2
55.1
20
5.1
25
log-s
licin
gbvurem
22
0oot
oot
oot
0.0
01.0
00.8
732
32
00.0
80
05.4
717.1
316.4
70.1
10.9
60.2
75.1
21
5.1
26
pip
epip
e-n
oabs.a
tla
s.q
fbv
20
1290.1
0oot
oot
0.0
01.0
00.7
56
419
45
01.0
81
01769.3
86305.1
26292.6
80.0
059.4
36.7
05.1
22
5.1
27
pspace
power2sum
.5500
20
496.1
6oot
oot
0.0
01.0
02.0
11
10
0.0
00
01208.0
91208.0
91207.7
21208.0
91208.0
91208.0
95.1
23
5.1
28
pspace
power2sum
.5898
20
343.5
7oot
oot
0.0
01.0
02.2
11
10
0.0
00
01471.2
21471.2
21470.8
31471.2
21471.2
21471.2
25.1
24
5.1
29
pspace
power2sum
.6296
20
434.7
6oot
oot
0.0
01.0
02.2
11
10
0.0
00
01780.2
01780.2
01779.7
61780.2
01780.2
01780.2
05.1
25
5.1
30
pspace
power2sum
.6495
20
504.5
1oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
26
5.1
31
pspace
power2sum
.6694
20
1005.8
1oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
27
5.1
32
pspace
power2sum
.6893
20
1102.0
2oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
28
Contin
ued
on
next
page
103
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.1
33
pspace
power2sum
.7092
20
452.5
0oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
29
5.1
34
pspace
power2sum
.7291
20
971.6
8oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
30
5.1
35
pspace
power2sum
.7490
20
505.6
9oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
31
5.1
36
pspace
power2sum
.7689
20
515.9
3oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
32
5.1
37
pspace
power2sum
.7888
20
1145.4
9oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
33
5.1
38
pspace
power2sum
.8087
0oot
oot
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0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
34
5.1
39
pspace
power2sum
.8286
20
898.4
1oot
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0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
35
5.1
40
pspace
power2sum
.8485
20
872.3
0oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
36
5.1
41
pspace
power2sum
.8684
20
1028.3
0oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
37
5.1
42
pspace
power2sum
.8883
20
1035.8
8oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
38
5.1
43
pspace
power2sum
.9082
20
1301.9
1oot
oot
0.0
01.0
02.1
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
39
5.1
44
pspace
power2sum
.9281
20
1338.5
3oot
oot
0.0
01.0
02.1
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
40
5.1
45
pspace
power2sum
.9480
20
751.8
3oot
oot
0.0
01.0
02.1
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
41
5.1
46
pspace
power2sum
.9679
0oot
oot
oot
0.0
01.0
02.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
42
5.1
47
pspace
power2sum
.9878
20
573.3
5oot
oot
0.0
01.0
02.1
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
43
5.1
48
RW
SExam
ple
15.txt
0oot
oot
oot
0.0
01.0
00.3
01
10
7.7
71
00.0
00.0
00.0
00.0
00.0
00.0
05.1
44
5.1
49
RW
SExam
ple
19.txt
0oot
oot
oot
0.0
01.0
00.3
01
10
35.5
10
00.0
00.0
00.0
00.0
00.0
00.0
05.1
45
5.1
50
RW
SExam
ple
20.txt
0oot
oot
oot
0.0
01.0
00.4
01
10
70.4
80
00.0
00.0
00.0
00.0
00.0
00.0
05.1
46
5.1
51
RW
SExam
ple
7.txt
0oot
oot
oot
0.0
01.0
00.1
01
10
1.7
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
47
5.1
52
uum
uum
12
0oot
oot
oot
0.0
01.0
00.6
10
0.0
00.0
00.0
00.0
00.0
05.1
48
5.1
53
uum
uum
16
0oot
oot
oot
0.0
01.0
01.5
50
368
238
02.1
21
01321.0
06864.7
06858.3
90.0
034.6
34.9
85.1
49
5.1
54
uum
uum
20
0oot
oot
oot
0.0
01.0
03.1
60
418
342
04.4
11
0783.6
46895.9
76883.7
80.0
038.6
83.4
45.1
50
5.1
55
uum
uum
24
0oot
oot
oot
0.0
01.0
03.4
50
250
248
06.3
51
01308.5
86816.2
76804.5
80.0
049.8
24.8
15.1
51
5.1
56
uum
uum
28
0oot
oot
oot
0.0
01.0
03.4
10
0.0
00.0
00.0
00.0
00.0
05.1
52
5.1
57
uum
uum
32
0oot
oot
oot
0.0
01.0
03.1
35
167
167
010.2
31
01776.3
16589.6
16579.0
40.3
0102.9
37.8
65.1
53
5.1
58
VS3
VS3-b
enchm
ark-A
10
0oot
oot
oot
0.0
01.0
03.5
45
207
207
01.1
71
01765.9
87004.5
16998.8
90.1
844.6
45.9
95.1
54
5.1
59
VS3
VS3-b
enchm
ark-A
11
0oot
oot
oot
0.0
01.0
03.5
47
213
211
01.1
11
01769.2
56964.7
16959.2
10.1
544.5
55.8
05.1
55
5.1
60
VS3
VS3-b
enchm
ark-A
12
0oot
oot
oot
0.0
01.0
03.3
34
176
176
01.1
91
01783.4
47073.0
07069.1
60.3
843.6
48.8
55.1
56
5.1
61
VS3
VS3-b
enchm
ark-A
13
0oot
oot
oot
0.0
01.0
03.2
30
111
111
01.5
71
0638.8
47007.2
97003.3
10.3
553.4
110.4
65.1
57
5.1
62
VS3
VS3-b
enchm
ark-A
80
oot
oot
oot
0.0
01.0
03.3
33
145
145
01.3
21
01782.5
87011.3
77007.7
80.3
745.3
89.4
65.1
58
5.1
63
VS3
VS3-b
enchm
ark-S
20
oot
oot
oot
0.0
01.0
03.5
20
42
42
01.3
71
01745.3
06793.8
86791.3
61.8
3110.2
722.9
55.1
59
5.1
64
...1
alt
true.B
V.c
.cil.c
.21
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
60
5.1
65
brum
mayerbie
re2
sm
ulo
v1bw48
0oot
oot
oot
0.0
01.0
01.3
48
311
67
40
0.8
60
01770.4
35466.7
45459.0
50.0
0104.9
56.6
15.1
61
5.1
66
...e
rbie
re3
isqrtaddeqcheck
0oot
oot
oot
0.0
01.0
03.3
60
413
248
13
0.1
90
0731.4
46722.5
56703.0
70.0
094.6
83.8
55.1
62
5.1
67
...m
ayerbie
re3
isqrtaddnoif
0oot
oot
oot
0.0
01.0
01.9
30
217
110
00.0
30
01353.7
86678.3
96669.4
60.0
2166.0
011.1
15.1
63
5.1
68
...m
ayerbie
re3
isqrteqcheck
0oot
oot
oot
0.0
01.0
03.1
50
337
228
00.1
90
01101.9
46614.9
56594.4
70.0
099.1
24.6
95.1
64
5.1
69
brum
mayerbie
re3
isqrtnoif
0oot
oot
oot
0.0
01.0
02.8
29
202
122
00.0
30
01717.0
56493.7
06483.7
00.0
0185.5
511.7
25.1
65
5.1
70
brum
mayerbie
re3
maxor128
0oot
oot
oot
0.0
01.0
00.9
10
11
38
24.0
40
01563.0
01694.9
21689.3
71.1
7266.0
458.4
55.1
66
5.1
71
brum
mayerbie
re3
maxor256
0oot
oot
oot
0.0
01.0
04.3
13
83
194.0
00
01296.5
12667.5
12644.7
80.7
1416.1
480.8
35.1
67
5.1
72
brum
mayerbie
re3
maxxor064
0oot
oot
oot
0.0
01.0
03.9
27
136
94
03.0
80
01760.7
56533.0
46517.8
70.0
2108.0
612.8
45.1
68
5.1
73
brum
mayerbie
re3
maxxor128
0oot
oot
oot
0.0
01.0
04.0
17
38
38
012.3
00
01771.2
66359.8
46343.4
40.4
2161.5
325.9
65.1
69
5.1
74
brum
mayerbie
re3
maxxor256
0oot
oot
oot
0.0
01.0
03.6
910
10
050.2
80
01203.8
35222.1
65206.8
41.2
5472.8
473.5
55.1
70
5.1
75
...b
iere3
maxxorm
axorand032
0oot
oot
oot
0.0
01.0
03.4
130
871
304
010.4
30
01713.6
76034.4
45997.9
60.0
046.3
11.9
75.1
71
5.1
76
...b
iere3
maxxorm
axorand064
0oot
oot
oot
0.0
01.0
03.8
66
419
82
041.6
70
01654.0
35214.5
05167.0
00.0
0109.4
94.5
75.1
72
5.1
77
...b
iere3
maxxorm
axorand128
0oot
oot
oot
0.0
01.0
03.8
15
31
25
0156.1
80
01434.0
65582.6
65548.8
20.0
8181.3
426.0
95.1
73
5.1
78
...b
iere3
maxxorm
axorand256
0oot
oot
oot
0.0
01.0
04.0
76
60
639.4
70
0877.7
02718.1
82688.0
921.6
8268.9
587.6
85.1
74
5.1
79
...y
erbie
re3
min
andm
axor128
0oot
oot
oot
0.0
01.0
04.5
17
67
17
198.0
50
01554.3
04846.0
94794.8
00.1
2372.8
827.3
85.1
75
5.1
80
...y
erbie
re3
min
andm
axor256
0oot
oot
oot
0.0
01.0
04.7
910
71
374.8
40
01194.2
93474.8
73448.5
68.1
6307.6
270.9
25.1
76
5.1
81
brum
mayerbie
re3
min
xor128
0oot
oot
oot
0.0
01.0
04.6
32
177
77
016.6
20
01739.9
76551.5
86517.9
80.0
8103.4
410.7
45.1
77
5.1
82
brum
mayerbie
re3
min
xor256
0oot
oot
oot
0.0
01.0
04.8
14
32
18
066.5
60
01406.8
35662.3
65625.1
60.1
7343.2
430.2
85.1
78
5.1
83
...e
rbie
re3
min
xorm
inand064
0oot
oot
oot
0.0
01.0
05.1
54
315
127
049.3
70
01728.1
96218.2
26155.6
70.0
157.7
95.4
55.1
79
Contin
ued
on
next
page
104
Table
C.2
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
with
FLR
(Parallel-In
crm
ental-FLR)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.1
84
...e
rbie
re3
min
xorm
inand128
0oot
oot
oot
0.0
01.0
04.8
16
47
43
0183.5
90
01234.4
05846.4
15818.9
70.0
7198.3
323.6
75.1
80
5.1
85
...e
rbie
re3
min
xorm
inand256
0oot
oot
oot
0.0
01.0
04.4
812
11
0720.9
40
0455.0
82255.0
82228.4
70.7
2121.5
832.6
85.1
81
5.1
86
brum
mayerbie
re3
mulh
s16
0oot
oot
oot
0.0
01.0
00.4
20
137
79
10.0
00
017.0
21676.5
41676.2
50.0
01605.0
44.6
35.1
82
5.1
87
brum
mayerbie
re3
mulh
s32
0oot
oot
oot
0.0
01.0
01.0
20
141
88
00.0
00
023.2
2101.3
0100.6
10.0
04.6
50.2
65.1
83
5.1
88
brum
mayerbie
re3
mulh
s64
0oot
oot
oot
0.0
01.0
03.7
38
141
77
00.0
00
01520.0
15346.4
25343.4
90.0
0434.8
47.3
45.1
84
5.1
89
...m
ayerbie
re
countbits1024
0oot
oot
oot
0.0
01.0
01.8
10
0.0
00.0
00.0
00.0
00.0
05.1
85
5.1
90
brum
mayerbie
re
countbits128
0oot
oot
oot
0.0
01.0
03.6
10
0.0
00.0
00.0
00.0
00.0
05.1
86
5.1
91
brum
mayerbie
re
countbits256
0oot
oot
oot
0.0
01.0
03.7
10
0.0
00.0
00.0
00.0
00.0
05.1
87
5.1
92
brum
mayerbie
re
countbits512
0oot
oot
oot
0.0
01.0
03.6
10
0.0
00.0
00.0
00.0
00.0
05.1
88
5.1
93
...b
iere
countbitsrotate032
0oot
oot
oot
0.0
01.0
00.2
00
0.0
00.0
00.0
00.0
00.0
05.1
89
5.1
94
...b
iere
countbitsrotate064
0oot
oot
oot
0.0
01.0
00.7
00
0.0
00.0
00.0
00.0
00.0
05.1
90
5.1
95
...b
iere
countbitsrotate128
0oot
oot
oot
0.0
01.0
01.6
00
0.0
00.0
00.0
00.0
00.0
05.1
91
5.1
96
...b
iere
countbitsrotate256
0oot
oot
oot
0.0
01.0
04.3
00
0.0
00.0
00.0
00.0
00.0
05.1
92
5.1
97
...y
erbie
re
countbitssrl0
64
0oot
oot
oot
0.0
01.0
01.1
10
0.0
00.0
00.0
00.0
00.0
05.1
93
5.1
98
...y
erbie
re
countbitssrl1
28
0oot
oot
oot
0.0
01.0
03.8
10
0.0
00.0
00.0
00.0
00.0
05.1
94
5.1
99
...y
erbie
re
countbitssrl2
56
0oot
oot
oot
0.0
01.0
04.7
10
0.0
00.0
00.0
00.0
00.0
05.1
95
5.2
00
challenge
integerO
verflow
0oot
oot
oot
0.0
01.0
00.4
512
80
0.0
00
013.5
039.9
439.4
40.0
09.7
01.7
45.1
96
5.2
01
...e
rbie
re4
unconstrain
ed03
0oom
oom
oom
-0.1
41.0
07.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
98
5.2
02
...e
rbie
re4
unconstrain
ed08
0oom
oom
oom
-0.1
81.0
06.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.5
5.2
03
...e
rbie
re4
unconstrain
ed09
0oom
oom
oom
-0.4
10.9
96.8
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.2
00
5.2
04
...e
rbie
re4
unconstrain
ed05
0oom
oom
oom
-0.5
60.9
96.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.6
5.2
05
...e
rbie
re4
unconstrain
ed10
0oom
oom
oom
-1.8
30.9
96.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.2
01
105
Tab
leC
.3:
Det
aile
dre
sult
sof
PB
oole
ctor
inn
on
-in
crem
enta
lm
od
e(P
ara
llel
-Non
-Incr
emen
tal)
com
pare
dto
the
refe
ren
ceti
mes
Bas
e-N
on-I
ncr
emen
tal.
Last
colu
mn
par-
nin
c-fl
rco
nta
ins
acr
oss
refe
ren
ceto
det
ail
edre
sult
sof
Para
llel
-Non
-In
crem
enta
l-F
LR
inT
ab.C
.4.
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
2.1
log-s
licin
gbvm
ul14
20
1477.5
01496.4
5885.1
8611.2
71.6
90.1
22
22
0.0
00.0
0884.5
91724.8
01724.7
20.2
7883.9
2344.9
62.1
2.2
log-s
licin
gbvsm
od
14
20
1414.5
41430.9
81147.2
0283.7
81.2
50.1
13
42
25
39
0.0
30.0
51146.4
93000.9
02998.5
00.2
1231.7
319.8
72.3
2.3
log-s
licin
gbvudiv
17
20
1270.3
61291.0
71042.8
3248.2
41.2
40.2
59
66
0.0
10.0
31042.1
02912.8
52911.6
20.3
3615.7
4126.6
52.4
2.4
...y
erbie
re
countbitssrl0
32
20
387.0
6395.1
1201.8
7193.2
41.9
60.1
34
123
14
41
0.0
40.0
2201.2
0480.1
5474.2
90.0
052.8
71.6
83.1
32.5
...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
71499.2
91326.7
8172.5
11.1
31.6
33
22
0.0
00.0
01326.1
11902.8
31841.3
46.6
61124.7
9271.8
32.7
2.6
log-s
licin
gbvm
ul13
20
428.6
3431.5
4290.6
9140.8
51.4
80.1
22
22
0.0
00.0
0290.0
7537.3
8537.3
00.2
4289.3
2107.4
82.6
2.7
challenge
multip
lyO
verflow
20
1035.6
11041.4
4941.8
799.5
71.1
10.1
22
22
0.0
00.0
0941.1
51453.4
31453.3
14.1
1930.7
5290.6
92.8
2.8
log-s
licin
gbvurem
14
20
757.1
6767.6
8704.6
863.0
01.0
90.1
33
33
0.0
00.0
0704.4
21009.3
41009.1
70.7
7528.5
0126.1
72.1
42.9
...y
erbie
re3
isqrtin
validvc
10
181.6
6183.0
6129.7
053.3
61.4
10.0
30
224
137
00.0
20.0
2129.0
2495.6
0482.9
10.0
29.8
20.7
52.1
52.1
0lo
g-s
licin
gbvm
ul12
20
129.1
9131.3
682.9
548.4
11.5
80.0
22
22
0.0
00.0
082.3
7136.8
9136.8
10.2
281.8
127.3
82.1
62.1
1...b
iere3
isqrtaddin
validvc
10
185.6
0181.9
9138.3
043.6
91.3
20.0
30
224
140
00.0
20.0
2137.6
5518.2
2503.6
00.0
29.6
80.7
72.1
02.1
2lo
g-s
licin
gbvudiv
15
20
162.7
0166.2
3150.3
215.9
11.1
10.0
33
33
0.0
00.0
0149.7
0199.0
6198.8
50.6
298.5
724.8
82.1
92.1
3lo
g-s
licin
gbvsm
od
12
20
138.5
8141.0
4127.2
613.7
81.1
10.1
13
42
25
39
0.0
30.0
4126.5
6337.9
5335.7
70.1
723.1
12.2
42.2
02.1
4lo
g-s
licin
gbvurem
13
20
147.9
7149.4
7137.8
511.6
21.0
80.0
33
33
0.0
00.0
0137.1
8186.8
8186.7
20.6
387.4
023.3
62.2
12.1
5lo
g-s
licin
gbvadd
27966
20
232.0
1236.9
9234.4
22.5
71.0
10.3
01
10
0.0
00.0
0234.2
3234.0
4215.6
7234.0
4234.0
4234.0
42.2
72.1
6lo
g-s
licin
gbvadd
26192
20
202.5
2207.6
7205.8
71.8
01.0
10.3
01
10
0.0
00.0
0205.6
4205.4
6190.5
0205.4
6205.4
6205.4
62.2
32.1
7lo
g-s
licin
gbvadd
23531
20
167.4
5173.1
8171.4
61.7
21.0
10.2
01
10
0.0
00.0
0170.8
2164.2
4154.5
6164.2
4164.2
4164.2
42.2
52.1
8lo
g-s
licin
gbvadd
21757
20
165.0
7169.8
5168.4
41.4
11.0
10.2
01
10
0.0
00.0
0168.3
0162.9
1155.3
0162.9
1162.9
1162.9
12.2
82.1
9lo
g-s
licin
gbvadd
24418
20
198.0
9204.1
8203.2
60.9
21.0
00.3
01
10
0.0
00.0
0202.6
6202.4
4191.4
8202.4
4202.4
4202.4
42.3
72.2
0lo
g-s
licin
gbvadd
29740
20
264.9
3273.8
6273.0
50.8
11.0
00.3
01
10
0.0
00.0
0272.3
8271.2
4244.6
8271.2
4271.2
4271.2
42.2
62.2
1lo
g-s
licin
gbvadd
25305
20
186.3
4190.4
8189.8
80.6
01.0
00.3
01
10
0.0
00.0
0189.2
3189.0
4177.1
5189.0
4189.0
4189.0
42.3
92.2
2lo
g-s
licin
gbvadd
19983
20
121.5
9124.6
6124.2
00.4
61.0
00.2
01
10
0.0
00.0
0124.0
2119.8
6113.0
1119.8
6119.8
6119.8
62.2
42.2
3lo
g-s
licin
gbvadd
18209
20
101.2
6102.5
1102.1
30.3
81.0
00.2
01
10
0.0
00.0
0101.5
598.6
794.4
498.6
798.6
798.6
73.5
2.2
4lo
g-s
licin
gbvadd
20870
20
132.0
0135.0
8134.7
00.3
81.0
00.2
01
10
0.0
00.0
0134.0
7129.3
5122.3
4129.3
5129.3
5129.3
52.2
92.2
5pspace
shift1add.2
2961
20
110.7
1113.3
2113.0
50.2
71.0
00.1
01
10
0.0
00.0
0112.4
9112.4
1111.9
3112.4
1112.4
1112.4
12.4
12.2
6pspace
shift1add.2
4955
20
128.4
7131.6
2131.3
70.2
51.0
00.1
01
10
0.0
00.0
0130.7
1130.6
2130.0
7130.6
2130.6
2130.6
22.3
02.2
7brum
mayerbie
re
bitrev8192
20
101.9
7101.5
6101.3
20.2
41.0
00.0
01
10
0.0
00.0
00.0
50.0
40.0
00.0
40.0
40.0
42.3
12.2
8pspace
shift1add.2
5952
20
143.2
8146.7
1146.5
30.1
81.0
00.1
01
10
0.0
00.0
0145.9
4145.8
5145.2
8145.8
5145.8
5145.8
52.3
82.2
9lo
g-s
licin
gbvadd
19096
20
109.7
8113.2
1113.0
50.1
61.0
00.2
01
10
0.0
00.0
0112.4
5108.9
5103.2
4108.9
5108.9
5108.9
52.4
02.3
0pspace
shift1add.2
9940
20
195.3
8200.4
5200.3
10.1
41.0
00.1
01
10
0.0
00.0
0199.6
9199.5
9198.9
6199.5
9199.5
9199.5
92.3
62.3
1pspace
shift1add.2
7946
20
167.8
1172.1
3172.0
00.1
31.0
00.2
01
10
0.0
00.0
0171.3
1171.2
2170.6
3171.2
2171.2
2171.2
22.3
22.3
2pspace
shift1add.2
8943
20
186.8
4191.4
6191.3
50.1
11.0
00.1
01
10
0.0
00.0
0190.7
2190.6
1190.0
0190.6
1190.6
1190.6
12.3
42.3
3lo
g-s
licin
gbvsub
05994
20
414.3
1415.5
7415.4
70.1
01.0
00.1
01
10
0.0
00.0
0414.8
6413.9
9412.0
0413.9
9413.9
9413.9
92.4
22.3
4pspace
shift1add.2
6949
20
152.3
3156.0
5155.9
50.1
01.0
00.1
01
10
0.0
00.0
0155.3
9155.3
0154.7
2155.3
0155.3
0155.3
02.3
52.3
5RW
SExam
ple
18.txt
10
160.6
1160.5
6160.4
70.0
91.0
00.1
01
10
0.0
00.0
0159.7
9159.6
9159.4
7159.6
9159.6
9159.6
93.1
12.3
6pspace
shift1add.2
3958
20
114.0
8116.8
1116.8
00.0
11.0
00.1
01
10
0.0
00.0
0116.0
2115.9
4115.4
3115.9
4115.9
4115.9
42.3
3
3.1
log-s
licin
gbvsub
04000
20
545.8
4545.6
2545.6
20.0
01.0
00.1
01
10
0.0
00.0
0545.0
1544.5
8543.6
3544.5
8544.5
8544.5
83.1
3.2
bm
c-b
vgraycode
10
442.5
4448.8
9448.9
3-0
.04
1.0
00.3
01
10
0.0
00.0
00.1
80.1
80.0
40.1
80.1
80.1
83.2
3.3
log-s
licin
gbvadd
22644
20
156.0
3165.9
3166.4
6-0
.53
1.0
00.2
01
10
0.0
00.0
0165.6
9159.6
8151.5
5159.6
8159.6
8159.6
83.3
3.4
bm
c-b
vm
agic
10
443.8
6453.0
8453.6
7-0
.59
1.0
00.3
01
10
0.0
00.0
00.6
00.6
00.4
50.6
00.6
00.6
03.7
3.5
log-s
licin
gbvadd
27079
20
216.4
1220.7
0221.9
0-1
.20
0.9
90.3
01
10
0.0
00.0
0221.3
3221.1
5203.3
3221.1
5221.1
5221.1
53.4
3.6
log-s
licin
gbvadd
28853
20
245.3
3251.6
5252.8
8-1
.23
1.0
00.3
01
10
0.0
00.0
0252.2
9252.1
0230.1
8252.1
0252.1
0252.1
03.6
3.7
brum
mayerbie
re3
isqrt
20
145.8
1146.4
6161.9
7-1
5.5
10.9
00.0
42
308
186
00.0
30.0
3161.2
5600.4
6580.3
60.0
29.7
90.5
72.2
23.8
log-s
licin
gbvsm
od
13
20
360.2
6371.8
1395.1
6-2
3.3
50.9
40.1
13
42
25
39
0.0
30.0
5394.4
71173.3
61170.9
70.1
970.8
07.7
73.8
Contin
ued
on
next
page
106
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
3.9
core
ext
con
052
002
1024
20
194.6
0205.8
0239.3
8-3
3.5
80.8
60.4
11
11
0.0
00.0
0238.8
2238.7
3238.0
142.4
6196.2
7119.3
72.1
73.1
0core
ext
con
044
002
1024
20
195.8
2197.3
3236.7
7-3
9.4
40.8
30.4
11
11
0.0
00.0
0236.1
1236.0
4235.4
739.2
4196.8
1118.0
22.9
3.1
1core
ext
con
048
002
1024
20
487.7
2492.4
8535.4
0-4
2.9
20.9
20.4
11
11
0.0
00.0
0534.7
2534.6
5534.0
443.7
0490.9
5267.3
22.2
3.1
2core
ext
con
056
002
1024
20
279.9
8284.3
3327.7
3-4
3.4
00.8
70.4
11
11
0.0
00.0
0327.0
7326.9
8326.2
145.6
3281.3
4163.4
92.1
23.1
3core
ext
con
060
002
1024
20
220.7
1220.8
6267.1
7-4
6.3
10.8
30.4
11
11
0.0
00.0
0266.4
8266.3
8265.5
845.4
6220.9
2133.1
93.9
3.1
4core
ext
con
036
002
1024
20
269.9
9268.4
2315.3
8-4
6.9
60.8
50.3
11
11
0.0
00.0
0314.7
4314.6
8314.2
146.1
1268.5
7157.3
42.5
3.1
5brum
mayerbie
re3
isqrtadd
20
144.7
3153.5
8206.3
6-5
2.7
80.7
40.0
37
255
157
00.0
30.0
2205.6
4738.9
0719.6
40.0
29.4
60.9
13.1
03.1
6core
ext
con
064
002
1024
20
344.2
9330.8
1385.4
3-5
4.6
20.8
60.5
11
11
0.0
00.0
0384.8
3384.7
2383.8
755.6
9329.0
3192.3
62.1
83.1
7core
ext
con
060
001
1024
20
110.6
0111.4
2173.6
9-6
2.2
70.6
40.4
11
11
0.0
00.0
0173.0
0172.9
0172.1
460.9
4111.9
786.4
52.1
33.1
8lo
g-s
licin
gbvudiv
16
20
390.9
9398.2
3471.5
7-7
3.3
40.8
40.1
715
66
0.0
10.0
2470.9
71422.2
91421.6
00.0
2339.1
340.6
43.1
23.1
9core
ext
con
064
001
1024
20
128.2
5129.3
2212.5
8-8
3.2
60.6
10.5
11
11
0.0
00.0
0211.8
6211.7
6210.9
483.8
2127.9
4105.8
82.1
13.2
0...a
yerbie
re2
sm
ulo
v4bw0512
10
128.7
8130.6
6224.3
6-9
3.7
00.5
80.5
33
22
0.0
00.0
0223.6
2350.7
8341.5
50.7
198.5
650.1
13.1
43.2
1pspace
power2sum
.5898
20
343.5
7351.3
4451.0
5-9
9.7
10.7
80.3
11
11
0.0
00.0
0450.4
5450.2
1449.4
2104.2
3345.9
8225.1
13.1
53.2
2pspace
power2sum
.5699
20
347.9
2348.9
0456.6
6-1
07.7
60.7
60.3
11
11
0.0
00.0
0456.0
4455.8
1455.0
5107.8
1348.0
0227.9
03.1
63.2
3lo
g-s
licin
gbvsub
06991
20
307.6
8315.9
4425.6
3-1
09.6
90.7
40.1
11
11
0.0
00.0
0425.0
5423.5
9418.3
3110.9
4312.6
6211.8
03.1
73.2
4pspace
power2sum
.6097
20
388.1
2387.3
7506.7
5-1
19.3
80.7
60.4
11
11
0.0
00.0
0506.1
2505.8
7505.0
3117.2
7388.6
0252.9
33.1
83.2
5pspace
power2sum
.6495
20
504.5
1505.7
8625.6
9-1
19.9
10.8
10.4
11
11
0.0
00.0
0625.1
3624.8
5623.9
4120.5
2504.3
3312.4
23.1
93.2
6pspace
power2sum
.8684
20
1028.3
01025.1
81154.7
3-1
29.5
50.8
90.4
11
11
0.0
00.0
01154.1
61154.0
41152.5
8115.5
11038.5
3577.0
23.2
03.2
7lo
g-s
licin
gbvsub
10979
20
161.2
1145.1
9276.5
6-1
31.3
70.5
20.1
11
11
0.0
00.0
0275.9
7270.1
4257.6
7124.0
9146.0
5135.0
73.2
13.2
8lo
g-s
licin
gbvsub
07988
20
321.6
9327.3
3470.3
9-1
43.0
60.7
00.1
11
11
0.0
00.0
0470.1
0466.8
5460.0
1139.2
9327.5
6233.4
23.2
23.2
9pspace
power2sum
.5500
20
496.1
6493.2
9642.7
9-1
49.5
00.7
70.4
11
11
0.0
00.0
0642.1
3641.9
2641.1
9150.9
4490.9
8320.9
63.2
33.3
0lo
g-s
licin
gbvsub
08985
20
378.5
6390.7
9555.3
7-1
64.5
80.7
00.1
11
11
0.0
00.0
0554.8
0551.1
5542.6
9162.5
3388.6
2275.5
83.2
43.3
1pspace
power2sum
.6296
20
434.7
6435.1
3600.3
8-1
65.2
50.7
20.4
11
11
0.0
00.0
0599.8
2599.5
5598.6
9156.9
6442.5
9299.7
83.2
53.3
2lfsr
lfsr
004
159
128
20
112.1
0152.7
8323.3
9-1
70.6
10.4
73.4
21
10
0.0
00.0
0319.3
2305.4
698.5
982.6
1138.7
9101.8
24.9
93.3
3pspace
power2sum
.7092
20
452.5
0454.2
4640.0
7-1
85.8
30.7
10.4
11
11
0.0
00.0
0639.5
0638.9
8637.9
7179.5
7459.4
2319.4
93.2
63.3
4pspace
power2sum
.7689
20
515.9
3517.5
8716.9
2-1
99.3
40.7
20.4
11
11
0.0
00.0
0716.3
2715.6
6714.5
1199.6
9515.9
8357.8
33.2
73.3
5pspace
power2sum
.9480
20
751.8
3748.6
6956.8
0-2
08.1
40.7
80.5
11
11
0.0
00.0
0956.1
8956.0
5954.4
0208.5
9747.4
6478.0
33.2
83.3
6pspace
power2sum
.9878
20
573.3
5578.3
7791.8
8-2
13.5
10.7
30.4
11
11
0.0
00.0
0791.3
1791.1
8789.4
1218.6
2572.5
6395.5
93.2
93.3
7lo
g-s
licin
gbvsub
11976
20
664.6
3696.1
1926.4
7-2
30.3
60.7
50.4
11
11
0.0
00.0
0925.4
6918.6
9904.3
0247.6
6671.0
3459.3
53.3
03.3
8pspace
power2sum
.8485
20
872.3
0872.2
61107.4
8-2
35.2
20.7
90.6
11
11
0.0
00.0
01106.8
91106.7
81105.3
6234.8
2871.9
5553.3
93.3
13.3
9pspace
power2sum
.9281
20
1338.5
31337.1
01572.5
1-2
35.4
10.8
50.8
11
11
0.0
00.0
01571.8
41571.7
21570.1
1235.1
61336.5
6785.8
63.3
23.4
0pspace
power2sum
.8883
20
1035.8
81037.8
51273.8
1-2
35.9
60.8
10.4
11
11
0.0
00.0
01273.2
51273.1
31271.6
0228.9
91044.1
4636.5
63.3
33.4
1lo
g-s
licin
gbvsub
09982
20
328.0
6333.7
4571.2
2-2
37.4
80.5
80.2
11
11
0.0
00.0
0570.5
6565.8
8555.7
8235.3
1330.5
7282.9
43.3
43.4
2pspace
power2sum
.7490
20
505.6
9505.5
4754.7
3-2
49.1
90.6
70.3
11
11
0.0
00.0
0754.0
2753.3
9752.2
8247.5
8505.8
1376.6
93.3
53.4
3...a
yerbie
re2
sm
ulo
v4bw0768
10
563.4
2566.2
5821.6
5-2
55.4
00.6
91.1
33
22
0.0
00.0
0820.9
81215.3
11185.2
82.9
3648.3
1173.6
23.3
63.4
4pspace
power2sum
.7291
20
971.6
8970.1
21228.7
2-2
58.6
00.7
90.6
11
11
0.0
00.0
01228.1
11227.5
11226.4
3257.6
8969.8
3613.7
63.3
73.4
5lfsr
lfsr
004
111
112
20
178.0
5198.9
3477.9
2-2
78.9
90.4
25.1
74
30
4.8
016.8
9448.3
41032.8
6433.5
144.6
673.6
751.6
44.8
63.4
6...a
yerbie
re2
sm
ulo
v4bw0896
10
1274.5
21280.0
11572.6
6-2
92.6
50.8
11.3
33
22
0.0
00.0
01571.9
51953.9
31909.8
84.6
61368.4
3279.1
33.3
83.4
7pspace
power2sum
.7888
20
1145.4
91133.2
31508.7
7-3
75.5
40.7
50.8
11
11
0.0
00.0
01508.1
41507.4
51506.2
3369.4
01138.0
5753.7
23.3
93.4
8brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3135.5
4533.5
1-3
97.9
70.2
50.0
74
496
124
00.0
70.0
6532.5
91863.2
51851.7
30.0
18.1
51.4
73.4
83.4
9brum
mayerbie
re
nlz
be256
20
117.5
1119.3
1540.3
4-4
21.0
30.2
20.4
15
81
41
00.1
10.0
8539.4
41641.2
41584.7
60.0
739.3
38.4
63.4
43.5
0lo
g-s
licin
gbvsrem
14
20
256.9
0263.7
6688.5
3-4
24.7
70.3
80.1
721
16
18
0.0
20.0
3687.8
21905.5
21904.3
50.1
9179.5
429.3
23.4
03.5
1pspace
power2sum
.6694
20
1005.8
11017.6
91465.4
1-4
47.7
20.6
90.6
11
11
0.0
00.0
01464.8
11464.5
21463.5
7456.9
11007.6
1732.2
63.4
23.5
2...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
3143.7
7594.5
2-4
50.7
50.2
40.7
33
22
0.0
00.0
0593.8
01023.0
01005.4
01.6
7410.6
4146.1
43.4
13.5
3pspace
power2sum
.8286
20
898.4
1893.2
41348.1
2-4
54.8
80.6
60.4
11
11
0.0
00.0
01347.4
81347.3
71345.9
9454.6
7892.7
0673.6
83.4
33.5
4lo
g-s
licin
gbvsdiv
16
20
182.3
5185.3
1716.3
8-5
31.0
70.2
60.1
10
38
19
18
0.0
20.0
4715.7
01811.3
71809.6
30.0
2146.4
717.9
33.4
53.5
5pspace
power2sum
.6893
20
1102.0
21106.4
11649.3
2-5
42.9
10.6
70.6
11
11
0.0
00.0
01648.6
41648.3
31647.3
5545.7
21102.6
0824.1
63.4
63.5
6lo
g-s
licin
gbvult
6000
20
406.3
8410.0
2967.3
7-5
57.3
50.4
20.7
510
910
0.9
60.1
7965.5
63021.0
13007.5
426.2
9372.5
5104.1
73.4
73.5
7brum
mayerbie
re3
maxor064
20
209.1
4209.4
81164.4
1-9
54.9
30.1
80.1
129
129
3109
0.1
90.1
31163.3
21384.6
61369.7
00.0
036.7
43.7
83.4
93.5
8lo
g-s
licin
gbvsdiv
17
20
502.6
4511.0
81511.0
3-9
99.9
50.3
40.2
927
16
24
0.0
30.0
81510.2
74988.4
64983.0
80.0
4334.9
158.6
93.5
03.5
9lo
g-s
licin
gbvult
6997
20
201.1
9204.8
11299.6
2-1
094.8
10.1
60.7
510
910
0.9
90.1
61297.7
63678.5
13664.8
627.0
5477.8
8126.8
53.5
1
Contin
ued
on
next
page
107
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
3.6
0lo
g-s
licin
gbvslt
5743
20
444.5
8444.9
21606.1
9-1
161.2
70.2
80.7
718
14
18
1.4
80.2
81603.7
05575.5
05553.2
824.9
3323.8
697.8
23.5
23.6
1lo
g-s
licin
gbvult
7994
20
230.1
0234.3
01565.6
1-1
331.3
10.1
50.7
510
910
0.9
60.1
81564.1
73894.6
63880.8
927.9
4613.8
0134.3
03.5
33.6
2lfsr
lfsr
008
159
048
20
100.5
6110.8
71462.9
1-1
352.0
40.0
84.1
912
12
03.6
610.5
41442.3
65348.9
83983.4
625.7
3105.6
061.4
84.1
22
3.6
3lfsr
lfsr
008
159
064
20
149.5
2166.9
31555.3
3-1
388.4
00.1
15.4
10
86
05.7
316.9
31524.0
14605.4
53027.2
742.1
0156.8
186.9
04.9
5
4.1
log-s
licin
gbvslt
17631
20
1750.5
61785.5
0oot
-14.5
00.9
96.4
414
14
040.8
11.0
3497.4
52424.1
02302.5
738.9
9182.6
089.7
84.1
4.2
...e
st
v7
r12
vr5
c1
s14336
10
933.1
71778.6
5oot
-21.3
50.9
92.5
19
87
80
04.5
44.1
21712.3
86029.4
75691.9
50.1
846.3
136.9
94.2
4.3
...sm
tcom
p09
lfbv
form
ula
20
180.2
4202.3
3sf
-39.1
80.8
40.2
0.0
00.0
00.0
00.0
00.0
04.8
54.4
...sm
tcom
p09
std
bv
form
ula
20
108.8
6128.5
9sf
-106.1
70.5
50.2
0.0
00.0
00.0
00.0
00.0
04.1
14
4.5
...m
tcom
p09
full
bv
form
ula
20
172.4
6159.0
4sf
-204.7
80.4
40.3
0.0
00.0
00.0
00.0
00.0
04.9
64.6
...test
v5
r15
vr5
c1
s8246
20
1106.6
41531.3
7oot
-268.6
30.8
52.3
18
93
88
04.1
93.8
31786.8
86233.2
45899.5
60.1
670.9
535.2
24.3
4.7
log-s
licin
gbvslt
16888
20
1502.5
61524.1
3oot
-275.8
70.8
56.4
414
14
037.8
91.0
1561.1
41992.9
01871.3
235.9
9208.3
773.8
14.4
4.8
log-s
licin
gbvsdiv
18
20
1485.5
21509.3
7oot
-290.6
30.8
40.2
734
24
40.0
20.1
01550.8
62281.9
92277.5
90.1
3896.9
330.4
34.5
4.9
...e
st
v5
r15
vr5
c1
s26657
20
oot
1480.6
0oot
-319.4
00.8
22.4
18
93
88
04.1
73.7
81729.7
06252.9
55912.0
70.1
643.9
234.5
54.6
4.1
0...e
st
v7
r7
vr10
c1
s32506
10
1754.9
51394.4
1oot
-405.5
90.7
71.7
23
119
112
03.1
23.0
91792.3
86400.6
16048.3
10.0
926.7
221.8
54.7
4.1
1...sm
tcom
p09
sw
bv
form
ula
20
202.4
2204.9
0sf
-455.5
80.3
10.4
0.0
00.0
00.0
00.0
00.0
04.8
34.1
2pspace
power2sum
.9082
20
1301.9
11303.6
6oot
-496.3
40.7
20.7
11
10
0.0
00.0
0508.1
8508.1
2507.3
4508.1
2508.1
2508.1
24.8
4.1
3pip
epip
e-n
oabs.a
tla
s.q
fbv
20
1290.1
01298.2
3oot
-501.7
70.7
20.1
54
416
62
00.1
20.0
81779.4
66478.5
56464.0
10.0
029.3
26.5
74.9
4.1
4lfsr
lfsr
008
159
128
20
285.7
4363.4
0oom
-550.5
60.4
06.8
51
10
0.0
00.0
0850.7
6800.9
8228.0
8125.8
8248.0
6160.2
04.5
24.1
5...st
v5
r10
vr10
c1
s21502
20
697.8
21247.1
6oot
-552.8
40.6
91.7
20
110
104
02.8
72.8
41337.1
96536.3
16183.3
40.0
928.0
122.7
04.1
04.1
6lo
g-s
licin
gbvslt
16145
20
1188.5
61208.8
1oot
-591.1
90.6
72.0
412
12
04.1
20.4
0503.3
81521.7
31489.0
130.3
0211.1
166.1
64.1
14.1
7lo
g-s
licin
gbvult
15970
20
1070.6
31091.1
1oot
-708.8
90.6
12.1
412
12
03.6
60.4
0436.4
91239.8
01204.3
831.5
3159.7
853.9
04.1
24.1
8brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
51078.0
7oot
-721.9
30.6
00.1
0.0
00.0
00.0
00.0
00.0
04.1
34.1
9lo
g-s
licin
gbvslt
15402
20
1048.3
41064.7
6oot
-735.2
40.5
92.0
412
12
03.9
90.4
0510.6
91710.2
31678.8
030.7
8154.5
374.3
64.1
44.2
0lo
g-s
licin
gbvult
14973
20
1019.5
01027.3
1oot
-772.6
90.5
72.0
412
12
04.0
10.3
8376.4
81151.5
61119.4
428.2
4137.0
450.0
74.1
54.2
1...test
v7
r7
vr5
c1
s19694
10
1370.1
1999.2
7oot
-800.7
30.5
61.7
22
111
104
02.8
82.8
91722.7
76404.9
36057.9
70.0
927.3
422.1
64.1
64.2
2VS3
VS3-b
enchm
ark-A
710
985.7
9987.8
8oot
-812.1
20.5
50.4
70
532
306
00.6
90.6
61625.3
16928.3
06737.7
90.2
616.0
23.1
64.1
74.2
3lo
g-s
licin
gbvslt
13916
20
908.9
9933.2
6oot
-866.7
40.5
22.0
412
12
34.0
00.4
0426.3
64759.9
34724.8
429.7
61310.6
6183.0
74.1
84.2
4lo
g-s
licin
gbvult
12979
20
893.4
9915.2
3oot
-884.7
70.5
12.0
412
12
03.7
60.4
0276.0
91017.1
7986.9
932.7
073.3
144.2
24.1
94.2
5lo
g-s
licin
gbvult
13976
20
893.7
3911.1
4oot
-888.8
60.5
12.0
412
12
03.8
10.3
9338.8
51108.6
31073.9
929.6
2112.1
648.2
04.2
04.2
6...e
st
v5
r15
vr5
c1
s23844
20
oot
875.1
0oot
-924.9
00.4
92.4
18
93
88
04.2
23.8
11738.2
06281.3
05936.5
90.1
745.5
934.7
04.2
14.2
7lo
g-s
licin
gbvslt
14659
20
838.9
9854.3
5oot
-945.6
50.4
72.0
412
12
33.8
70.3
9370.2
24394.0
34356.5
729.4
41386.4
0169.0
04.2
24.2
8lo
g-s
licin
gbvslt
13173
20
801.4
6815.3
4oot
-984.6
60.4
52.0
412
12
14.0
00.4
0307.3
12010.4
11977.7
527.7
21000.8
483.7
74.2
34.2
9lo
g-s
licin
gbvsrem
15
20
791.1
0801.9
2oot
-998.0
80.4
50.1
721
16
16
0.0
10.0
31046.5
24738.9
74737.8
10.2
2445.1
675.2
24.2
44.3
0float
newton.6
.3.i
10
564.0
4787.5
3oot
-1012.4
70.4
40.8
32
227
46
02.4
82.6
61756.0
56329.5
95900.7
30.0
536.5
012.1
04.2
54.3
1...test
v7
r7
vr1
c1
s4574
10
609.3
3754.5
2oot
-1045.4
80.4
21.9
20
144
144
03.3
03.3
91386.5
76772.5
86366.3
610.6
631.2
716.2
44.2
64.3
2...e
st
v5
r10
vr10
c1
s7608
20
1199.1
4715.0
0oot
-1085.0
00.4
01.7
20
109
104
02.8
92.8
41334.8
76485.0
96127.8
40.0
929.2
422.6
84.2
74.3
3...e
st
v5
r10
vr5
c1
s13679
10
899.6
4696.3
3oot
-1103.6
70.3
91.7
20
111
104
02.7
92.7
91281.2
86598.9
06236.8
60.1
027.4
121.9
24.2
84.3
4lo
g-s
licin
gbvslt
11687
20
665.8
6672.3
9oot
-1127.6
10.3
72.0
412
12
23.9
50.4
0282.4
53424.5
43390.9
027.4
71288.0
0136.9
84.2
94.3
5...st
v5
r15
vr10
c1
s11127
10
346.0
3669.6
9oot
-1130.3
10.3
72.4
18
93
88
04.2
03.8
21716.3
56356.9
36009.2
60.1
641.2
734.3
64.3
04.3
6...tcom
p09
stall
bv
form
ula
20
609.5
3615.5
9oot
-1184.4
10.3
40.7
20
142
49
02.5
42.1
31526.5
86483.4
46221.1
00.0
373.7
919.8
94.3
14.3
7...test
v7
r7
vr1
c1
s22845
10
316.6
3605.4
9oot
-1194.5
10.3
41.9
20
144
144
03.2
93.3
81295.9
66762.7
86345.4
96.7
628.4
514.8
64.3
24.3
8lo
g-s
licin
gbvult
9988
20
555.0
3572.9
3oot
-1227.0
70.3
22.2
512
12
45.3
90.4
11553.6
35690.2
55654.2
427.6
41297.9
9210.7
54.3
34.3
9lo
g-s
licin
gbvslt
10201
20
552.9
2571.4
3oot
-1228.5
70.3
21.9
412
12
33.9
40.4
0229.0
04100.1
54063.0
130.3
61392.0
6157.7
04.3
44.4
0lo
g-s
licin
gbvult
10985
20
556.6
2561.1
4oot
-1238.8
60.3
12.0
412
12
03.9
60.4
1283.0
01005.8
9972.2
430.5
373.2
843.7
34.3
54.4
1lo
g-s
licin
gbvult
8991
20
542.6
1554.9
3oot
-1245.0
70.3
11.9
412
12
03.8
60.4
1240.2
7907.9
7876.9
629.6
162.3
639.4
84.3
64.4
2lo
g-s
licin
gbvslt
9458
20
539.1
6553.4
3oot
-1246.5
70.3
11.9
412
12
33.9
40.4
0238.6
33290.0
43254.3
728.1
3925.1
1126.5
44.3
74.4
3lo
g-s
licin
gbvult
11982
20
518.1
6528.4
4oot
-1271.5
60.2
92.0
412
12
03.7
70.4
0251.8
8960.9
6928.0
028.7
882.0
741.7
84.3
84.4
4float
mul03
30
420
514.8
0520.1
8oot
-1279.8
20.2
90.3
103
808
264
00.9
40.7
41795.6
87067.2
26838.3
20.0
012.6
32.8
54.3
94.4
5lo
g-s
licin
gbvslt
10944
20
505.6
8510.4
4oot
-1289.5
60.2
82.0
412
12
33.9
20.4
1288.3
93349.2
23315.3
829.3
4993.7
1128.8
24.4
04.4
6...test
v5
r10
vr5
c1
s8690
20
353.9
3459.7
9oot
-1340.2
10.2
61.7
20
109
104
02.8
52.8
31328.5
36519.3
26165.6
80.1
028.2
022.4
84.4
1
Contin
ued
on
next
page
108
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.4
7float
newton.3
.3.i
20
355.4
9450.8
5oot
-1349.1
50.2
50.8
32
232
60
02.5
22.7
71791.8
16225.2
25815.1
60.0
434.1
412.4
54.4
24.4
8float
newton.1
.3.i
20
385.3
4445.6
2oot
-1354.3
80.2
50.8
37
263
66
02.8
73.1
01773.3
86498.0
76056.4
10.0
336.9
511.7
94.4
34.4
9float
newton.4
.3.i
20
334.8
4418.4
4oot
-1381.5
60.2
30.8
31
224
48
02.4
02.6
31782.7
16088.2
85693.2
50.0
436.9
612.7
14.4
44.5
0...e
st
v5
r15
vr1
c1
s26845
20
619.0
9415.2
0oot
-1384.8
00.2
32.3
19
136
136
05.1
04.9
91780.6
56614.4
66199.5
510.0
751.8
321.8
34.4
54.5
1float
newton.5
.3.i
20
415.2
8404.4
7oot
-1395.5
30.2
20.8
33
238
44
02.6
82.8
81715.3
36346.6
25915.4
40.0
538.1
511.6
94.4
64.5
2lfsr
lfsr
008
015
112
20
388.4
4399.4
9oot
-1400.5
10.2
23.1
30
81
81
06.5
811.8
21707.1
36880.2
64513.2
65.4
517.6
812.2
94.4
74.5
3...e
st
v7
r7
vr10
c1
s10625
10
852.7
6391.8
4oot
-1408.1
60.2
21.6
22
111
104
02.9
02.9
01734.1
36329.5
95984.7
50.0
928.7
622.2
14.4
84.5
4float
newton.2
.3.i
20
289.3
8385.6
1oot
-1414.3
90.2
10.8
33
243
40
02.5
92.7
71768.0
46122.8
55708.1
70.0
236.9
611.9
84.4
94.5
5...test
v7
r7
vr5
c1
s3582
10
326.7
8384.0
3oot
-1415.9
70.2
11.7
22
111
104
02.9
02.9
11734.9
66353.7
26012.5
90.0
927.3
322.2
94.5
04.5
6...sm
tcom
p09
nt
bv
form
ula
20
156.7
2137.2
7oom
-1418.1
20.0
96.8
0.0
00.0
00.0
00.0
00.0
04.1
09
4.5
7...test
v7
r12
vr5
c1
s8938
10
615.1
6367.1
4oot
-1432.8
60.2
02.5
19
87
80
04.6
54.2
01730.5
86043.7
65705.7
20.3
046.2
137.5
44.5
14.5
8lo
g-s
licin
gbvslt
8715
20
342.5
6350.6
8oot
-1449.3
20.1
92.0
518
14
55.5
30.6
31281.4
84729.6
94683.4
127.6
51003.3
2118.2
44.5
34.5
9lfsr
lfsr
008
159
112
20
275.9
4338.9
8oot
-1461.0
20.1
94.9
91
10
0.0
00.0
01626.8
41567.2
3816.6
0115.6
1198.1
2174.1
44.5
44.6
0lfsr
lfsr
008
143
128
20
261.7
9336.9
6oot
-1463.0
40.1
96.2
13
11
00.0
00.0
01796.2
21686.6
7455.3
3119.2
5225.1
0129.7
44.5
54.6
1...test
v7
r7
vr1
c1
s24449
20
679.7
0330.9
5oot
-1469.0
50.1
82.0
20
144
144
03.3
33.3
71460.4
66790.5
66387.9
113.0
631.1
917.1
54.5
64.6
2...servers
sla
pd
avc149923
10
325.6
4330.0
6oot
-1469.9
40.1
81.4
40
304
304
00.2
32.2
51569.5
27015.2
26743.3
65.5
013.4
46.2
64.5
74.6
3...servers
sla
pd
avc149922
10
325.6
8329.3
7oot
-1470.6
30.1
81.4
40
304
304
00.2
42.2
21603.7
76939.6
76671.7
55.3
313.4
16.3
34.5
84.6
4lfsr
lfsr
008
159
096
20
275.8
6320.4
7oot
-1479.5
30.1
86.5
11
22
03.8
010.7
31760.6
23263.1
61941.7
094.4
7183.3
5155.3
94.5
94.6
5...e
st
v5
r10
vr5
c1
s13195
10
337.5
5314.0
0oot
-1486.0
00.1
71.7
20
109
104
02.8
32.8
31329.0
66505.5
86149.0
90.0
529.0
522.5
14.6
14.6
6lo
g-s
licin
gbvslt
7972
20
305.2
5307.9
8oot
-1492.0
20.1
70.7
618
14
91.4
90.2
91421.5
35691.9
05671.9
627.4
8677.2
1129.3
64.6
24.6
7lo
g-s
licin
gbvslt
5000
20
296.4
0297.7
4oot
-1502.2
60.1
70.7
618
14
13
1.3
60.2
91319.6
75047.7
65028.7
423.5
8451.8
6103.0
24.6
34.6
8brum
mayerbie
re3
min
and256
20
289.6
6295.2
1oot
-1504.7
90.1
60.3
18
19
316
0.1
00.0
61724.3
22649.9
92631.7
80.1
2280.1
550.0
04.6
44.6
9lfsr
lfsr
008
015
128
20
277.3
6286.7
4oot
-1513.2
60.1
63.1
28
65
65
05.9
310.5
71733.8
86835.8
44381.6
16.1
519.2
013.5
64.6
54.7
0...e
st
v5
r10
vr1
c1
s32538
20
208.1
6285.1
4oot
-1514.8
60.1
61.9
22
160
160
03.3
73.5
01792.3
76805.8
46397.8
97.0
728.2
714.2
14.6
64.7
1lfsr
lfsr
008
127
128
20
212.8
0282.8
6oot
-1517.1
40.1
66.2
13
11
00.0
00.0
01742.3
51628.7
6489.0
9109.5
0214.9
7125.2
94.6
74.7
2lfsr
lfsr
008
015
096
20
272.7
5280.3
4oot
-1519.6
60.1
63.1
35
94
94
06.3
111.2
91754.4
96886.5
74430.1
24.4
015.1
99.5
24.6
84.7
3VS3
VS3-b
enchm
ark-A
910
268.4
5271.7
7oot
-1528.2
30.1
50.5
70
543
388
00.7
80.7
11742.6
67030.7
16815.3
20.2
96.4
23.0
84.6
94.7
4lfsr
lfsr
008
143
112
20
205.3
9260.6
0oot
-1539.4
00.1
44.6
14
11
00.0
00.0
01694.1
61604.5
0575.2
2102.6
3182.3
2114.6
14.7
04.7
5...e
st
v5
r10
vr1
c1
s13516
10
231.3
6254.9
6oot
-1545.0
40.1
41.9
20
144
144
03.2
03.2
71314.5
86744.3
86342.1
810.2
627.8
815.2
24.7
14.7
6...e
st
v5
r15
vr1
c1
s32559
20
163.7
5250.6
0oot
-1549.4
00.1
42.3
17
113
111
04.4
44.2
11709.0
16529.2
26141.8
40.1
041.8
325.0
24.7
24.7
7...test
v5
r5
vr10
c1
s7194
10
141.8
4249.3
6oot
-1550.6
40.1
40.9
32
200
118
02.0
62.0
71729.3
36529.6
06213.3
40.0
250.0
311.1
44.7
34.7
8lfsr
lfsr
008
127
112
20
176.0
0226.4
7oot
-1573.5
30.1
34.5
13
11
00.0
00.0
01775.9
71697.6
9830.8
495.7
5167.1
6130.5
94.7
44.7
9lfsr
lfsr
008
159
080
20
194.3
3224.5
2oot
-1575.4
80.1
25.8
13
33
04.3
512.4
91671.5
64559.4
82828.1
376.2
5158.5
9126.6
54.7
54.8
0...test
v5
r5
vr5
c1
s2800
10
228.1
4223.6
7oot
-1576.3
30.1
20.9
39
261
108
02.7
02.7
41768.6
06472.7
16094.5
20.0
216.5
59.2
34.7
64.8
1lo
g-s
licin
gbvslt
7229
20
217.0
1221.1
9oot
-1578.8
10.1
20.7
618
14
10
1.3
80.2
91447.4
45043.3
85023.8
227.5
5709.7
5112.0
84.7
74.8
2lfsr
lfsr
008
143
096
20
173.9
4217.0
4oot
-1582.9
60.1
26.1
13
22
03.5
410.2
21652.5
13313.8
31701.6
089.0
4136.6
8122.7
34.7
84.8
3...test
v7
r7
vr5
c1
s14675
10
364.5
5212.1
9oot
-1587.8
10.1
21.6
22
111
104
02.9
02.9
01718.6
56392.2
96041.6
60.0
928.1
122.1
24.7
94.8
4lfsr
lfsr
008
111
128
20
160.6
8210.2
8oot
-1589.7
20.1
24.2
16
11
00.0
00.0
01732.2
91645.7
5593.7
888.8
8153.0
0102.8
64.8
04.8
5...y
erbie
re3
min
andm
axor064
20
207.7
9206.6
6oot
-1593.3
40.1
10.1
138
649
23
00.6
20.5
01785.4
43488.6
23420.4
60.0
121.3
92.6
44.8
14.8
6...e
st
v7
r7
vr10
c1
s24535
10
237.8
8205.9
0oot
-1594.1
00.1
11.7
23
119
112
03.1
73.1
31749.3
86372.7
46016.6
00.0
926.6
721.3
14.8
24.8
7lfsr
lfsr
008
095
128
20
148.8
9203.3
5oot
-1596.6
50.1
15.6
13
22
03.2
18.8
71748.5
33219.4
11109.5
3118.2
1158.1
0128.7
84.8
44.8
8...m
ayerbie
re2
sm
ulo
v2bw512
20
198.3
8198.7
8oot
-1601.2
20.1
12.7
68
40
0.0
90.0
21500.3
92048.2
41928.1
711.6
2402.5
9186.2
04.8
74.8
9lfsr
lfsr
008
111
112
20
148.6
0194.6
4oot
-1605.3
60.1
13.6
17
11
00.0
00.0
01738.7
01655.7
8615.6
083.0
3140.6
197.4
04.8
84.9
0lo
g-s
licin
gbvslt
6486
20
191.6
5194.4
0oot
-1605.6
00.1
10.7
718
14
14
1.4
70.3
01611.9
15727.8
05706.3
326.9
8445.8
0108.0
74.8
94.9
1lfsr
lfsr
008
095
112
20
129.0
8179.9
1oot
-1620.0
90.1
05.0
13
22
02.7
98.3
81778.8
23298.1
81346.2
4118.3
9157.0
6131.9
34.9
04.9
2float
newton.5
.2.i
20
249.5
3175.9
2oot
-1624.0
80.1
00.7
49
366
86
02.6
12.4
71790.1
26622.7
76212.4
00.0
127.2
77.6
94.9
14.9
3lfsr
lfsr
008
047
096
20
160.4
0173.7
1oot
-1626.2
90.1
04.0
15
12
12
04.0
19.7
71721.4
16681.7
13429.1
829.6
150.1
944.2
54.9
24.9
4lfsr
lfsr
008
111
096
20
133.3
2172.2
2oot
-1627.7
80.1
05.9
18
22
03.2
39.6
71773.8
03267.8
41315.6
178.8
1120.9
193.3
74.9
34.9
5float
newton.4
.2.i
20
143.2
5170.4
6oot
-1629.5
40.0
90.6
44
273
58
01.8
71.8
21779.7
36479.3
86163.6
70.0
131.1
110.3
74.9
44.9
6...test
v5
r5
vr1
c1
s14623
10
143.2
0158.2
0oot
-1641.8
00.0
91.1
40
304
208
02.2
12.4
01563.3
66735.0
96390.6
20.0
258.2
26.9
14.9
74.9
7lfsr
lfsr
004
143
128
20
116.2
1156.0
3oot
-1643.9
70.0
96.0
18
22
03.2
610.2
41777.8
23260.8
41353.6
180.9
8130.6
693.1
74.9
8
Contin
ued
on
next
page
109
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.9
8...m
ayerbie
re2
sm
ulo
v2bw448
20
150.6
3150.3
4oot
-1649.6
60.0
82.4
0.0
00.0
00.0
00.0
00.0
04.1
00
4.9
9lfsr
lfsr
008
127
096
20
113.0
4149.6
1oot
-1650.3
90.0
85.8
16
22
03.1
79.8
61749.6
93235.5
71650.1
986.9
9130.0
5104.3
74.1
01
4.1
00
lfsr
lfsr
008
079
128
20
107.7
6149.6
0oot
-1650.4
00.0
85.8
14
33
04.3
511.4
51694.4
54548.7
61834.8
8102.2
2133.9
7116.6
34.1
02
4.1
01
brum
mayerbie
re3
maxxor032
20
143.9
5148.0
4oot
-1651.9
60.0
80.1
150
1171
205
00.3
20.2
31788.0
56722.3
56675.3
50.0
512.6
71.8
94.1
03
4.1
02
lfsr
lfsr
008
095
096
20
105.5
2147.6
8oot
-1652.3
20.0
86.4
15
43
06.9
821.0
61740.4
24647.4
81939.6
495.8
5129.7
8110.6
54.1
04
4.1
03
tacas07
Y86
btnft
20
135.7
7146.5
9oot
-1653.4
10.0
80.7
29
208
168
00.5
40.5
31785.7
56852.6
06494.5
87.8
838.7
711.5
94.6
04.1
04
...m
ayerbie
re2
um
ulo
v1bw256
20
143.9
4145.2
2oot
-1654.7
80.0
81.3
0.0
00.0
00.0
00.0
00.0
04.1
05
4.1
05
...e
st
v5
r10
vr1
c1
s19145
10
316.4
9144.1
6oot
-1655.8
40.0
81.9
22
160
157
03.4
03.5
61768.3
76784.8
76374.3
17.7
527.7
013.9
34.1
06
4.1
06
float
newton.3
.2.i
20
186.5
6140.6
8oot
-1659.3
20.0
80.6
41
265
80
01.8
41.7
01655.8
56148.5
15838.6
10.0
161.6
210.0
64.1
07
4.1
07
lfsr
lfsr
008
143
080
20
112.1
3139.6
4oot
-1660.3
60.0
85.4
16
33
04.1
711.9
11711.1
94639.5
12571.0
864.2
5118.6
9103.1
04.1
08
4.1
08
float
newton.1
.2.i
20
139.9
8134.2
2oot
-1665.7
80.0
70.6
41
302
72
02.0
01.9
21702.4
36485.1
36137.9
80.0
149.2
99.2
04.1
10
4.1
09
...test
v5
r5
vr5
c1
s24018
10
182.8
1131.0
6oot
-1668.9
40.0
70.9
40
268
114
02.7
42.7
41790.9
06468.4
26083.1
10.0
215.8
59.0
74.1
11
4.1
10
brum
mayerbie
re
countbits064
20
129.3
4130.2
0oot
-1669.8
00.0
70.2
92
713
631
00.2
10.1
61782.1
27098.3
87004.2
90.0
018.4
01.5
54.1
12
4.1
11
tacas07
Y86
std
20
129.6
5129.5
0oot
-1670.5
00.0
70.7
26
184
144
00.4
50.4
41779.9
16820.2
96509.4
10.0
037.4
012.5
64.1
13
4.1
12
...m
ayerbie
re2
sm
ulo
v2bw384
20
122.0
6122.2
7oot
-1677.7
30.0
71.9
0.0
00.0
00.0
00.0
00.0
04.1
15
4.1
13
...test
v5
r15
vr1
c1
s8236
20
927.6
6120.2
6oot
-1679.7
40.0
72.3
17
120
120
04.7
24.4
81729.4
86607.1
66205.0
113.0
741.8
823.6
84.1
16
4.1
14
lfsr
lfsr
008
143
064
20
103.4
9119.6
5oot
-1680.3
50.0
74.8
13
66
03.8
611.3
31621.5
15328.9
23477.1
335.9
3111.5
075.0
64.1
17
4.1
15
brum
mayerbie
re3
min
or256
20
116.1
3118.5
5oot
-1681.4
50.0
70.7
29
216
216
00.4
80.4
01784.5
86182.2
75943.3
92.6
355.3
510.3
24.1
18
4.1
16
float
newton.2
.2.i
20
143.1
8116.2
9oot
-1683.7
10.0
60.7
47
293
98
02.0
21.9
41743.1
06480.4
46107.7
50.0
227.9
28.9
64.1
19
4.1
17
brum
mayerbie
re3
min
xor064
20
111.8
4115.9
8oot
-1684.0
20.0
60.2
0.0
00.0
00.0
00.0
00.0
04.1
20
4.1
18
VS3
VS3-b
enchm
ark-A
610
111.9
3112.6
8oot
-1687.3
20.0
60.2
50
371
176
00.4
70.4
11470.6
06861.6
76759.8
10.2
716.8
65.5
24.1
21
4.1
19
core
ext
con
060
256
1024
20
101.7
3110.0
6oot
-1689.9
40.0
63.6
940
40
053.7
012.2
51566.3
06094.0
04498.1
322.6
6134.3
361.5
64.1
23
4.1
20
...m
ayerbie
re2
um
ulo
v1bw224
20
108.4
2109.1
8oot
-1690.8
20.0
61.0
26
162
22
00.2
30.1
21743.2
36169.8
65802.6
81.1
286.8
418.4
24.1
24
4.1
21
float
mult2.c
.50
10
109.3
9108.0
5oot
-1691.9
50.0
61.2
20
137
136
01.2
21.2
71508.1
46716.6
96386.4
46.3
141.8
418.8
14.1
25
4.1
22
brum
mayerbie
re3
maxand256
20
102.0
5103.8
3oot
-1696.1
70.0
60.8
37
280
280
00.6
20.5
31793.2
26364.2
56035.6
70.9
539.2
46.8
74.1
26
5.1
...m
ayerbie
re
countbits1024
0oot
oot
oom
253.6
91.1
66.9
37
60
0.1
60.0
31531.0
12291.1
02217.3
70.0
3625.0
4327.3
05.1
5.2
...e
rbie
re4
unconstrain
ed10
0oom
oom
oom
1.3
31.0
17.0
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
5.3
...e
rbie
re4
unconstrain
ed03
0oom
oom
oom
0.8
21.0
26.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.3
5.4
...e
rbie
re4
unconstrain
ed07
0oom
oom
oom
0.6
21.0
16.8
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.7
5.5
...e
rbie
re4
unconstrain
ed06
0oom
oom
oom
0.5
11.0
16.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.4
5.6
...e
rbie
re4
unconstrain
ed04
0oom
oom
oom
0.3
61.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
50
5.7
...e
rbie
re4
unconstrain
ed05
0oom
oom
oom
0.2
71.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
49
5.8
...e
rbie
re4
unconstrain
ed01
0oom
oom
oom
0.1
11.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.6
5.9
fft
Sz1024
34824
0oot
oot
oot
0.0
01.0
01.1
90
704
704
02.0
71.7
31754.1
27065.5
06613.4
01.5
54.9
61.7
25.8
5.1
0fft
Sz512
15128
00
oot
oot
oot
0.0
01.0
00.7
0.0
00.0
00.0
00.0
00.0
05.9
5.1
1fft
Sz512
15128
10
oot
oot
oot
0.0
01.0
00.5
0.0
00.0
00.0
00.0
00.0
05.1
05.1
2fft
Sz512
15128
20
oot
oot
oot
0.0
01.0
00.7
170
1336
1256
01.3
30.9
71594.0
97096.8
26612.4
50.0
72.3
20.5
25.1
15.1
3fft
Sz512
15128
40
oot
oot
oot
0.0
01.0
00.8
0.0
00.0
00.0
00.0
00.0
05.1
25.1
4fft
Sz512
15128
50
oot
oot
oot
0.0
01.0
00.7
0.0
00.0
00.0
00.0
00.0
05.1
35.1
5fft
Sz512
15128
60
oot
oot
oot
0.0
01.0
00.6
160
1232
1016
01.2
50.8
91762.6
47104.1
66696.4
90.0
72.2
90.6
25.1
45.1
6...st
v5
r10
vr10
c1
s15708
0oot
oot
oot
0.0
01.0
01.7
20
110
104
02.8
42.8
41310.6
26544.9
06187.1
80.1
027.5
222.0
45.1
55.1
7...st
v7
r12
vr10
c1
s15994
10
1367.7
1oot
oot
0.0
01.0
02.4
19
87
80
04.5
24.1
01721.4
25967.6
35637.2
00.1
747.1
937.0
75.1
65.1
8...e
st
v7
r12
vr1
c1
s10576
0oot
oot
oot
0.0
01.0
02.3
14
96
96
04.5
44.2
21701.3
46663.1
26233.0
721.1
443.5
728.3
55.1
75.1
9...test
v7
r12
vr1
c1
s703
0oot
oot
oot
0.0
01.0
02.2
14
96
96
04.5
24.1
51720.3
26678.4
26242.9
121.1
646.5
828.9
15.1
85.2
0...e
st
v7
r12
vr5
c1
s29826
10
896.5
0oot
oot
0.0
01.0
02.6
19
87
80
04.5
64.1
41673.4
76087.4
75743.7
90.1
844.4
136.2
35.1
95.2
1...e
st
v7
r17
vr1
c1
s23882
0oot
oot
oot
0.0
01.0
03.1
10
64
64
04.3
84.1
91267.6
66651.6
26198.4
533.1
357.1
741.5
75.2
05.2
2...e
st
v7
r17
vr1
c1
s30331
0oot
oot
oot
0.0
01.0
03.0
10
64
64
04.2
94.1
71248.2
66663.3
66206.7
130.1
657.6
440.8
85.2
15.2
3...fyIn
terle
avedM
odM
ult-1
28
0oot
oot
oot
0.0
01.0
02.6
10
62
62
00.1
41.0
41232.1
96712.6
96176.8
734.6
249.6
640.4
45.2
25.2
4...ffy
Interle
avedM
odM
ult-3
20
oot
oot
oot
0.0
01.0
01.1
60
442
442
00.2
51.7
71686.8
56768.1
66507.5
72.5
513.0
43.3
35.2
35.2
5...iffyIn
terle
avedM
odM
ult-8
0oot
oot
oot
0.0
01.0
00.1
114
869
170
00.1
40.5
21780.1
96792.1
66777.0
30.0
08.8
82.7
05.2
4
Contin
ued
on
next
page
110
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.2
6lfsr
lfsr
008
031
064
0oot
oot
oot
0.0
01.0
03.2
20
41
41
04.7
011.4
81627.8
46812.7
74725.6
68.6
131.7
021.5
65.2
55.2
7lfsr
lfsr
008
063
096
0oot
oot
oot
0.0
01.0
05.3
14
66
04.4
112.4
11741.3
05031.5
32271.4
452.9
879.2
568.9
35.2
65.2
8lfsr
lfsr
008
063
128
0oot
oot
oot
0.0
01.0
05.8
18
44
03.9
210.2
11738.6
16288.3
32639.3
076.8
799.2
093.8
65.2
75.2
9lo
g-s
licin
gbvm
ul15
0oot
oot
oot
0.0
01.0
00.1
22
20
0.0
00.0
00.8
61.1
51.0
90.2
90.5
30.3
85.2
85.3
0lo
g-s
licin
gbvm
ul16
0oot
oot
oot
0.0
01.0
00.2
48
80
0.0
00.0
11.6
54.2
94.0
50.2
00.4
80.2
95.2
95.3
1lo
g-s
licin
gbvm
ul17
0oot
oot
oot
0.0
01.0
00.1
22
20
0.0
00.0
01.2
21.6
61.3
40.4
70.6
70.5
55.3
05.3
2lo
g-s
licin
gbvm
ul18
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00.0
01.3
01.7
51.4
30.4
90.7
60.5
85.3
15.3
3lo
g-s
licin
gbvsdiv
19
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
46.4
919.5
217.4
20.4
32.2
20.7
25.3
25.3
4lo
g-s
licin
gbvsdiv
20
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
47.0
821.6
219.4
40.4
62.1
70.8
05.3
35.3
5lo
g-s
licin
gbvsdiv
21
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
46.6
222.1
920.3
80.4
22.4
40.8
25.3
45.3
6lo
g-s
licin
gbvsdiv
22
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
47.4
124.1
321.9
30.5
42.6
90.8
95.3
55.3
7lo
g-s
licin
gbvsdiv
23
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
59.3
827.1
824.9
20.5
83.3
81.0
15.3
65.3
8lo
g-s
licin
gbvsdiv
24
0oot
oot
oot
0.0
01.0
00.3
414
14
00.0
10.0
49.6
729.4
127.1
20.6
13.4
21.0
95.3
75.3
9lo
g-s
licin
gbvsdiv
25
0oot
oot
oot
0.0
01.0
00.3
414
14
00.0
10.0
410.3
331.0
929.0
60.5
73.5
91.1
55.3
85.4
0lo
g-s
licin
gbvslt
12430
0oot
oot
oot
0.0
01.0
02.2
412
12
04.2
60.4
0261.9
81037.1
01003.9
129.5
076.5
245.0
95.3
95.4
1lo
g-s
licin
gbvslt
18374
0oot
oot
oot
0.0
01.0
06.4
414
14
038.6
81.0
3785.4
33061.2
72947.6
139.6
5236.6
5113.3
85.4
05.4
2lo
g-s
licin
gbvslt
19117
0oot
oot
oot
0.0
01.0
06.4
414
14
038.2
21.0
2584.5
72395.1
02273.9
737.3
8236.6
388.7
15.4
15.4
3lo
g-s
licin
gbvslt
19860
0oot
oot
oot
0.0
01.0
06.4
414
14
038.8
81.0
1609.6
12217.5
42093.1
134.0
5257.0
682.1
35.4
25.4
4lo
g-s
licin
gbvsm
od
15
0oot
oot
oot
0.0
01.0
00.2
11
42
25
21
0.0
30.0
51787.8
14840.0
94837.9
30.2
3613.8
239.3
55.4
35.4
5lo
g-s
licin
gbvsm
od
16
0oot
oot
oot
0.0
01.0
00.1
944
30
20.0
20.0
51744.8
72249.1
92247.2
10.0
21735.0
822.9
55.4
45.4
6lo
g-s
licin
gbvsm
od
17
0oot
oot
oot
0.0
01.0
00.3
945
31
70.0
30.1
4101.8
93933.9
83926.0
90.0
41129.7
836.4
35.4
55.4
7lo
g-s
licin
gbvsm
od
18
0oot
oot
oot
0.0
01.0
00.1
415
15
00.0
10.0
45.9
319.5
617.5
70.3
51.9
60.6
75.4
65.4
8lo
g-s
licin
gbvsm
od
19
0oot
oot
oot
0.0
01.0
00.1
415
15
00.0
10.0
56.7
622.4
920.2
20.4
62.4
30.7
85.4
75.4
9lo
g-s
licin
gbvsm
od
20
0oot
oot
oot
0.0
01.0
00.2
415
15
00.0
10.0
58.1
126.5
024.2
20.4
72.7
20.9
15.4
85.5
0lo
g-s
licin
gbvsrem
16
0oot
oot
oot
0.0
01.0
00.2
728
18
50.0
10.0
3543.2
03605.9
03604.6
00.0
21161.3
556.3
45.4
95.5
1lo
g-s
licin
gbvsrem
17
0oot
oot
oot
0.0
01.0
00.3
627
21
40.0
20.0
88.8
63855.1
83851.0
10.0
4957.5
462.1
85.5
05.5
2lo
g-s
licin
gbvsrem
18
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
56.5
421.6
519.7
10.2
01.8
80.8
05.5
15.5
3lo
g-s
licin
gbvsrem
19
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
47.4
423.0
520.8
60.4
32.1
30.8
55.5
25.5
4lo
g-s
licin
gbvsrem
20
0oot
oot
oot
0.0
01.0
00.2
414
14
00.0
10.0
48.2
825.2
723.0
30.4
62.5
50.9
45.5
35.5
5lo
g-s
licin
gbvsub
12973
0oot
oot
oot
0.0
01.0
00.5
11
10
0.0
00.0
0320.9
8318.3
3309.7
7318.3
3318.3
3318.3
35.5
45.5
6lo
g-s
licin
gbvsub
13970
0oot
oot
oot
0.0
01.0
00.2
11
10
0.0
00.0
0394.2
9391.4
8381.5
9391.4
8391.4
8391.4
85.5
55.5
7lo
g-s
licin
gbvsub
14967
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
0313.9
6310.9
8299.8
1310.9
8310.9
8310.9
85.5
65.5
8lo
g-s
licin
gbvsub
15964
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
0373.4
6366.9
3353.8
3366.9
3366.9
3366.9
35.5
75.5
9lo
g-s
licin
gbvsub
16961
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
0574.4
8567.7
0552.8
5567.7
0567.7
0567.7
05.5
85.6
0lo
g-s
licin
gbvsub
17958
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
0392.6
8385.6
3369.2
4385.6
3385.6
3385.6
35.5
95.6
1lo
g-s
licin
gbvsub
18955
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
0499.9
8491.9
1474.1
4491.9
1491.9
1491.9
15.6
05.6
2lo
g-s
licin
gbvsub
19952
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00.0
0532.1
2522.8
3501.8
8522.8
3522.8
3522.8
35.6
15.6
3lo
g-s
licin
gbvsub
20949
0oot
oot
oot
0.0
01.0
00.5
11
10
0.0
00.0
0675.5
1665.2
5643.2
3665.2
5665.2
5665.2
55.6
25.6
4lo
g-s
licin
gbvsub
21946
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
0894.4
0882.8
6858.7
4882.8
6882.8
6882.8
65.6
35.6
5lo
g-s
licin
gbvsub
22943
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
0828.1
2815.5
5786.2
3815.5
5815.5
5815.5
55.6
45.6
6lo
g-s
licin
gbvsub
23940
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
0722.1
1708.5
4676.6
0708.5
4708.5
4708.5
45.6
55.6
7lo
g-s
licin
gbvudiv
18
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
12.9
05.8
05.4
80.9
41.9
31.1
65.6
65.6
8lo
g-s
licin
gbvudiv
19
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
13.6
47.2
26.8
91.2
02.2
71.4
45.6
75.6
9lo
g-s
licin
gbvudiv
20
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
14.1
78.4
48.1
01.4
12.3
41.6
95.6
85.7
0lo
g-s
licin
gbvudiv
21
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
14.3
08.8
08.4
51.4
82.6
71.7
65.6
95.7
1lo
g-s
licin
gbvudiv
22
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
14.5
78.6
58.3
11.3
52.6
81.7
35.7
05.7
2lo
g-s
licin
gbvudiv
23
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
14.8
59.7
79.4
11.5
83.1
41.9
55.7
15.7
3lo
g-s
licin
gbvudiv
24
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
15.2
110.5
010.1
51.7
53.3
62.1
05.7
25.7
4lo
g-s
licin
gbvudiv
25
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
15.6
011.6
011.2
41.9
73.4
32.3
25.7
35.7
5lo
g-s
licin
gbvult
16967
0oot
oot
oot
0.0
01.0
06.1
36
60
19.0
50.4
8399.5
2938.4
1874.5
539.6
8190.6
685.3
15.7
45.7
6lo
g-s
licin
gbvult
17964
0oot
oot
oot
0.0
01.0
06.1
36
60
18.1
50.4
8456.0
9928.8
1864.6
045.8
3218.2
584.4
45.7
5
Contin
ued
on
next
page
111
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.7
7lo
g-s
licin
gbvult
18961
0oot
oot
oot
0.0
01.0
06.1
36
60
18.6
20.4
7559.0
91043.3
5978.6
538.4
6287.9
994.8
55.7
65.7
8lo
g-s
licin
gbvult
19958
0oot
oot
oot
0.0
01.0
06.1
36
60
21.1
70.4
8449.9
4873.8
5808.3
042.5
6268.2
079.4
45.7
75.7
9lo
g-s
licin
gbvult
20955
0oot
oot
oot
0.0
01.0
06.1
36
60
18.7
80.4
9556.0
71148.9
01084.2
754.4
0268.7
9104.4
55.7
85.8
0lo
g-s
licin
gbvult
21952
0oot
oot
oot
0.0
01.0
06.1
36
60
18.8
20.4
7555.4
7991.5
4926.1
140.5
6276.6
790.1
45.7
95.8
1lo
g-s
licin
gbvult
22949
0oot
oot
oot
0.0
01.0
06.1
36
60
18.7
70.4
9698.0
01359.6
31293.5
640.5
0317.9
9123.6
05.8
05.8
2lo
g-s
licin
gbvult
23946
0oot
oot
oot
0.0
01.0
06.2
36
60
18.8
20.4
9522.2
61207.6
11144.1
943.6
7317.9
8109.7
85.8
15.8
3lo
g-s
licin
gbvult
24943
0oot
oot
oot
0.0
01.0
06.2
36
60
19.2
00.4
9589.7
01221.9
61154.4
848.7
8300.4
3111.0
95.8
25.8
4lo
g-s
licin
gbvult
25940
0oot
oot
oot
0.0
01.0
06.2
36
60
18.8
00.4
8632.2
01302.6
91240.6
052.3
4273.8
6118.4
35.8
35.8
5lo
g-s
licin
gbvurem
15
0oot
oot
oot
0.0
01.0
00.1
33
32
0.0
00.0
0587.8
01767.6
51767.4
20.8
81177.1
3252.5
25.8
45.8
6lo
g-s
licin
gbvurem
16
0oot
oot
oot
0.0
01.0
00.2
725
16
00.0
10.0
314.5
843.1
341.8
80.1
03.0
30.8
85.8
55.8
7lo
g-s
licin
gbvurem
17
0oot
oot
oot
0.0
01.0
00.2
624
14
00.0
20.0
711.1
334.0
931.3
40.3
12.8
90.7
35.8
65.8
8lo
g-s
licin
gbvurem
18
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
14.7
59.5
89.2
21.5
92.9
91.9
25.8
75.8
9lo
g-s
licin
gbvurem
19
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
15.4
010.9
610.5
91.8
63.3
92.1
95.8
85.9
0lo
g-s
licin
gbvurem
20
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
15.8
711.8
911.5
11.9
53.7
22.3
85.8
95.9
1lo
g-s
licin
gbvurem
21
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
16.9
513.8
013.4
22.2
64.5
12.7
65.9
05.9
2lo
g-s
licin
gbvurem
22
0oot
oot
oot
0.0
01.0
00.1
23
30
0.0
00.0
17.1
914.3
814.0
02.3
54.5
62.8
85.9
15.9
3pspace
power2sum
.8087
0oot
oot
oot
0.0
01.0
00.8
11
10
0.0
00.0
0707.8
8707.5
2706.8
9707.5
2707.5
2707.5
25.9
25.9
4pspace
power2sum
.9679
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00.0
0187.6
6187.5
9186.7
3187.5
9187.5
9187.5
95.9
35.9
5RW
SExam
ple
15.txt
0oot
oot
oot
0.0
01.0
00.4
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.9
45.9
6RW
SExam
ple
19.txt
0oot
oot
oot
0.0
01.0
00.6
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.9
55.9
7RW
SExam
ple
20.txt
0oot
oot
oot
0.0
01.0
00.8
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.9
65.9
8RW
SExam
ple
7.txt
0oot
oot
oot
0.0
01.0
00.1
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.9
75.9
9uum
uum
12
0oot
oot
oot
0.0
01.0
00.5
241
1865
309
03.0
84.5
01788.3
86911.9
06505.3
60.0
06.7
30.9
55.9
85.1
00
uum
uum
16
0oot
oot
oot
0.0
01.0
01.2
132
1035
453
02.9
66.2
21781.0
76950.4
76496.8
10.0
19.4
81.3
35.9
95.1
01
uum
uum
20
0oot
oot
oot
0.0
01.0
01.9
79
603
280
06.1
612.1
61776.8
06860.8
96261.7
50.2
311.9
92.6
35.1
00
5.1
02
uum
uum
24
0oot
oot
oot
0.0
01.0
02.9
57
283
283
03.9
39.2
11771.8
66942.0
96375.9
80.6
216.4
73.8
95.1
01
5.1
03
uum
uum
28
0oot
oot
oot
0.0
01.0
02.9
44
159
159
04.3
99.7
91783.6
56954.4
06329.9
11.0
215.9
06.2
35.1
02
5.1
04
uum
uum
32
0oot
oot
oot
0.0
01.0
02.9
41
153
153
03.9
19.4
81773.6
66887.2
56333.1
71.1
727.9
36.5
05.1
03
5.1
05
VS3
VS3-b
enchm
ark-A
10
0oot
oot
oot
0.0
01.0
00.6
70
544
497
00.7
00.7
01764.1
47064.0
96823.8
30.4
67.3
02.8
45.1
04
5.1
06
VS3
VS3-b
enchm
ark-A
11
0oot
oot
oot
0.0
01.0
00.5
60
463
393
00.5
50.5
61631.7
37015.8
06813.4
70.3
79.2
43.2
65.1
05
5.1
07
VS3
VS3-b
enchm
ark-A
12
0oot
oot
oot
0.0
01.0
00.6
70
544
452
00.7
50.7
01657.7
07035.7
16789.3
70.4
68.0
52.7
55.1
06
5.1
08
VS3
VS3-b
enchm
ark-A
13
0oot
oot
oot
0.0
01.0
00.6
53
408
384
00.5
80.6
11784.5
27033.1
76802.3
32.5
511.6
53.8
25.1
07
5.1
09
VS3
VS3-b
enchm
ark-A
80
oot
oot
oot
0.0
01.0
00.6
60
464
410
00.6
50.6
31630.5
27055.2
96834.1
62.6
17.8
13.4
15.1
08
5.1
10
VS3
VS3-b
enchm
ark-S
20
oot
oot
oot
0.0
01.0
00.6
20
136
136
00.1
90.1
51570.3
86716.0
26544.9
615.7
837.5
618.1
05.1
09
5.1
11
...1
alt
true.B
V.c
.cil.c
.17
0oot
oot
oot
0.0
01.0
02.4
31
171
140
00.6
92.2
21724.1
16806.7
56517.7
60.0
334.8
59.5
35.1
10
5.1
12
...1
alt
true.B
V.c
.cil.c
.21
0oot
oot
oot
0.0
01.0
03.1
26
134
133
00.7
02.5
01701.5
26714.5
36403.8
77.4
170.8
813.0
65.1
11
5.1
13
brum
mayerbie
re2
sm
ulo
v1bw48
0oot
oot
oot
0.0
01.0
00.2
0.0
00.0
00.0
00.0
00.0
05.1
12
5.1
14
...e
rbie
re3
isqrtaddeqcheck
0oot
oot
oot
0.0
01.0
00.1
50
375
241
00.0
90.0
61261.7
96852.9
96810.5
90.0
428.4
34.8
35.1
13
5.1
15
...m
ayerbie
re3
isqrtaddnoif
0oot
oot
oot
0.0
01.0
00.1
60
458
184
00.0
80.0
51665.9
16836.6
26817.8
90.0
027.8
24.8
65.1
14
5.1
16
...m
ayerbie
re3
isqrteqcheck
0oot
oot
oot
0.0
01.0
00.1
48
368
198
00.0
90.0
61779.2
46862.9
06825.9
70.0
031.5
75.7
05.1
15
5.1
17
brum
mayerbie
re3
isqrtnoif
0oot
oot
oot
0.0
01.0
00.0
142
1076
131
00.1
40.1
11799.0
16893.8
36870.8
60.0
016.3
02.3
15.1
16
5.1
18
brum
mayerbie
re3
maxor128
0oot
oot
oot
0.0
01.0
00.2
20
21
318
0.0
60.0
31736.2
91956.6
31943.5
72.5
8131.1
833.1
65.1
17
5.1
19
brum
mayerbie
re3
maxor256
0oot
oot
oot
0.0
01.0
00.6
19
20
317
0.1
00.0
71750.5
22311.1
02276.1
62.6
9142.8
341.2
75.1
18
5.1
20
brum
mayerbie
re3
maxxor064
0oot
oot
oot
0.0
01.0
00.3
94
736
593
00.3
30.2
51759.3
16988.3
76814.4
00.1
829.8
21.5
35.1
19
5.1
21
brum
mayerbie
re3
maxxor128
0oot
oot
oot
0.0
01.0
00.5
50
384
384
00.3
70.2
41793.5
76953.2
46631.8
13.3
28.3
44.8
95.1
20
5.1
22
brum
mayerbie
re3
maxxor256
0oot
oot
oot
0.0
01.0
01.2
17
120
120
00.2
20.1
71740.3
56760.7
26428.4
414.9
834.0
324.2
35.1
21
5.1
23
...b
iere3
maxxorm
axorand032
0oot
oot
oot
0.0
01.0
00.3
0.0
00.0
00.0
00.0
00.0
05.1
22
5.1
24
...b
iere3
maxxorm
axorand064
0oot
oot
oot
0.0
01.0
00.7
91
712
712
01.0
20.7
11792.4
76830.7
06281.7
81.0
116.2
11.4
45.1
23
5.1
25
...b
iere3
maxxorm
axorand128
0oot
oot
oot
0.0
01.0
00.8
33
248
248
00.7
20.6
41788.4
76649.1
86200.1
64.9
635.6
47.6
85.1
24
5.1
26
...b
iere3
maxxorm
axorand256
0oot
oot
oot
0.0
01.0
02.0
11
72
72
00.6
00.5
51693.5
76558.7
56139.3
623.5
873.4
935.4
55.1
25
5.1
27
...y
erbie
re3
min
andm
axor128
0oot
oot
oot
0.0
01.0
00.4
62
469
26
00.7
70.5
21751.1
16099.5
25914.2
60.0
629.8
46.3
05.1
26
Contin
ued
on
next
page
112
Table
C.3
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental)
cm
p
base-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-n
inc-flr
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.1
28
...y
erbie
re3
min
andm
axor256
0oot
oot
oot
0.0
01.0
01.8
18
117
19
00.4
70.3
91785.7
45835.1
25500.6
51.3
282.3
726.2
85.1
27
5.1
29
brum
mayerbie
re3
min
xor128
0oot
oot
oot
0.0
01.0
00.7
90
704
704
00.7
10.5
71584.4
76888.2
96360.1
91.0
318.6
11.4
95.1
28
5.1
30
brum
mayerbie
re3
min
xor256
0oot
oot
oot
0.0
01.0
00.7
29
216
216
00.4
70.4
01769.9
56467.9
16215.4
57.1
042.2
410.7
35.1
29
5.1
31
...e
rbie
re3
min
xorm
inand064
0oot
oot
oot
0.0
01.0
00.5
70
544
419
00.8
20.5
61197.4
86887.9
96556.7
50.0
618.1
62.5
75.1
30
5.1
32
...e
rbie
re3
min
xorm
inand128
0oot
oot
oot
0.0
01.0
00.8
30
224
224
00.6
60.6
01369.9
06897.7
36552.7
07.0
929.3
18.4
95.1
31
5.1
33
...e
rbie
re3
min
xorm
inand256
0oot
oot
oot
0.0
01.0
01.7
10
64
64
00.5
40.5
41464.5
06647.0
16394.3
543.6
780.3
648.8
85.1
32
5.1
34
brum
mayerbie
re3
mulh
s16
0oot
oot
oot
0.0
01.0
00.2
20
137
92
10.0
10.0
123.2
487.5
985.2
50.0
01.3
30.2
35.1
33
5.1
35
brum
mayerbie
re3
mulh
s32
0oot
oot
oot
0.0
01.0
00.2
18
121
86
20.0
10.0
166.3
7239.4
4231.9
20.0
26.7
00.8
55.1
34
5.1
36
brum
mayerbie
re3
mulh
s64
0oot
oot
oot
0.0
01.0
00.2
18
125
101
00.0
10.0
1292.2
91035.6
11001.7
40.0
98.8
63.7
75.1
35
5.1
37
...e
rbie
re4
unconstrain
ed02
0oot
oot
oot
0.0
01.0
05.2
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
36
5.1
38
brum
mayerbie
re
countbits128
0oot
oot
oot
0.0
01.0
00.3
50
381
380
00.2
90.1
51510.2
46971.3
46738.7
30.0
020.7
04.1
45.1
37
5.1
39
brum
mayerbie
re
countbits256
0oot
oot
oot
0.0
01.0
00.7
20
141
140
00.2
70.1
11527.1
86825.4
26539.0
10.0
138.7
818.7
05.1
38
5.1
40
brum
mayerbie
re
countbits512
0oot
oot
oot
0.0
01.0
02.3
10
61
60
00.2
70.1
21530.4
96470.1
46051.3
40.0
281.7
951.7
65.1
39
5.1
41
...b
iere
countbitsrotate032
0oot
oot
oot
0.0
01.0
00.1
51
391
285
00.2
50.1
61766.2
66947.9
76930.1
20.0
029.4
44.3
05.1
40
5.1
42
...b
iere
countbitsrotate064
0oot
oot
oot
0.0
01.0
00.9
57
434
432
01.1
60.8
41795.5
96754.0
16650.0
50.0
215.7
33.6
55.1
41
5.1
43
...b
iere
countbitsrotate128
0oot
oot
oot
0.0
01.0
01.3
18
128
128
03.9
01.8
21716.4
06507.3
96401.1
07.3
352.9
821.7
65.1
42
5.1
44
...b
iere
countbitsrotate256
0oot
oot
oot
0.0
01.0
03.9
10
58
58
035.4
34.4
41236.8
76307.5
95953.8
034.2
4100.1
437.7
75.1
43
5.1
45
...y
erbie
re
countbitssrl0
64
0oot
oot
oot
0.0
01.0
00.1
47
294
24
21
0.1
10.0
61743.0
55340.2
25318.5
90.0
0445.6
78.3
25.1
44
5.1
46
...y
erbie
re
countbitssrl1
28
0oot
oot
oot
0.0
01.0
00.4
51
331
23
39
0.2
60.1
21751.6
94219.9
64157.6
40.0
0228.3
55.6
75.1
45
5.1
47
...y
erbie
re
countbitssrl2
56
0oot
oot
oot
0.0
01.0
01.4
21
141
43
00.2
40.1
01701.4
25947.1
55726.9
50.0
069.5
618.6
45.1
46
5.1
48
challenge
integerO
verflow
0oot
oot
oot
0.0
01.0
00.2
512
82
0.0
00.0
017.8
82023.5
02022.7
10.0
31066.9
680.9
45.1
47
5.1
49
...e
rbie
re4
unconstrain
ed08
0oom
oom
oom
-0.0
51.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
48
5.1
50
...e
rbie
re4
unconstrain
ed09
0oom
oom
oom
-0.0
91.0
06.9
01
10
0.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
05.5
113
Tab
leC
.4:
Det
aile
dre
sult
sof
PB
oole
ctor
inn
on
-in
crem
enta
lm
od
esw
ith
FL
R(P
ara
llel
-Non
-In
crem
enta
l-F
LR
)co
mp
are
dto
the
refe
ren
ceti
mes
Bas
e-N
on
-In
crem
enta
l.L
ast
colu
mn
par-
nin
cco
nta
ins
cross
refe
ren
ceto
det
ail
edre
sult
sof
Para
llel
-Non
-In
crem
enta
lin
Tab
.C.3
.
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
2.1
log-s
licin
gbvm
ul14
20
1477.5
01496.4
5884.5
0611.9
51.6
90.1
22
22
0.0
00
0883.9
91727.4
31727.3
40.2
7883.3
2345.4
92.1
2.2
core
ext
con
048
002
1024
20
487.7
2492.4
8127.6
9364.7
93.8
60.0
01
10
127.2
247
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
12.3
log-s
licin
gbvsm
od
14
20
1414.5
41430.9
81148.5
2282.4
61.2
50.1
13
42
25
39
1.2
50
01146.6
22999.0
72996.5
50.2
1233.2
719.8
62.2
2.4
log-s
licin
gbvudiv
17
20
1270.3
61291.0
71051.9
4239.1
31.2
30.2
59
66
4.0
70
01047.2
92923.9
82922.7
30.3
3623.2
1127.1
32.3
2.5
core
ext
con
036
002
1024
20
269.9
9268.4
252.2
9216.1
35.1
30.0
01
10
51.8
235
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
42.6
log-s
licin
gbvm
ul13
20
428.6
3431.5
4291.2
1140.3
31.4
80.1
22
22
0.0
00
0290.6
9538.9
2538.8
40.2
4289.9
4107.7
82.6
2.7
...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
71499.2
91364.7
4134.5
51.1
01.6
33
22
0.0
41
01364.1
31945.9
21883.5
16.8
41161.6
2277.9
92.5
2.8
challenge
multip
lyO
verflow
20
1035.6
11041.4
4928.1
9113.2
51.1
20.1
22
22
0.0
00
0927.6
71427.6
41427.5
24.0
8917.3
3285.5
32.7
2.9
core
ext
con
044
002
1024
20
195.8
2197.3
398.6
798.6
62.0
00.0
01
10
98.1
743
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
02.1
0...b
iere3
isqrtaddin
validvc
10
185.6
0181.9
986.9
395.0
62.0
90.0
30
209
179
00.1
60
086.2
0330.4
9316.7
80.0
22.6
20.4
42.1
12.1
1core
ext
con
064
001
1024
20
128.2
5129.3
243.9
485.3
82.9
40.0
01
10
43.4
363
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
92.1
2core
ext
con
056
002
1024
20
279.9
8284.3
3205.2
679.0
71.3
90.0
01
10
204.7
855
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
22.1
3core
ext
con
060
001
1024
20
110.6
0111.4
236.0
875.3
43.0
90.0
01
10
35.6
259
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
72.1
4lo
g-s
licin
gbvurem
14
20
757.1
6767.6
8700.5
767.1
11.1
00.1
33
33
0.7
00
0699.3
51003.6
41003.4
70.7
7523.7
9125.4
52.8
2.1
5...y
erbie
re3
isqrtin
validvc
10
181.6
6183.0
6124.5
358.5
31.4
70.0
30
222
141
00.1
60
0123.7
8476.1
8463.6
20.0
29.7
10.7
02.9
2.1
6lo
g-s
licin
gbvm
ul12
20
129.1
9131.3
683.0
848.2
81.5
80.0
22
22
0.0
00
082.4
8137.4
8137.4
00.2
281.9
227.5
02.1
02.1
7core
ext
con
052
002
1024
20
194.6
0205.8
0166.0
839.7
21.2
40.0
01
10
165.5
251
20
0.0
00.0
00.0
00.0
00.0
00.0
03.9
2.1
8core
ext
con
064
002
1024
20
344.2
9330.8
1313.3
317.4
81.0
60.0
01
10
312.8
463
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
62.1
9lo
g-s
licin
gbvudiv
15
20
162.7
0166.2
3150.6
115.6
21.1
00.0
33
33
0.6
80
0149.4
0198.7
1198.5
50.5
998.3
524.8
42.1
22.2
0lo
g-s
licin
gbvsm
od
12
20
138.5
8141.0
4129.3
811.6
61.0
90.1
13
42
25
39
1.2
50
0127.4
6338.6
0336.3
50.1
723.1
22.2
42.1
32.2
1lo
g-s
licin
gbvurem
13
20
147.9
7149.4
7138.7
810.6
91.0
80.0
33
33
0.6
90
0137.4
3187.2
9187.1
30.6
487.4
423.4
12.1
42.2
2brum
mayerbie
re3
isqrt
20
145.8
1146.4
6143.8
02.6
61.0
20.0
32
219
162
00.4
70
0142.7
6531.1
7516.9
40.0
19.7
80.7
23.7
2.2
3lo
g-s
licin
gbvadd
26192
20
202.5
2207.6
7206.1
71.5
01.0
10.3
01
10
0.0
00
0205.6
6205.4
8190.4
9205.4
8205.4
8205.4
82.1
62.2
4lo
g-s
licin
gbvadd
19983
20
121.5
9124.6
6123.6
21.0
41.0
10.2
01
10
0.0
00
0123.1
1118.9
5112.9
9118.9
5118.9
5118.9
52.2
22.2
5lo
g-s
licin
gbvadd
23531
20
167.4
5173.1
8172.1
81.0
01.0
10.2
01
10
0.0
00
0171.6
1165.0
3154.5
6165.0
3165.0
3165.0
32.1
72.2
6lo
g-s
licin
gbvadd
29740
20
264.9
3273.8
6273.0
00.8
61.0
00.3
01
10
0.0
00
0272.5
3271.4
0244.6
4271.4
0271.4
0271.4
02.2
02.2
7lo
g-s
licin
gbvadd
27966
20
232.0
1236.9
9236.1
70.8
21.0
00.3
01
10
0.0
00
0235.7
1235.5
2215.7
6235.5
2235.5
2235.5
22.1
52.2
8lo
g-s
licin
gbvadd
21757
20
165.0
7169.8
5169.1
90.6
61.0
00.2
01
10
0.0
00
0168.6
6163.2
7156.0
0163.2
7163.2
7163.2
72.1
82.2
9lo
g-s
licin
gbvadd
20870
20
132.0
0135.0
8134.4
40.6
41.0
00.2
01
10
0.0
00
0133.8
8129.1
6122.4
8129.1
6129.1
6129.1
62.2
42.3
0pspace
shift1add.2
4955
20
128.4
7131.6
2131.2
00.4
21.0
00.1
01
10
0.0
00
0130.6
7130.5
9130.0
4130.5
9130.5
9130.5
92.2
62.3
1brum
mayerbie
re
bitrev8192
20
101.9
7101.5
6101.1
90.3
71.0
00.0
01
10
0.0
00
00.0
50.0
40.0
00.0
40.0
40.0
42.2
72.3
2pspace
shift1add.2
7946
20
167.8
1172.1
3171.8
10.3
21.0
00.2
01
10
0.0
00
0171.2
8171.1
9170.6
0171.1
9171.1
9171.1
92.3
12.3
3pspace
shift1add.2
3958
20
114.0
8116.8
1116.5
20.2
91.0
00.1
01
10
0.0
00
0115.9
8115.9
0115.3
9115.9
0115.9
0115.9
02.3
62.3
4pspace
shift1add.2
8943
20
186.8
4191.4
6191.2
10.2
51.0
00.1
01
10
0.0
00
0190.7
3190.6
3190.0
2190.6
3190.6
3190.6
32.3
22.3
5pspace
shift1add.2
6949
20
152.3
3156.0
5155.8
20.2
31.0
00.1
01
10
0.0
00
0155.3
4155.2
5154.6
7155.2
5155.2
5155.2
52.3
42.3
6pspace
shift1add.2
9940
20
195.3
8200.4
5200.2
20.2
31.0
00.1
01
10
0.0
00
0199.7
4199.6
3199.0
0199.6
3199.6
3199.6
32.3
02.3
7lo
g-s
licin
gbvadd
24418
20
198.0
9204.1
8203.9
60.2
21.0
00.3
01
10
0.0
00
0203.4
4203.2
2191.5
0203.2
2203.2
2203.2
22.1
92.3
8pspace
shift1add.2
5952
20
143.2
8146.7
1146.4
90.2
21.0
00.1
01
10
0.0
00
0145.9
4145.8
5145.2
9145.8
5145.8
5145.8
52.2
82.3
9lo
g-s
licin
gbvadd
25305
20
186.3
4190.4
8190.2
90.1
91.0
00.3
01
10
0.0
00
0189.8
0189.6
2177.2
2189.6
2189.6
2189.6
22.2
12.4
0lo
g-s
licin
gbvadd
19096
20
109.7
8113.2
1113.0
40.1
71.0
00.2
01
10
0.0
00
0112.5
6109.0
6103.2
3109.0
6109.0
6109.0
62.2
92.4
1pspace
shift1add.2
2961
20
110.7
1113.3
2113.1
50.1
71.0
00.1
01
10
0.0
00
0112.5
0112.4
2111.9
4112.4
2112.4
2112.4
22.2
52.4
2lo
g-s
licin
gbvsub
05994
20
414.3
1415.5
7415.5
20.0
51.0
00.1
01
10
0.0
00
0414.9
9414.1
2412.1
3414.1
2414.1
2414.1
22.3
3
3.1
log-s
licin
gbvsub
04000
20
545.8
4545.6
2545.6
8-0
.06
1.0
00.1
01
10
0.0
00
0545.1
1544.6
8543.7
3544.6
8544.6
8544.6
83.1
3.2
bm
c-b
vgraycode
10
442.5
4448.8
9449.1
4-0
.25
1.0
00.3
01
10
0.2
11
00.1
70.1
70.0
40.1
70.1
70.1
73.2
Contin
ued
on
next
page
114
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
3.3
log-s
licin
gbvadd
22644
20
156.0
3165.9
3166.2
0-0
.27
1.0
00.2
01
10
0.0
00
0165.6
7159.6
7151.4
7159.6
7159.6
7159.6
73.3
3.4
log-s
licin
gbvadd
27079
20
216.4
1220.7
0221.1
0-0
.40
1.0
00.3
01
10
0.0
00
0220.5
8220.4
0203.2
9220.4
0220.4
0220.4
03.5
3.5
log-s
licin
gbvadd
18209
20
101.2
6102.5
1103.3
2-0
.81
0.9
90.2
01
10
0.0
00
0102.8
299.9
494.4
499.9
499.9
499.9
42.2
33.6
log-s
licin
gbvadd
28853
20
245.3
3251.6
5253.2
8-1
.63
0.9
90.3
01
10
0.0
00
0252.6
8252.4
8230.1
7252.4
8252.4
8252.4
83.6
3.7
bm
c-b
vm
agic
10
443.8
6453.0
8477.9
3-2
4.8
50.9
50.3
01
10
25.0
01
00.6
10.6
00.4
60.6
00.6
00.6
03.4
3.8
log-s
licin
gbvsm
od
13
20
360.2
6371.8
1396.6
9-2
4.8
80.9
40.1
13
42
25
39
1.2
30
0394.8
41172.3
81170.0
90.1
971.0
47.7
63.8
3.9
core
ext
con
060
002
1024
20
220.7
1220.8
6254.8
1-3
3.9
50.8
70.0
01
10
254.3
259
20
0.0
00.0
00.0
00.0
00.0
00.0
03.1
33.1
0brum
mayerbie
re3
isqrtadd
20
144.7
3153.5
8207.1
8-5
3.6
00.7
40.0
37
255
158
00.4
70
0206.1
4742.9
4723.7
90.0
29.3
40.9
13.1
53.1
1RW
SExam
ple
18.txt
10
160.6
1160.5
6222.8
7-6
2.3
10.7
20.1
01
10
62.0
40
0160.1
9160.1
0159.8
8160.1
0160.1
0160.1
02.3
53.1
2lo
g-s
licin
gbvudiv
16
20
390.9
9398.2
3470.2
5-7
2.0
20.8
50.1
715
66
0.6
80
0468.9
91420.4
11419.6
90.0
1337.0
340.5
83.1
83.1
3...y
erbie
re
countbitssrl0
32
20
387.0
6395.1
1485.0
2-8
9.9
10.8
10.1
40
164
19
42
1.2
82
0483.0
71042.7
41036.4
00.0
0150.7
22.6
32.4
3.1
4...a
yerbie
re2
sm
ulo
v4bw0512
10
128.7
8130.6
6224.7
4-9
4.0
80.5
80.4
33
22
0.0
21
0224.1
0350.2
7340.7
10.8
098.4
450.0
43.2
03.1
5pspace
power2sum
.5898
20
343.5
7351.3
4451.3
3-9
9.9
90.7
80.3
11
11
0.0
00
0450.8
0450.5
6449.7
6104.2
5346.3
0225.2
83.2
13.1
6pspace
power2sum
.5699
20
347.9
2348.9
0456.5
2-1
07.6
20.7
60.3
11
11
0.0
00
0456.0
1455.7
8455.0
2107.7
6348.0
2227.8
93.2
23.1
7lo
g-s
licin
gbvsub
06991
20
307.6
8315.9
4425.5
2-1
09.5
80.7
40.1
11
11
0.0
00
0425.0
1423.5
5418.2
9110.8
3312.7
2211.7
73.2
33.1
8pspace
power2sum
.6097
20
388.1
2387.3
7506.7
3-1
19.3
60.7
60.4
11
11
0.0
00
0506.1
8505.9
3505.0
9117.2
2388.7
1252.9
63.2
43.1
9pspace
power2sum
.6495
20
504.5
1505.7
8626.2
1-1
20.4
30.8
10.4
11
11
0.0
00
0625.6
2625.3
4624.4
3120.5
4504.8
0312.6
73.2
53.2
0pspace
power2sum
.8684
20
1028.3
01025.1
81155.1
9-1
30.0
10.8
90.4
11
11
0.0
00
01154.7
01154.5
81153.1
1115.5
81039.0
0577.2
93.2
63.2
1lo
g-s
licin
gbvsub
10979
20
161.2
1145.1
9275.9
4-1
30.7
50.5
30.2
11
11
0.0
00
0275.4
6269.6
2257.1
4123.3
7146.2
5134.8
13.2
73.2
2lo
g-s
licin
gbvsub
07988
20
321.6
9327.3
3471.2
5-1
43.9
20.6
90.1
11
11
0.0
00
0470.7
9467.5
4460.6
9139.5
0328.0
4233.7
73.2
83.2
3pspace
power2sum
.5500
20
496.1
6493.2
9642.9
1-1
49.6
20.7
70.4
11
11
0.0
00
0642.3
3642.1
2641.4
0150.8
9491.2
2321.0
63.2
93.2
4lo
g-s
licin
gbvsub
08985
20
378.5
6390.7
9554.8
8-1
64.0
90.7
00.1
11
11
0.0
00
0554.4
2550.7
6542.2
8162.2
0388.5
7275.3
83.3
03.2
5pspace
power2sum
.6296
20
434.7
6435.1
3600.5
5-1
65.4
20.7
20.4
11
11
0.0
00
0599.9
9599.7
2598.8
5156.9
7442.7
5299.8
63.3
13.2
6pspace
power2sum
.7092
20
452.5
0454.2
4640.6
6-1
86.4
20.7
10.4
11
11
0.0
00
0640.1
2639.6
1638.5
9179.5
2460.1
0319.8
13.3
33.2
7pspace
power2sum
.7689
20
515.9
3517.5
8716.2
5-1
98.6
70.7
20.4
11
11
0.0
00
0715.7
5715.1
0713.9
4199.6
7515.4
2357.5
53.3
43.2
8pspace
power2sum
.9480
20
751.8
3748.6
6956.7
3-2
08.0
70.7
80.5
11
11
0.0
00
0956.2
3956.1
0954.4
4208.7
1747.3
9478.0
53.3
53.2
9pspace
power2sum
.9878
20
573.3
5578.3
7792.2
2-2
13.8
50.7
30.4
11
11
0.0
00
0791.6
7791.5
4789.7
8218.6
2572.9
2395.7
73.3
63.3
0lo
g-s
licin
gbvsub
11976
20
664.6
3696.1
1927.1
2-2
31.0
10.7
50.4
11
11
0.0
00
0926.6
1919.8
4905.4
7247.8
0672.0
3459.9
23.3
73.3
1pspace
power2sum
.8485
20
872.3
0872.2
61106.8
0-2
34.5
40.7
90.6
11
11
0.0
00
01106.2
71106.1
51104.7
3234.3
4871.8
1553.0
83.3
83.3
2pspace
power2sum
.9281
20
1338.5
31337.1
01572.0
9-2
34.9
90.8
50.8
11
11
0.0
00
01571.5
61571.4
31569.8
2234.7
61336.6
7785.7
13.3
93.3
3pspace
power2sum
.8883
20
1035.8
81037.8
51273.7
1-2
35.8
60.8
10.4
11
11
0.0
00
01273.1
91273.0
71271.5
5228.6
91044.3
8636.5
43.4
03.3
4lo
g-s
licin
gbvsub
09982
20
328.0
6333.7
4570.8
6-2
37.1
20.5
80.1
11
11
0.0
00
0570.2
4565.5
5555.4
5235.2
0330.3
5282.7
83.4
13.3
5pspace
power2sum
.7490
20
505.6
9505.5
4754.0
7-2
48.5
30.6
70.3
11
11
0.0
00
0753.5
9752.9
6751.8
5247.5
9505.3
7376.4
83.4
23.3
6...a
yerbie
re2
sm
ulo
v4bw0768
10
563.4
2566.2
5821.7
1-2
55.4
60.6
91.0
33
22
0.0
31
0821.1
61200.9
21170.8
93.1
0648.7
0171.5
63.4
33.3
7pspace
power2sum
.7291
20
971.6
8970.1
21227.6
2-2
57.5
00.7
90.5
11
11
0.0
00
01227.0
41226.4
41225.3
6257.5
7968.8
6613.2
23.4
43.3
8...a
yerbie
re2
sm
ulo
v4bw0896
10
1274.5
21280.0
11566.2
6-2
86.2
50.8
21.3
33
22
0.0
31
01565.6
71943.6
91899.6
54.8
11363.0
2277.6
73.4
63.3
9pspace
power2sum
.7888
20
1145.4
91133.2
31508.4
2-3
75.1
90.7
50.8
11
11
0.0
00
01507.8
91507.2
01505.9
7369.4
31137.7
7753.6
03.4
73.4
0lo
g-s
licin
gbvsrem
14
20
256.9
0263.7
6692.0
7-4
28.3
10.3
80.1
721
16
18
1.0
20
0690.4
71908.9
41907.6
50.2
0179.9
029.3
73.5
03.4
1...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
3143.7
7588.9
7-4
45.2
00.2
40.7
33
22
0.0
21
0588.3
71018.7
31001.2
21.6
8405.1
6145.5
33.5
23.4
2pspace
power2sum
.6694
20
1005.8
11017.6
91464.7
0-4
47.0
10.6
90.6
11
11
0.0
00
01464.1
61463.8
71462.9
2456.7
61007.1
1731.9
33.5
13.4
3pspace
power2sum
.8286
20
898.4
1893.2
41347.4
6-4
54.2
20.6
60.5
11
11
0.0
00
01346.8
31346.7
21345.3
3454.6
3892.0
8673.3
63.5
33.4
4brum
mayerbie
re
nlz
be256
20
117.5
1119.3
1632.7
4-5
13.4
30.1
91.3
15
75
31
066.8
62
0565.0
61487.3
01439.8
40.0
754.3
98.6
53.4
93.4
5lo
g-s
licin
gbvsdiv
16
20
182.3
5185.3
1716.6
0-5
31.2
90.2
60.1
10
38
19
18
1.0
50
0714.9
41809.0
11807.2
80.0
2148.0
717.9
13.5
43.4
6pspace
power2sum
.6893
20
1102.0
21106.4
11648.6
6-5
42.2
50.6
70.6
11
11
0.0
00
01648.1
41647.8
31646.8
4545.8
01102.0
3823.9
13.5
53.4
7lo
g-s
licin
gbvult
6000
20
406.3
8410.0
2996.5
9-5
86.5
70.4
10.6
510
910
65.7
50
0929.1
22941.2
72927.7
426.0
0361.6
7101.4
23.5
63.4
8brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3135.5
4774.6
9-6
39.1
50.1
70.0
113
836
174
00.2
60
0773.6
02774.7
22753.1
00.0
18.0
41.1
03.4
83.4
9brum
mayerbie
re3
maxor064
20
209.1
4209.4
81175.6
0-9
66.1
20.1
80.1
129
129
3109
5.9
90
01168.7
01391.2
91376.1
30.0
036.9
73.8
03.5
73.5
0lo
g-s
licin
gbvsdiv
17
20
502.6
4511.0
81516.8
1-1
005.7
30.3
40.2
927
16
24
5.8
50
01510.2
14993.1
14987.6
70.0
4334.9
858.7
43.5
83.5
1lo
g-s
licin
gbvult
6997
20
201.1
9204.8
11374.0
3-1
169.2
20.1
50.7
510
910
65.8
10
01306.5
23696.7
33683.5
827.3
9481.2
8127.4
73.5
93.5
2lo
g-s
licin
gbvslt
5743
20
444.5
8444.9
21687.8
5-1
242.9
30.2
60.7
718
14
18
66.9
41
01618.4
65575.3
75552.3
425.0
7355.4
597.8
13.6
03.5
3lo
g-s
licin
gbvult
7994
20
230.1
0234.3
01635.7
2-1
401.4
20.1
40.7
510
910
65.7
30
01568.3
03914.6
23900.8
327.9
8618.7
3134.9
93.6
1
Contin
ued
on
next
page
115
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.1
log-s
licin
gbvslt
17631
20
1750.5
61785.5
0oot
-14.5
00.9
90.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
4.2
...e
st
v7
r12
vr5
c1
s14336
10
933.1
71778.6
5oot
-21.3
50.9
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.2
4.3
...test
v5
r15
vr5
c1
s8246
20
1106.6
41531.3
7oot
-268.6
30.8
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
4.4
log-s
licin
gbvslt
16888
20
1502.5
61524.1
3oot
-275.8
70.8
50.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
4.5
log-s
licin
gbvsdiv
18
20
1485.5
21509.3
7oot
-290.6
30.8
40.2
734
24
45.8
20
01527.2
52266.2
52261.7
90.1
4877.0
430.2
24.8
4.6
...e
st
v5
r15
vr5
c1
s26657
20
oot
1480.6
0oot
-319.4
00.8
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
4.7
...e
st
v7
r7
vr10
c1
s32506
10
1754.9
51394.4
1oot
-405.5
90.7
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
04.8
pspace
power2sum
.9082
20
1301.9
11303.6
6oot
-496.3
40.7
20.7
11
10
0.0
00
0508.8
2508.7
6507.9
8508.7
6508.7
6508.7
64.1
24.9
pip
epip
e-n
oabs.a
tla
s.q
fbv
20
1290.1
01298.2
3oot
-501.7
70.7
20.1
98
745
44
01.1
81
01796.3
26523.4
26508.9
70.0
022.5
03.9
84.1
34.1
0...st
v5
r10
vr10
c1
s21502
20
697.8
21247.1
6oot
-552.8
40.6
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
54.1
1lo
g-s
licin
gbvslt
16145
20
1188.5
61208.8
1oot
-591.1
90.6
72.0
412
12
0265.6
31
0500.6
11513.0
41482.2
630.3
2210.7
965.7
84.1
64.1
2lo
g-s
licin
gbvult
15970
20
1070.6
31091.1
1oot
-708.8
90.6
12.0
412
12
0263.3
50
0432.1
31230.7
31195.2
032.3
0156.7
953.5
14.1
74.1
3brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
51078.0
7oot
-721.9
30.6
00.1
137
1052
317
00.4
40
01786.1
76738.6
96682.1
00.0
215.6
81.7
04.1
84.1
4lo
g-s
licin
gbvslt
15402
20
1048.3
41064.7
6oot
-735.2
40.5
92.1
412
12
0267.8
91
0505.2
81695.2
61662.5
832.4
2153.7
773.7
14.1
94.1
5lo
g-s
licin
gbvult
14973
20
1019.5
01027.3
1oot
-772.6
90.5
72.0
412
12
0263.0
70
0377.1
41132.9
51099.6
728.3
2137.1
749.2
64.2
04.1
6...test
v7
r7
vr5
c1
s19694
10
1370.1
1999.2
7oot
-800.7
30.5
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.2
14.1
7VS3
VS3-b
enchm
ark-A
710
985.7
9987.8
8oot
-812.1
20.5
50.4
65
481
290
01.1
41
01774.8
36968.8
76794.0
00.2
714.6
03.4
64.2
24.1
8lo
g-s
licin
gbvslt
13916
20
908.9
9933.2
6oot
-866.7
40.5
22.0
412
12
1267.3
01
0422.3
22496.4
02462.6
030.3
0824.2
5104.0
24.2
34.1
9lo
g-s
licin
gbvult
12979
20
893.4
9915.2
3oot
-884.7
70.5
12.0
412
12
0261.8
10
0280.6
31038.3
21002.6
234.1
273.2
845.1
44.2
44.2
0lo
g-s
licin
gbvult
13976
20
893.7
3911.1
4oot
-888.8
60.5
12.0
412
12
0263.1
90
0334.6
61093.9
51063.3
329.3
0112.6
247.5
64.2
54.2
1...e
st
v5
r15
vr5
c1
s23844
20
oot
875.1
0oot
-924.9
00.4
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.2
64.2
2lo
g-s
licin
gbvslt
14659
20
838.9
9854.3
5oot
-945.6
50.4
72.0
412
12
2269.9
31
0367.7
52933.0
12898.2
130.0
51054.3
5117.3
24.2
74.2
3lo
g-s
licin
gbvslt
13173
20
801.4
6815.3
4oot
-984.6
60.4
52.0
412
12
1266.5
31
0306.8
81959.8
51930.8
827.4
8957.1
581.6
64.2
84.2
4lo
g-s
licin
gbvsrem
15
20
791.1
0801.9
2oot
-998.0
80.4
50.1
721
16
16
1.0
20
01046.6
14740.1
74738.8
80.2
2446.0
775.2
44.2
94.2
5float
newton.6
.3.i
10
564.0
4787.5
3oot
-1012.4
70.4
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
04.2
6...test
v7
r7
vr1
c1
s4574
10
609.3
3754.5
2oot
-1045.4
80.4
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
14.2
7...e
st
v5
r10
vr10
c1
s7608
20
1199.1
4715.0
0oot
-1085.0
00.4
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
24.2
8...e
st
v5
r10
vr5
c1
s13679
10
899.6
4696.3
3oot
-1103.6
70.3
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
34.2
9lo
g-s
licin
gbvslt
11687
20
665.8
6672.3
9oot
-1127.6
10.3
72.0
412
12
1266.1
01
0275.9
92131.7
52101.2
027.7
81042.5
588.8
24.3
44.3
0...st
v5
r15
vr10
c1
s11127
10
346.0
3669.6
9oot
-1130.3
10.3
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
54.3
1...tcom
p09
stall
bv
form
ula
20
609.5
3615.5
9oot
-1184.4
10.3
41.2
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
64.3
2...test
v7
r7
vr1
c1
s22845
10
316.6
3605.4
9oot
-1194.5
10.3
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
74.3
3lo
g-s
licin
gbvult
9988
20
555.0
3572.9
3oot
-1227.0
70.3
22.0
412
12
2263.0
20
0194.3
43041.7
63006.7
127.8
11132.4
3121.6
74.3
84.3
4lo
g-s
licin
gbvslt
10201
20
552.9
2571.4
3oot
-1228.5
70.3
22.0
412
12
2265.8
81
0228.8
82771.5
92739.7
029.0
61039.5
7110.8
64.3
94.3
5lo
g-s
licin
gbvult
10985
20
556.6
2561.1
4oot
-1238.8
60.3
12.0
412
12
0265.7
50
0288.6
81015.5
8981.7
029.3
173.9
144.1
64.4
04.3
6lo
g-s
licin
gbvult
8991
20
542.6
1554.9
3oot
-1245.0
70.3
11.9
412
12
0263.3
50
0246.3
1951.2
0918.3
430.2
861.5
141.3
64.4
14.3
7lo
g-s
licin
gbvslt
9458
20
539.1
6553.4
3oot
-1246.5
70.3
12.0
412
12
3266.2
01
0234.6
73238.5
43205.4
228.6
5893.5
3124.5
64.4
24.3
8lo
g-s
licin
gbvult
11982
20
518.1
6528.4
4oot
-1271.5
60.2
92.0
412
12
0262.9
70
0250.9
8962.3
6928.8
729.6
782.0
941.8
44.4
34.3
9float
mul03
30
420
514.8
0520.1
8oot
-1279.8
20.2
90.2
110
856
192
0139.9
220
01634.1
16515.3
86307.8
80.0
118.8
72.8
04.4
44.4
0lo
g-s
licin
gbvslt
10944
20
505.6
8510.4
4oot
-1289.5
60.2
82.0
412
12
3269.8
51
0292.1
83356.4
23325.5
829.2
2990.2
7129.0
94.4
54.4
1...test
v5
r10
vr5
c1
s8690
20
353.9
3459.7
9oot
-1340.2
10.2
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
64.4
2float
newton.3
.3.i
20
355.4
9450.8
5oot
-1349.1
50.2
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
74.4
3float
newton.1
.3.i
20
385.3
4445.6
2oot
-1354.3
80.2
50.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
84.4
4float
newton.4
.3.i
20
334.8
4418.4
4oot
-1381.5
60.2
30.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
94.4
5...e
st
v5
r15
vr1
c1
s26845
20
619.0
9415.2
0oot
-1384.8
00.2
30.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
04.4
6float
newton.5
.3.i
20
415.2
8404.4
7oot
-1395.5
30.2
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
14.4
7lfsr
lfsr
008
015
112
20
388.4
4399.4
9oot
-1400.5
10.2
20.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
24.4
8...e
st
v7
r7
vr10
c1
s10625
10
852.7
6391.8
4oot
-1408.1
60.2
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
34.4
9float
newton.2
.3.i
20
289.3
8385.6
1oot
-1414.3
90.2
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
44.5
0...test
v7
r7
vr5
c1
s3582
10
326.7
8384.0
3oot
-1415.9
70.2
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
5
Contin
ued
on
next
page
116
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.5
1...test
v7
r12
vr5
c1
s8938
10
615.1
6367.1
4oot
-1432.8
60.2
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
74.5
2lfsr
lfsr
008
159
128
20
285.7
4363.4
0oot
-1436.6
00.2
06.5
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
44.5
3lo
g-s
licin
gbvslt
8715
20
342.5
6350.6
8oot
-1449.3
20.1
92.0
518
14
4266.2
91
01279.0
94341.4
74296.7
027.4
8998.9
0111.3
24.5
84.5
4lfsr
lfsr
008
159
112
20
275.9
4338.9
8oot
-1461.0
20.1
94.5
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
94.5
5lfsr
lfsr
008
143
128
20
261.7
9336.9
6oot
-1463.0
40.1
95.8
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
04.5
6...test
v7
r7
vr1
c1
s24449
20
679.7
0330.9
5oot
-1469.0
50.1
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
14.5
7...servers
sla
pd
avc149923
10
325.6
4330.0
6oot
-1469.9
40.1
81.4
40
304
304
00.5
81
01579.1
36997.1
96722.9
35.2
813.5
36.3
24.6
24.5
8...servers
sla
pd
avc149922
10
325.6
8329.3
7oot
-1470.6
30.1
81.3
40
304
304
00.7
11
01587.6
16989.4
96717.6
45.2
913.3
76.3
34.6
34.5
9lfsr
lfsr
008
159
096
20
275.8
6320.4
7oot
-1479.5
30.1
83.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
44.6
0tacas07
Y86
btnft
20
135.7
7146.5
9oom
-1483.1
30.0
96.8
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
03
4.6
1...e
st
v5
r10
vr5
c1
s13195
10
337.5
5314.0
0oot
-1486.0
00.1
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
54.6
2lo
g-s
licin
gbvslt
7972
20
305.2
5307.9
8oot
-1492.0
20.1
70.7
618
14
666.9
11
01429.2
94676.0
64658.8
127.7
6688.7
6114.0
54.6
64.6
3lo
g-s
licin
gbvslt
5000
20
296.4
0297.7
4oot
-1502.2
60.1
70.7
618
14
10
67.0
61
01311.6
24137.7
04120.8
022.9
9451.7
591.9
54.6
74.6
4brum
mayerbie
re3
min
and256
20
289.6
6295.2
1oot
-1504.7
90.1
60.3
17
18
315
92.6
80
01672.8
22484.5
62467.7
90.1
2275.8
250.7
14.6
84.6
5lfsr
lfsr
008
015
128
20
277.3
6286.7
4oot
-1513.2
60.1
60.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.6
94.6
6...e
st
v5
r10
vr1
c1
s32538
20
208.1
6285.1
4oot
-1514.8
60.1
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
04.6
7lfsr
lfsr
008
127
128
20
212.8
0282.8
6oot
-1517.1
40.1
66.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
14.6
8lfsr
lfsr
008
015
096
20
272.7
5280.3
4oot
-1519.6
60.1
60.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
24.6
9VS3
VS3-b
enchm
ark-A
910
268.4
5271.7
7oot
-1528.2
30.1
50.5
70
543
369
01.2
71
01730.2
96992.8
76778.5
80.2
86.2
53.0
74.7
34.7
0lfsr
lfsr
008
143
112
20
205.3
9260.6
0oot
-1539.4
00.1
44.3
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
44.7
1...e
st
v5
r10
vr1
c1
s13516
10
231.3
6254.9
6oot
-1545.0
40.1
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
54.7
2...e
st
v5
r15
vr1
c1
s32559
20
163.7
5250.6
0oot
-1549.4
00.1
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
64.7
3...test
v5
r5
vr10
c1
s7194
10
141.8
4249.3
6oot
-1550.6
40.1
40.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
74.7
4lfsr
lfsr
008
127
112
20
176.0
0226.4
7oot
-1573.5
30.1
34.2
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
84.7
5lfsr
lfsr
008
159
080
20
194.3
3224.5
2oot
-1575.4
80.1
21.9
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.7
94.7
6...test
v5
r5
vr5
c1
s2800
10
228.1
4223.6
7oot
-1576.3
30.1
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
04.7
7lo
g-s
licin
gbvslt
7229
20
217.0
1221.1
9oot
-1578.8
10.1
20.7
618
14
667.2
71
01444.8
23738.9
93722.1
927.6
7711.0
991.1
94.8
14.7
8lfsr
lfsr
008
143
096
20
173.9
4217.0
4oot
-1582.9
60.1
22.9
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
24.7
9...test
v7
r7
vr5
c1
s14675
10
364.5
5212.1
9oot
-1587.8
10.1
20.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
34.8
0lfsr
lfsr
008
111
128
20
160.6
8210.2
8oot
-1589.7
20.1
23.8
0.0
00.0
00.0
00.0
00.0
04.8
44.8
1...y
erbie
re3
min
andm
axor064
20
207.7
9206.6
6oot
-1593.3
40.1
10.1
137
641
23
025.4
40
01764.6
23426.0
43358.9
80.0
121.3
92.6
34.8
54.8
2...e
st
v7
r7
vr10
c1
s24535
10
237.8
8205.9
0oot
-1594.1
00.1
10.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
64.8
3...sm
tcom
p09
sw
bv
form
ula
20
202.4
2204.9
0oot
-1595.1
00.1
11.5
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
14.8
4lfsr
lfsr
008
095
128
20
148.8
9203.3
5oot
-1596.6
50.1
12.7
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
74.8
5...sm
tcom
p09
lfbv
form
ula
20
180.2
4202.3
3oot
-1597.6
70.1
10.9
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.3
4.8
6lfsr
lfsr
004
111
112
20
178.0
5198.9
3oot
-1601.0
70.1
11.7
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.4
54.8
7...m
ayerbie
re2
sm
ulo
v2bw512
20
198.3
8198.7
8oot
-1601.2
20.1
12.5
68
40
132.4
80
01497.4
42047.5
51927.3
412.1
8403.8
4186.1
44.8
84.8
8lfsr
lfsr
008
111
112
20
148.6
0194.6
4oot
-1605.3
60.1
13.4
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.8
94.8
9lo
g-s
licin
gbvslt
6486
20
191.6
5194.4
0oot
-1605.6
00.1
10.7
718
14
14
67.1
21
01597.2
35703.3
15682.7
027.2
7445.2
1107.6
14.9
04.9
0lfsr
lfsr
008
095
112
20
129.0
8179.9
1oot
-1620.0
90.1
02.3
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
14.9
1float
newton.5
.2.i
20
249.5
3175.9
2oot
-1624.0
80.1
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
24.9
2lfsr
lfsr
008
047
096
20
160.4
0173.7
1oot
-1626.2
90.1
00.6
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
34.9
3lfsr
lfsr
008
111
096
20
133.3
2172.2
2oot
-1627.7
80.1
02.9
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
44.9
4float
newton.4
.2.i
20
143.2
5170.4
6oot
-1629.5
40.0
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
54.9
5lfsr
lfsr
008
159
064
20
149.5
2166.9
3oot
-1633.0
70.0
91.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.6
34.9
6...m
tcom
p09
full
bv
form
ula
20
172.4
6159.0
4oot
-1640.9
60.0
93.4
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
4.9
7...test
v5
r5
vr1
c1
s14623
10
143.2
0158.2
0oot
-1641.8
00.0
90.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
64.9
8lfsr
lfsr
004
143
128
20
116.2
1156.0
3oot
-1643.9
70.0
92.9
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
74.9
9lfsr
lfsr
004
159
128
20
112.1
0152.7
8oot
-1647.2
20.0
83.3
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.3
24.1
00
...m
ayerbie
re2
sm
ulo
v2bw448
20
150.6
3150.3
4oot
-1649.6
60.0
82.2
710
40
102.9
10
01591.5
83123.7
63001.9
98.1
2458.1
2208.2
54.9
84.1
01
lfsr
lfsr
008
127
096
20
113.0
4149.6
1oot
-1650.3
90.0
82.8
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.9
9
Contin
ued
on
next
page
117
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
4.1
02
lfsr
lfsr
008
079
128
20
107.7
6149.6
0oot
-1650.4
00.0
81.8
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
00
4.1
03
brum
mayerbie
re3
maxxor032
20
143.9
5148.0
4oot
-1651.9
60.0
80.1
160
1254
284
00.7
90
01764.4
76747.9
06694.2
90.0
612.5
51.6
34.1
01
4.1
04
lfsr
lfsr
008
095
096
20
105.5
2147.6
8oot
-1652.3
20.0
82.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
02
4.1
05
...m
ayerbie
re2
um
ulo
v1bw256
20
143.9
4145.2
2oot
-1654.7
80.0
81.2
00
0.0
00.0
00.0
00.0
00.0
04.1
04
4.1
06
...e
st
v5
r10
vr1
c1
s19145
10
316.4
9144.1
6oot
-1655.8
40.0
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
05
4.1
07
float
newton.3
.2.i
20
186.5
6140.6
8oot
-1659.3
20.0
80.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
06
4.1
08
lfsr
lfsr
008
143
080
20
112.1
3139.6
4oot
-1660.3
60.0
81.8
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
07
4.1
09
...sm
tcom
p09
nt
bv
form
ula
20
156.7
2137.2
7oot
-1662.7
30.0
80.7
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.5
64.1
10
float
newton.1
.2.i
20
139.9
8134.2
2oot
-1665.7
80.0
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
08
4.1
11
...test
v5
r5
vr5
c1
s24018
10
182.8
1131.0
6oot
-1668.9
40.0
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
09
4.1
12
brum
mayerbie
re
countbits064
20
129.3
4130.2
0oot
-1669.8
00.0
70.2
20
0.0
00.0
00.0
00.0
00.0
04.1
10
4.1
13
tacas07
Y86
std
20
129.6
5129.5
0oot
-1670.5
00.0
70.8
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
11
4.1
14
...sm
tcom
p09
std
bv
form
ula
20
108.8
6128.5
9oot
-1671.4
10.0
70.7
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.4
4.1
15
...m
ayerbie
re2
sm
ulo
v2bw384
20
122.0
6122.2
7oot
-1677.7
30.0
71.9
10
16
40
79.3
70
01670.5
74180.3
04023.9
15.2
2299.0
7154.8
34.1
12
4.1
16
...test
v5
r15
vr1
c1
s8236
20
927.6
6120.2
6oot
-1679.7
40.0
70.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
13
4.1
17
lfsr
lfsr
008
143
064
20
103.4
9119.6
5oot
-1680.3
50.0
71.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
14
4.1
18
brum
mayerbie
re3
min
or256
20
116.1
3118.5
5oot
-1681.4
50.0
70.7
29
216
216
067.8
10
01725.6
36015.8
35785.3
82.5
955.8
710.1
84.1
15
4.1
19
float
newton.2
.2.i
20
143.1
8116.2
9oot
-1683.7
10.0
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
16
4.1
20
brum
mayerbie
re3
min
xor064
20
111.8
4115.9
8oot
-1684.0
20.0
60.2
108
847
364
04.3
90
01790.3
96738.6
26550.4
90.0
221.4
41.7
84.1
17
4.1
21
VS3
VS3-b
enchm
ark-A
610
111.9
3112.6
8oot
-1687.3
20.0
60.3
60
453
237
01.0
31
01711.0
86919.6
26791.2
20.2
816.3
24.3
24.1
18
4.1
22
lfsr
lfsr
008
159
048
20
100.5
6110.8
7oot
-1689.1
30.0
60.5
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
03.6
24.1
23
core
ext
con
060
256
1024
20
101.7
3110.0
6oot
-1689.9
40.0
60.3
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
19
4.1
24
...m
ayerbie
re2
um
ulo
v1bw224
20
108.4
2109.1
8oot
-1690.8
20.0
61.0
26
162
22
023.4
00
01743.3
86085.5
75726.9
61.0
587.4
918.2
74.1
20
4.1
25
float
mult2.c
.50
10
109.3
9108.0
5oot
-1691.9
50.0
60.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
04.1
21
4.1
26
brum
mayerbie
re3
maxand256
20
102.0
5103.8
3oot
-1696.1
70.0
60.7
34
256
256
065.1
20
01705.0
76131.7
95815.1
90.9
639.2
86.8
74.1
22
5.1
...m
ayerbie
re
countbits1024
0oot
oot
oom
723.6
61.6
76.8
24
40
376.8
32
0684.5
8908.7
1876.1
0227.2
2436.3
3302.9
05.1
5.2
...e
rbie
re4
unconstrain
ed10
0oom
oom
oom
1.3
31.0
17.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.2
5.3
...e
rbie
re4
unconstrain
ed03
0oom
oom
oom
1.1
01.0
27.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.3
5.4
...e
rbie
re4
unconstrain
ed06
0oom
oom
oom
0.8
11.0
26.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.5
5.5
...e
rbie
re4
unconstrain
ed09
0oom
oom
oom
0.6
71.0
16.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
50
5.6
...e
rbie
re4
unconstrain
ed01
0oom
oom
oom
0.3
81.0
16.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.8
5.7
...e
rbie
re4
unconstrain
ed07
0oom
oom
oom
0.2
01.0
07.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.4
5.8
fft
Sz1024
34824
0oot
oot
oot
0.0
01.0
01.0
80
624
624
0122.6
40
01422.1
56586.4
86166.2
31.5
54.9
51.7
15.9
5.9
fft
Sz512
15128
00
oot
oot
oot
0.0
01.0
00.7
150
1168
1080
023.4
80
01655.5
27004.7
26579.4
80.0
52.7
00.6
75.1
05.1
0fft
Sz512
15128
10
oot
oot
oot
0.0
01.0
00.5
00
0.0
00.0
00.0
00.0
00.0
05.1
15.1
1fft
Sz512
15128
20
oot
oot
oot
0.0
01.0
00.7
170
1336
1256
021.0
81
01591.6
37014.3
46531.0
20.0
72.3
10.5
25.1
25.1
2fft
Sz512
15128
40
oot
oot
oot
0.0
01.0
00.8
00
0.0
00.0
00.0
00.0
00.0
05.1
35.1
3fft
Sz512
15128
50
oot
oot
oot
0.0
01.0
00.7
00
0.0
00.0
00.0
00.0
00.0
05.1
45.1
4fft
Sz512
15128
60
oot
oot
oot
0.0
01.0
00.6
10
0.0
00.0
00.0
00.0
00.0
05.1
55.1
5...st
v5
r10
vr10
c1
s15708
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
65.1
6...st
v7
r12
vr10
c1
s15994
10
1367.7
1oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
75.1
7...e
st
v7
r12
vr1
c1
s10576
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
85.1
8...test
v7
r12
vr1
c1
s703
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
95.1
9...e
st
v7
r12
vr5
c1
s29826
10
896.5
0oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
05.2
0...e
st
v7
r17
vr1
c1
s23882
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
15.2
1...e
st
v7
r17
vr1
c1
s30331
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
25.2
2...fyIn
terle
avedM
odM
ult-1
28
0oot
oot
oot
0.0
01.0
02.7
10
61
61
039.2
51
01251.4
56486.8
45977.9
734.3
550.5
640.8
05.2
35.2
3...ffy
Interle
avedM
odM
ult-3
20
oot
oot
oot
0.0
01.0
01.2
60
446
446
01.5
21
01563.6
96997.7
56712.3
32.5
712.9
83.1
45.2
45.2
4...iffyIn
terle
avedM
odM
ult-8
0oot
oot
oot
0.0
01.0
00.1
150
1141
205
00.1
11
01746.7
56813.4
16791.2
70.0
08.9
71.8
35.2
55.2
5lfsr
lfsr
008
031
064
0oot
oot
oot
0.0
01.0
00.2
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
6
Contin
ued
on
next
page
118
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.2
6lfsr
lfsr
008
063
096
0oot
oot
oot
0.0
01.0
01.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
75.2
7lfsr
lfsr
008
063
128
0oot
oot
oot
0.0
01.0
01.4
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.2
85.2
8lo
g-s
licin
gbvm
ul15
0oot
oot
oot
0.0
01.0
00.1
22
20
0.0
00
00.8
71.1
61.0
90.3
00.5
40.3
95.2
95.2
9lo
g-s
licin
gbvm
ul16
0oot
oot
oot
0.0
01.0
00.2
48
80
0.0
00
01.6
54.3
14.0
80.2
10.4
70.2
95.3
05.3
0lo
g-s
licin
gbvm
ul17
0oot
oot
oot
0.0
01.0
00.1
22
20
0.0
00
01.2
11.6
41.3
20.4
60.6
80.5
55.3
15.3
1lo
g-s
licin
gbvm
ul18
0oot
oot
oot
0.0
01.0
00.2
22
20
0.0
00
01.3
01.7
51.4
30.4
90.7
60.5
85.3
25.3
2lo
g-s
licin
gbvsdiv
19
0oot
oot
oot
0.0
01.0
00.2
414
14
05.8
10
06.5
019.6
217.5
10.4
42.2
20.7
35.3
35.3
3lo
g-s
licin
gbvsdiv
20
0oot
oot
oot
0.0
01.0
00.2
414
14
05.8
20
07.0
721.6
019.4
30.4
72.1
60.8
05.3
45.3
4lo
g-s
licin
gbvsdiv
21
0oot
oot
oot
0.0
01.0
00.3
414
14
05.8
20
06.6
222.2
720.4
50.4
22.4
30.8
25.3
55.3
5lo
g-s
licin
gbvsdiv
22
0oot
oot
oot
0.0
01.0
00.2
414
14
05.7
90
07.3
924.0
621.8
20.5
42.6
90.8
95.3
65.3
6lo
g-s
licin
gbvsdiv
23
0oot
oot
oot
0.0
01.0
00.2
414
14
05.8
10
09.3
827.2
024.9
20.5
83.3
81.0
15.3
75.3
7lo
g-s
licin
gbvsdiv
24
0oot
oot
oot
0.0
01.0
00.2
414
14
05.7
80
09.6
329.3
927.1
60.6
13.4
21.0
95.3
85.3
8lo
g-s
licin
gbvsdiv
25
0oot
oot
oot
0.0
01.0
00.2
414
14
05.8
40
010.3
531.2
429.1
70.5
83.5
91.1
65.3
95.3
9lo
g-s
licin
gbvslt
12430
0oot
oot
oot
0.0
01.0
02.0
412
12
0264.4
01
0260.6
31040.1
01007.0
830.1
574.2
645.2
25.4
05.4
0lo
g-s
licin
gbvslt
18374
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.4
15.4
1lo
g-s
licin
gbvslt
19117
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.4
25.4
2lo
g-s
licin
gbvslt
19860
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.4
35.4
3lo
g-s
licin
gbvsm
od
15
0oot
oot
oot
0.0
01.0
00.2
11
42
25
21
1.2
50
01784.2
64839.7
94837.6
30.2
3613.0
239.3
55.4
45.4
4lo
g-s
licin
gbvsm
od
16
0oot
oot
oot
0.0
01.0
00.1
944
30
21.2
30
01757.7
92264.2
52262.3
20.0
21748.0
723.1
05.4
55.4
5lo
g-s
licin
gbvsm
od
17
0oot
oot
oot
0.0
01.0
00.3
945
31
76.6
60
0101.9
53935.1
83927.2
10.0
41122.2
236.4
45.4
65.4
6lo
g-s
licin
gbvsm
od
18
0oot
oot
oot
0.0
01.0
00.1
415
15
06.5
70
05.9
319.6
317.5
70.3
71.9
40.6
85.4
75.4
7lo
g-s
licin
gbvsm
od
19
0oot
oot
oot
0.0
01.0
00.1
415
15
06.6
20
06.7
322.5
520.2
30.4
52.4
30.7
85.4
85.4
8lo
g-s
licin
gbvsm
od
20
0oot
oot
oot
0.0
01.0
00.2
415
15
06.5
70
08.0
626.5
724.3
00.4
82.7
30.9
25.4
95.4
9lo
g-s
licin
gbvsrem
16
0oot
oot
oot
0.0
01.0
00.2
728
18
51.0
20
0582.8
63620.1
13618.7
80.0
21149.9
056.5
65.5
05.5
0lo
g-s
licin
gbvsrem
17
0oot
oot
oot
0.0
01.0
00.3
627
21
45.6
30
08.8
83859.0
33854.7
30.0
4968.5
262.2
45.5
15.5
1lo
g-s
licin
gbvsrem
18
0oot
oot
oot
0.0
01.0
00.2
414
14
05.6
20
06.5
321.5
919.7
20.2
01.8
70.8
05.5
25.5
2lo
g-s
licin
gbvsrem
19
0oot
oot
oot
0.0
01.0
00.2
414
14
05.6
80
07.4
123.2
020.9
50.4
52.1
20.8
65.5
35.5
3lo
g-s
licin
gbvsrem
20
0oot
oot
oot
0.0
01.0
00.2
414
14
05.6
10
08.2
825.2
823.0
60.4
62.5
40.9
45.5
45.5
4lo
g-s
licin
gbvsub
12973
0oot
oot
oot
0.0
01.0
00.5
11
10
0.0
00
0321.6
9319.0
4310.4
7319.0
4319.0
4319.0
45.5
55.5
5lo
g-s
licin
gbvsub
13970
0oot
oot
oot
0.0
01.0
00.2
11
10
0.0
00
0394.9
1392.1
0382.2
1392.1
0392.1
0392.1
05.5
65.5
6lo
g-s
licin
gbvsub
14967
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00
0314.1
1311.1
2299.9
5311.1
2311.1
2311.1
25.5
75.5
7lo
g-s
licin
gbvsub
15964
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
0373.4
2366.8
9353.9
2366.8
9366.8
9366.8
95.5
85.5
8lo
g-s
licin
gbvsub
16961
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
0573.3
3566.5
6551.8
5566.5
6566.5
6566.5
65.5
95.5
9lo
g-s
licin
gbvsub
17958
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
0392.4
3385.3
9369.2
2385.3
9385.3
9385.3
95.6
05.6
0lo
g-s
licin
gbvsub
18955
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
0498.7
1490.6
3472.0
2490.6
3490.6
3490.6
35.6
15.6
1lo
g-s
licin
gbvsub
19952
0oot
oot
oot
0.0
01.0
00.3
11
10
0.0
00
0531.5
9522.3
0501.9
7522.3
0522.3
0522.3
05.6
25.6
2lo
g-s
licin
gbvsub
20949
0oot
oot
oot
0.0
01.0
00.5
11
10
0.0
00
0675.5
4665.2
9643.1
9665.2
9665.2
9665.2
95.6
35.6
3lo
g-s
licin
gbvsub
21946
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00
0896.8
5885.3
1859.7
7885.3
1885.3
1885.3
15.6
45.6
4lo
g-s
licin
gbvsub
22943
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00
0843.8
4831.2
8803.1
0831.2
8831.2
8831.2
85.6
55.6
5lo
g-s
licin
gbvsub
23940
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00
0718.5
1704.9
4675.5
9704.9
4704.9
4704.9
45.6
65.6
6lo
g-s
licin
gbvudiv
18
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
70
02.8
95.7
85.4
60.9
31.9
21.1
65.6
75.6
7lo
g-s
licin
gbvudiv
19
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
70
03.6
37.2
16.8
91.2
02.2
71.4
45.6
85.6
8lo
g-s
licin
gbvudiv
20
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
70
04.1
88.4
38.1
01.4
22.3
41.6
95.6
95.6
9lo
g-s
licin
gbvudiv
21
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
90
04.2
88.7
98.4
61.4
72.6
51.7
65.7
05.7
0lo
g-s
licin
gbvudiv
22
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
80
04.5
58.6
58.3
11.3
62.6
61.7
35.7
15.7
1lo
g-s
licin
gbvudiv
23
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
90
04.8
59.7
79.4
21.5
73.1
41.9
55.7
25.7
2lo
g-s
licin
gbvudiv
24
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
90
05.2
010.4
910.1
51.7
53.3
42.1
05.7
35.7
3lo
g-s
licin
gbvudiv
25
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
80
05.6
011.5
611.2
01.9
53.4
22.3
15.7
45.7
4lo
g-s
licin
gbvult
16967
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.7
55.7
5lo
g-s
licin
gbvult
17964
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.7
65.7
6lo
g-s
licin
gbvult
18961
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.7
7
Contin
ued
on
next
page
119
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.7
7lo
g-s
licin
gbvult
19958
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.7
85.7
8lo
g-s
licin
gbvult
20955
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.7
95.7
9lo
g-s
licin
gbvult
21952
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.8
05.8
0lo
g-s
licin
gbvult
22949
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.8
15.8
1lo
g-s
licin
gbvult
23946
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.8
25.8
2lo
g-s
licin
gbvult
24943
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.8
35.8
3lo
g-s
licin
gbvult
25940
0oot
oot
oot
0.0
01.0
00.1
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.8
45.8
4lo
g-s
licin
gbvurem
15
0oot
oot
oot
0.0
01.0
00.1
33
32
0.6
80
0584.1
21767.5
41767.3
60.8
71180.7
2252.5
15.8
55.8
5lo
g-s
licin
gbvurem
16
0oot
oot
oot
0.0
01.0
00.2
725
16
00.6
90
014.5
843.2
441.9
20.1
13.0
30.8
85.8
65.8
6lo
g-s
licin
gbvurem
17
0oot
oot
oot
0.0
01.0
00.2
624
14
04.0
80
011.2
134.1
631.3
70.3
22.8
90.7
35.8
75.8
7lo
g-s
licin
gbvurem
18
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
60
04.7
69.5
99.2
41.5
93.0
01.9
25.8
85.8
8lo
g-s
licin
gbvurem
19
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
60
05.4
210.9
810.6
11.8
53.4
12.2
05.8
95.8
9lo
g-s
licin
gbvurem
20
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
80
05.8
811.9
011.5
31.9
73.7
22.3
85.9
05.9
0lo
g-s
licin
gbvurem
21
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
80
06.9
413.8
213.4
32.2
74.4
92.7
65.9
15.9
1lo
g-s
licin
gbvurem
22
0oot
oot
oot
0.0
01.0
00.1
23
30
4.0
70
07.2
114.4
214.0
42.3
34.5
82.8
85.9
25.9
2pspace
power2sum
.8087
0oot
oot
oot
0.0
01.0
00.8
11
10
0.0
00
0707.9
9707.6
2706.9
9707.6
2707.6
2707.6
25.9
35.9
3pspace
power2sum
.9679
0oot
oot
oot
0.0
01.0
00.4
11
10
0.0
00
0187.6
4187.5
7186.7
1187.5
7187.5
7187.5
75.9
45.9
4RW
SExam
ple
15.txt
0oot
oot
oot
0.0
01.0
00.4
01
10
66.4
01
00.0
00.0
00.0
00.0
00.0
00.0
05.9
55.9
5RW
SExam
ple
19.txt
0oot
oot
oot
0.0
01.0
00.5
01
10
475.5
70
00.0
00.0
00.0
00.0
00.0
00.0
05.9
65.9
6RW
SExam
ple
20.txt
0oot
oot
oot
0.0
01.0
00.7
01
10
1029.9
20
00.0
00.0
00.0
00.0
00.0
00.0
05.9
75.9
7RW
SExam
ple
7.txt
0oot
oot
oot
0.0
01.0
00.1
01
10
15.4
30
00.0
00.0
00.0
00.0
00.0
00.0
05.9
85.9
8uum
uum
12
0oot
oot
oot
0.0
01.0
00.4
177
1367
213
0208.0
12
01574.0
86033.8
55786.0
10.0
07.7
81.3
55.9
95.9
9uum
uum
16
0oot
oot
oot
0.0
01.0
01.3
93
720
469
0496.4
82
01292.3
75047.8
74700.7
00.0
610.6
01.3
05.1
00
5.1
00
uum
uum
20
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
01
5.1
01
uum
uum
24
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
02
5.1
02
uum
uum
28
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
03
5.1
03
uum
uum
32
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
04
5.1
04
VS3
VS3-b
enchm
ark-A
10
0oot
oot
oot
0.0
01.0
00.6
70
544
499
01.3
61
01756.6
17066.6
06827.5
10.4
77.2
22.8
45.1
05
5.1
05
VS3
VS3-b
enchm
ark-A
11
0oot
oot
oot
0.0
01.0
00.5
60
463
391
01.2
71
01632.7
67006.6
46804.9
70.3
69.2
53.2
65.1
06
5.1
06
VS3
VS3-b
enchm
ark-A
12
0oot
oot
oot
0.0
01.0
00.5
70
536
397
01.3
91
01686.6
16956.4
66727.7
10.4
520.7
12.9
05.1
07
5.1
07
VS3
VS3-b
enchm
ark-A
13
0oot
oot
oot
0.0
01.0
00.6
53
408
384
01.8
01
01783.8
07012.1
16783.4
32.5
29.4
13.8
05.1
08
5.1
08
VS3
VS3-b
enchm
ark-A
80
oot
oot
oot
0.0
01.0
00.5
60
464
410
01.5
31
01648.1
87046.0
96820.5
52.6
37.7
43.4
55.1
09
5.1
09
VS3
VS3-b
enchm
ark-S
20
oot
oot
oot
0.0
01.0
00.6
20
136
136
01.5
71
01567.2
66734.2
06561.5
815.8
737.5
418.1
55.1
10
5.1
10
...1
alt
true.B
V.c
.cil.c
.17
0oot
oot
oot
0.0
01.0
02.2
20
137
124
01178.1
124
0473.3
82226.0
12056.3
00.0
423.5
85.6
85.1
11
5.1
11
...1
alt
true.B
V.c
.cil.c
.21
0oot
oot
oot
0.0
01.0
00.0
01
00
0.0
00.0
00.0
00.0
00.0
00.0
00.0
05.1
12
5.1
12
brum
mayerbie
re2
sm
ulo
v1bw48
0oot
oot
oot
0.0
01.0
00.1
00
0.0
00.0
00.0
00.0
00.0
05.1
13
5.1
13
...e
rbie
re3
isqrtaddeqcheck
0oot
oot
oot
0.0
01.0
00.1
50
376
240
00.3
70
01266.0
56852.8
56810.9
20.0
328.0
64.8
35.1
14
5.1
14
...m
ayerbie
re3
isqrtaddnoif
0oot
oot
oot
0.0
01.0
00.1
51
378
159
00.0
50
01780.5
26778.8
96762.6
90.0
027.7
26.0
75.1
15
5.1
15
...m
ayerbie
re3
isqrteqcheck
0oot
oot
oot
0.0
01.0
00.1
48
368
198
00.3
70
01781.4
16865.8
86828.8
90.0
032.0
25.7
05.1
16
5.1
16
brum
mayerbie
re3
isqrtnoif
0oot
oot
oot
0.0
01.0
00.0
145
1100
131
00.0
50
01777.6
36846.1
46822.8
20.0
116.2
82.2
65.1
17
5.1
17
brum
mayerbie
re3
maxor128
0oot
oot
oot
0.0
01.0
00.2
20
21
318
24.0
60
01741.6
81963.3
91950.2
42.8
7130.3
633.2
85.1
18
5.1
18
brum
mayerbie
re3
maxor256
0oot
oot
oot
0.0
01.0
00.6
18
19
316
93.6
20
01690.1
62195.9
42164.1
23.1
2140.6
842.2
35.1
19
5.1
19
brum
mayerbie
re3
maxxor064
0oot
oot
oot
0.0
01.0
00.3
93
728
588
03.1
00
01743.5
56985.8
16812.4
60.1
729.6
11.5
45.1
20
5.1
20
brum
mayerbie
re3
maxxor128
0oot
oot
oot
0.0
01.0
00.5
50
384
384
012.3
00
01746.0
16949.9
06628.9
63.0
98.3
64.8
05.1
21
5.1
21
brum
mayerbie
re3
maxxor256
0oot
oot
oot
0.0
01.0
01.3
17
120
120
050.3
60
01742.4
86574.1
86251.2
218.8
832.7
724.2
65.1
22
5.1
22
...b
iere3
maxxorm
axorand032
0oot
oot
oot
0.0
01.0
00.2
00
0.0
00.0
00.0
00.0
00.0
05.1
23
5.1
23
...b
iere3
maxxorm
axorand064
0oot
oot
oot
0.0
01.0
00.7
90
704
704
041.5
40
01676.5
46695.1
56177.3
81.0
316.1
41.5
95.1
24
5.1
24
...b
iere3
maxxorm
axorand128
0oot
oot
oot
0.0
01.0
00.8
30
224
224
0155.4
80
01297.5
56043.0
95636.1
34.8
735.5
87.7
25.1
25
5.1
25
...b
iere3
maxxorm
axorand256
0oot
oot
oot
0.0
01.0
02.0
10
64
64
0639.7
30
01122.8
84075.3
13807.4
623.5
774.3
736.0
65.1
26
5.1
26
...y
erbie
re3
min
andm
axor128
0oot
oot
oot
0.0
01.0
00.4
61
461
26
098.0
20
01668.8
25766.3
55585.6
80.0
629.4
46.0
85.1
27
5.1
27
...y
erbie
re3
min
andm
axor256
0oot
oot
oot
0.0
01.0
01.7
14
85
19
0375.0
90
01377.8
04277.9
94037.6
51.3
282.4
427.0
85.1
28
Contin
ued
on
next
page
120
Table
C.4
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-Non-Increm
ental-FLR)
cm
p
par-n
inc
tot(wc)
diff
su
mem
rnds
subproble
ms
flr
search
par-n
inc
tot
max
fin
tot(wc)
flres
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[]
5.1
28
brum
mayerbie
re3
min
xor128
0oot
oot
oot
0.0
01.0
00.6
90
704
704
016.6
40
01612.9
26767.0
06257.1
31.0
518.5
81.5
25.1
29
5.1
29
brum
mayerbie
re3
min
xor256
0oot
oot
oot
0.0
01.0
00.7
28
208
208
067.4
00
01730.1
36222.7
35983.1
87.1
942.6
510.8
25.1
30
5.1
30
...e
rbie
re3
min
xorm
inand064
0oot
oot
oot
0.0
01.0
00.5
80
624
434
049.4
10
01572.5
36530.7
56148.6
80.0
018.1
82.0
85.1
31
5.1
31
...e
rbie
re3
min
xorm
inand128
0oot
oot
oot
0.0
01.0
00.8
30
224
224
0183.5
10
01371.6
46194.5
85882.8
77.4
929.3
18.5
85.1
32
5.1
32
...e
rbie
re3
min
xorm
inand256
0oot
oot
oot
0.0
01.0
01.6
740
40
0721.9
40
0893.9
33701.3
63562.4
743.6
974.8
649.3
55.1
33
5.1
33
brum
mayerbie
re3
mulh
s16
0oot
oot
oot
0.0
01.0
00.1
19
129
89
00.0
00
021.4
975.5
773.6
90.0
10.9
20.2
55.1
34
5.1
34
brum
mayerbie
re3
mulh
s32
0oot
oot
oot
0.0
01.0
00.2
18
123
87
00.0
00
065.5
5229.4
8222.0
00.0
22.3
80.8
15.1
35
5.1
35
brum
mayerbie
re3
mulh
s64
0oot
oot
oot
0.0
01.0
00.2
18
125
101
00.0
00
0293.6
91034.3
51000.8
30.1
08.8
43.7
65.1
36
5.1
36
...e
rbie
re4
unconstrain
ed02
0oot
oot
oot
0.0
01.0
05.2
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
37
5.1
37
brum
mayerbie
re
countbits128
0oot
oot
oot
0.0
01.0
00.3
20
0.0
00.0
00.0
00.0
00.0
05.1
38
5.1
38
brum
mayerbie
re
countbits256
0oot
oot
oot
0.0
01.0
00.8
20
0.0
00.0
00.0
00.0
00.0
05.1
39
5.1
39
brum
mayerbie
re
countbits512
0oot
oot
oot
0.0
01.0
02.3
20
0.0
00.0
00.0
00.0
00.0
05.1
40
5.1
40
...b
iere
countbitsrotate032
0oot
oot
oot
0.0
01.0
00.1
51
391
285
00.3
20
01760.9
96936.9
26919.1
00.0
029.7
24.2
95.1
41
5.1
41
...b
iere
countbitsrotate064
0oot
oot
oot
0.0
01.0
00.9
56
426
424
03.9
20
01785.2
86743.2
46639.3
30.0
215.9
53.6
55.1
42
5.1
42
...b
iere
countbitsrotate128
0oot
oot
oot
0.0
01.0
01.3
18
128
128
052.0
10
01734.5
76236.8
76135.1
47.2
852.4
421.7
35.1
43
5.1
43
...b
iere
countbitsrotate256
0oot
oot
oot
0.0
01.0
02.6
00
0.0
00.0
00.0
00.0
00.0
05.1
44
5.1
44
...y
erbie
re
countbitssrl0
64
0oot
oot
oot
0.0
01.0
00.1
39
185
14
48
2.8
02
01496.0
22936.7
12920.4
20.0
0394.3
27.0
45.1
45
5.1
45
...y
erbie
re
countbitssrl1
28
0oot
oot
oot
0.0
01.0
00.5
57
405
46
27
7.0
72
01572.3
15508.7
95408.2
10.0
055.3
65.6
45.1
46
5.1
46
...y
erbie
re
countbitssrl2
56
0oot
oot
oot
0.0
01.0
01.1
40
282
38
16
20.2
32
01631.0
55961.1
25781.3
40.0
097.4
59.3
05.1
47
5.1
47
challenge
integerO
verflow
0oot
oot
oot
0.0
01.0
00.2
512
82
0.0
10
017.9
22021.3
02020.4
90.0
31064.3
580.8
55.1
48
5.1
48
...e
rbie
re4
unconstrain
ed08
0oom
oom
oom
-0.0
61.0
06.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.1
49
5.1
49
...e
rbie
re4
unconstrain
ed05
0oom
oom
oom
-0.1
01.0
07.0
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.7
5.1
50
...e
rbie
re4
unconstrain
ed04
0oom
oom
oom
-0.5
80.9
96.9
01
10
0.0
00
00.0
00.0
00.0
00.0
00.0
00.0
05.6
121
Tab
leC
.5:
Det
aile
dre
sult
sof
PB
oole
ctor
wit
hlo
ok-a
hea
dfo
rsa
tisfi
ab
lein
stan
ces
(Para
llel
-In
crem
enta
l-S
at)
com
pare
dto
the
refe
ren
ceti
mes
Bas
e-In
crem
enta
l.L
ast
colu
mn
par-
inc
conta
ins
cross
refe
ren
ceto
det
ail
edre
sult
sof
Para
llel
-In
crem
enta
lin
Tab
.C.1
.
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental-Sat)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
1.1
log-s
licin
gbvurem
16
20
oot
oot
1004.8
5795.1
51.7
90.2
824
23
22
0.0
10.3
41003.8
72534.6
22534.4
30.0
0340.9
830.5
41.2
1.2
brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
5oot
1019.3
9780.6
11.7
70.9
107
599
83
106
0.0
85.6
61012.1
53066.8
73056.0
40.0
028.2
61.8
01.1
1.3
...iffyIn
terle
avedM
odM
ult-8
20
oot
oot
1283.7
9516.2
11.4
00.2
78
506
78
00.0
91.4
31281.0
24664.2
24662.6
70.0
021.9
63.8
01.3
1.4
core
ext
con
056
002
1024
20
279.9
8oot
1461.8
6338.1
41.2
30.9
11
11
0.0
00.0
01461.2
81461.2
81460.8
9155.0
61306.2
3730.6
41.4
1.5
core
ext
con
060
002
1024
20
220.7
1oot
1465.9
5334.0
51.2
30.6
11
11
0.0
00.0
01465.4
61465.4
61465.0
5165.9
31299.5
3732.7
31.5
1.6
core
ext
con
052
002
1024
20
194.6
0oot
1496.5
1303.4
91.2
00.7
11
11
0.0
00.0
01496.0
01496.0
01495.6
3138.2
11357.7
9748.0
01.6
1.7
log-s
licin
gbvudiv
18
20
oot
oot
1542.5
4257.4
61.1
70.9
11
40
31
23
0.0
37.5
51530.5
64253.0
14251.8
00.0
0530.9
934.5
81.7
1.8
core
ext
con
064
002
1024
20
344.2
9oot
1769.0
031.0
01.0
20.9
11
11
0.0
00.0
01768.4
61768.4
61768.0
2164.7
41603.7
2884.2
31.8
2.1
log-s
licin
gbvudiv
17
20
1270.3
61373.3
4621.9
9751.3
52.2
10.7
10
33
23
21
0.0
25.9
7612.4
61573.2
91572.1
70.0
0186.0
215.7
32.1
2.2
log-s
licin
gbvurem
15
20
oot
1147.7
1412.3
8735.3
32.7
80.2
824
23
20
0.0
10.4
1411.3
7984.9
3984.7
50.0
0136.5
412.3
12.2
2.3
log-s
licin
gbvsm
od
14
20
1414.5
41350.5
9802.2
8548.3
11.6
80.3
42
174
46
111
0.1
22.7
1797.9
51926.1
91924.6
50.0
062.3
43.1
12.3
2.4
log-s
licin
gbvm
ul12
20
129.1
9962.3
1421.2
8541.0
32.2
80.1
22
22
0.0
00.0
2420.5
5814.1
6814.1
40.2
8419.7
5162.8
32.4
2.5
core
ext
con
048
002
1024
20
487.7
21716.6
41295.0
0421.6
41.3
30.7
11
11
0.0
00.0
01294.5
01294.5
01294.1
9124.7
31169.7
7647.2
52.5
2.6
core
ext
con
044
002
1024
20
195.8
21571.3
51267.2
0304.1
51.2
40.7
11
11
0.0
00.0
01266.6
91266.6
91266.4
0107.6
01159.0
9633.3
52.6
2.7
log-s
licin
gbvudiv
16
20
390.9
9521.1
2250.4
4270.6
82.0
80.2
824
22
20
0.0
10.5
1249.2
2739.0
8738.8
50.0
086.2
79.2
42.8
2.8
core
ext
con
036
002
1024
20
269.9
91156.1
9896.2
1259.9
81.2
90.5
11
11
0.0
00.0
0895.7
2895.7
2895.4
864.5
4831.1
9447.8
62.9
2.9
...sm
tcom
p09
lfbv
form
ula
20
180.2
4207.7
120.5
0187.2
110.1
30.8
840
32
00.0
52.4
117.2
855.6
650.2
20.0
22.8
90.5
92.1
02.1
0lo
g-s
licin
gbvurem
14
20
757.1
6270.8
8128.6
0142.2
82.1
10.1
824
22
19
0.0
10.3
0127.6
5443.7
3443.5
70.0
060.1
65.6
92.1
12.1
1float
newton.6
.3.i
10
564.0
4437.0
6303.7
5133.3
11.4
44.0
10
30
21
00.4
118.5
6279.7
8881.3
7846.1
00.3
854.6
67.6
04.4
72.1
2...sm
tcom
p09
nt
bv
form
ula
20
156.7
2175.2
882.1
993.0
92.1
31.8
14
80
56
00.1
57.8
871.9
2274.2
4260.7
40.0
620.7
31.3
22.1
22.1
3challenge
multip
lyO
verflow
20
1035.6
11002.1
1923.8
878.2
31.0
80.1
22
22
0.0
00.0
2923.3
31507.2
81507.2
50.5
6921.5
5301.4
62.1
42.1
4lo
g-s
licin
gbvudiv
15
20
162.7
0192.9
5117.3
575.6
01.6
40.2
823
21
18
0.0
10.5
0116.2
1339.6
1339.3
70.0
040.8
54.5
32.1
52.1
5brum
mayerbie
re
nlz
be256
20
117.5
1157.9
797.0
860.8
91.6
30.7
35
40
0.0
14.1
992.1
4127.5
4123.5
70.1
134.3
114.1
72.1
62.1
6lo
g-s
licin
gbvurem
13
20
147.9
7116.6
757.5
759.1
02.0
30.1
721
17
17
0.0
10.2
656.7
2191.8
2191.6
60.0
025.8
72.8
62.1
72.1
7lo
g-s
licin
gbvsm
od
13
20
360.2
6438.2
6387.3
750.8
91.1
30.3
40
174
46
105
0.1
12.4
8383.2
5860.0
4858.6
60.0
028.2
31.4
42.1
32.1
8brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3132.0
191.0
041.0
11.4
50.2
28
136
48
10
0.0
20.6
089.7
6246.1
1244.7
00.0
015.9
30.7
72.1
82.1
9...y
erbie
re3
isqrtin
validvc
10
181.6
615.8
94.1
111.7
83.8
70.0
511
80
0.0
00.0
43.5
76.4
46.3
20.0
01.0
20.3
12.2
02.2
0...b
iere3
isqrtaddin
validvc
10
185.6
016.0
04.8
311.1
73.3
10.0
511
80
0.0
00.0
44.3
17.1
67.0
40.0
01.3
60.3
43.1
22.2
1bm
c-b
vm
agic
10
443.8
6453.4
4452.5
70.8
71.0
00.1
01
10
0.0
00.0
00.4
60.4
60.3
90.4
60.4
60.4
62.2
12.2
2...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
710.9
110.4
00.5
11.0
50.5
01
10
0.0
00.0
09.3
19.3
10.7
19.3
19.3
19.3
12.2
22.2
3...m
ayerbie
re2
sm
ulo
v2bw384
20
122.0
615.6
415.1
70.4
71.0
30.4
01
10
0.0
00.0
014.3
314.3
38.8
014.3
314.3
314.3
32.2
32.2
4...a
yerbie
re2
sm
ulo
v4bw0512
10
128.7
82.5
72.1
70.4
01.1
80.1
01
10
0.0
00.0
01.5
81.5
80.1
71.5
81.5
81.5
82.2
62.2
5...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
33.9
63.5
60.4
01.1
10.2
01
10
0.0
00.0
02.8
92.8
90.2
72.8
92.8
92.8
92.2
72.2
6...a
yerbie
re2
sm
ulo
v4bw0896
10
1274.5
28.2
67.8
80.3
81.0
50.5
01
10
0.0
00.0
06.9
76.9
70.5
46.9
76.9
76.9
72.2
92.2
7...a
yerbie
re2
sm
ulo
v4bw0768
10
563.4
25.8
05.5
40.2
61.0
50.4
01
10
0.0
00.0
04.7
04.7
00.4
04.7
04.7
04.7
02.2
82.2
8brum
mayerbie
re
bitrev8192
20
101.9
7101.4
9101.2
40.2
51.0
00.0
01
10
0.0
00.0
00.0
40.0
40.0
00.0
40.0
40.0
42.3
02.2
9...m
ayerbie
re2
sm
ulo
v2bw448
20
150.6
343.6
943.5
40.1
51.0
00.5
01
10
0.0
00.0
042.5
942.5
934.8
542.5
942.5
942.5
92.2
42.3
0...m
ayerbie
re2
sm
ulo
v2bw512
20
198.3
836.6
836.5
90.0
91.0
00.7
01
10
0.0
00.0
035.2
935.2
922.9
835.2
935.2
935.2
92.2
5
122
Tab
leC
.6:
Det
aile
dre
sult
sof
PB
oole
ctor
wit
hh
igh
lim
itco
nfi
gu
rati
on
1(P
ara
llel
-In
crem
enta
l-H
igh
Lim
it1)
com
pare
dto
the
refe
ren
ceti
mes
Bas
e-In
crem
enta
l.L
ast
colu
mn
par-
inc
conta
ins
cross
refe
ren
ceto
det
ail
edre
sult
sof
Para
llel
-In
crem
enta
lin
Tab
.C.1
.
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental-Hig
hLim
it1)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
1.1
brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
5oot
342.1
01457.9
05.2
60.2
27
81
16
00.0
11.0
5340.4
6828.1
9826.5
50.0
023.2
35.1
41.1
1.2
log-s
licin
gbvurem
16
20
oot
oot
958.4
5841.5
51.8
80.2
824
23
21
0.0
20.3
3957.6
22579.5
92579.4
30.0
1389.9
231.4
61.2
1.3
log-s
licin
gbvudiv
18
20
oot
oot
1341.0
4458.9
61.3
40.5
928
22
21
0.0
25.2
21332.9
84178.8
94178.1
70.0
0477.8
546.9
51.7
1.4
...iffyIn
terle
avedM
odM
ult-8
20
oot
oot
1461.8
6338.1
41.2
30.3
32
204
103
00.0
30.6
21460.5
45515.6
65515.0
50.0
039.7
09.7
11.3
1.5
log-s
licin
gbvsm
od
15
20
oot
oot
1629.7
6170.2
41.1
00.3
36
156
51
106
0.1
02.3
61626.1
04011.0
64010.1
30.0
0143.2
16.9
91.9
2.1
log-s
licin
gbvudiv
17
20
1270.3
61373.3
4552.3
8820.9
62.4
90.5
926
22
20
0.0
24.8
0544.9
71592.2
31591.5
60.0
0172.2
219.1
82.1
2.2
log-s
licin
gbvurem
15
20
oot
1147.7
1357.4
4790.2
73.2
10.1
824
22
19
0.0
10.3
8356.6
11044.6
41044.5
10.0
0134.7
013.2
22.2
2.3
log-s
licin
gbvsm
od
14
20
1414.5
41350.5
9708.5
6642.0
31.9
10.3
30
139
46
60
0.0
81.9
5705.4
52047.6
12046.8
00.0
039.6
84.6
92.3
2.4
...e
st
v5
r10
vr1
c1
s32538
20
208.1
6996.7
5415.4
3581.3
22.4
01.8
13
20
80
0.5
524.5
2385.7
4442.5
4433.7
10.0
466.9
111.3
54.1
72.5
log-s
licin
gbvm
ul12
20
129.1
9962.3
1400.8
3561.4
82.4
00.1
22
22
0.0
00.0
2400.4
3792.3
5792.3
31.2
6396.5
5158.4
72.4
2.6
log-s
licin
gbvudiv
16
20
390.9
9521.1
2253.5
6267.5
62.0
60.2
823
21
17
0.0
10.4
7252.5
3705.0
5704.8
90.0
183.4
19.6
62.8
2.7
log-s
licin
gbvsm
od
13
20
360.2
6438.2
6259.4
3178.8
31.6
90.3
25
121
45
53
0.0
71.6
0256.7
8813.8
5813.1
90.0
018.7
92.1
92.1
32.8
log-s
licin
gbvurem
14
20
757.1
6270.8
8148.8
8122.0
01.8
20.1
824
22
17
0.0
10.3
0147.6
4453.3
0453.1
70.0
060.1
06.0
42.1
12.9
...e
st
v5
r10
vr5
c1
s13195
10
337.5
5662.6
0559.7
9102.8
11.1
82.1
16
29
80
0.7
634.2
9518.1
3693.5
4680.3
60.0
5142.6
512.1
74.3
22.1
0...sm
tcom
p09
lfbv
form
ula
20
180.2
4207.7
1113.3
894.3
31.8
31.0
840
32
00.0
62.7
4109.8
5342.0
7334.8
40.4
923.9
63.6
02.1
02.1
1...sm
tcom
p09
std
bv
form
ula
20
108.8
6169.6
177.6
291.9
92.1
90.9
624
24
00.0
52.4
474.5
2219.6
3212.7
20.1
223.6
33.9
93.1
32.1
2pspace
power2sum
.5699
20
347.9
21748.1
21659.5
188.6
11.0
52.0
11
11
0.0
00.0
01658.9
71658.9
71658.5
9111.3
21547.6
4829.4
84.4
2.1
3float
newton.6
.3.i
10
564.0
4437.0
6351.2
185.8
51.2
41.6
917
60
0.2
411.9
7336.5
6562.3
6552.1
60.0
074.1
817.0
44.4
72.1
4brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3132.0
154.0
777.9
42.4
40.1
25
74
12
00.0
10.4
753.1
2153.5
8152.6
40.0
04.1
41.0
42.1
82.1
5lo
g-s
licin
gbvudiv
15
20
162.7
0192.9
5116.5
976.3
61.6
50.1
823
20
15
0.0
10.4
6115.6
2317.1
3316.9
60.0
030.6
74.6
02.1
52.1
6lo
g-s
licin
gbvurem
13
20
147.9
7116.6
757.7
658.9
12.0
20.1
721
17
16
0.0
10.2
657.0
1181.2
4181.1
20.0
022.8
22.7
92.1
72.1
7lo
g-s
licin
gbvsm
od
12
20
138.5
8174.8
0117.0
657.7
41.4
90.2
14
80
36
20.0
30.9
7115.3
6408.7
4408.3
30.0
013.4
42.1
42.1
92.1
8brum
mayerbie
re3
isqrtadd
20
144.7
3148.8
8105.5
643.3
21.4
10.2
944
32
00.0
10.2
4104.9
1347.3
3344.2
10.0
315.8
13.3
73.2
82.1
9lo
g-s
licin
gbvslt
7972
20
305.2
5931.6
7891.5
140.1
61.0
51.0
734
23
00.0
110.0
5877.4
62646.0
62643.4
60.2
3168.6
035.7
64.2
02.2
0brum
mayerbie
re3
isqrt
20
145.8
1148.1
1122.4
625.6
51.2
10.2
10
51
34
00.0
10.2
9121.6
6423.7
6420.0
90.1
816.2
03.5
63.2
22.2
1lo
g-s
licin
gbvsub
04000
20
545.8
4842.7
7817.5
725.2
01.0
30.1
11
11
0.0
00.0
0816.6
5816.6
5815.7
3179.1
9637.4
6408.3
33.6
2.2
2lfsr
lfsr
004
111
112
20
178.0
5181.6
1176.9
64.6
51.0
30.2
23
20
0.0
11.6
2174.8
1210.0
2209.3
415.8
6107.3
042.0
03.3
22.2
3...y
erbie
re3
isqrtin
validvc
10
181.6
615.8
912.2
13.6
81.3
00.0
24
40
0.0
00.0
211.8
620.0
119.8
80.7
56.1
32.8
62.2
02.2
4bm
c-b
vgraycode
10
442.5
4449.0
3447.9
81.0
51.0
00.1
01
10
0.0
00.0
00.0
40.0
40.0
10.0
40.0
40.0
43.1
2.2
5lo
g-s
licin
gbvsub
05994
20
414.3
11726.2
51725.2
41.0
11.0
00.2
11
11
0.0
00.0
01723.8
61723.8
61721.8
7435.4
81288.3
8861.9
34.5
2.2
6...m
ayerbie
re2
sm
ulo
v2bw512
20
198.3
836.6
835.9
00.7
81.0
20.7
01
10
0.0
00.0
034.8
434.8
422.5
334.8
434.8
434.8
42.2
52.2
7bm
c-b
vm
agic
10
443.8
6453.4
4452.7
50.6
91.0
00.1
01
10
0.0
00.0
00.4
60.4
60.3
90.4
60.4
60.4
62.2
12.2
8...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
710.9
110.2
30.6
81.0
70.5
01
10
0.0
00.0
09.3
49.3
40.7
19.3
49.3
49.3
42.2
22.2
9...m
ayerbie
re2
sm
ulo
v2bw448
20
150.6
343.6
943.0
40.6
51.0
20.5
01
10
0.0
00.0
042.2
642.2
634.4
942.2
642.2
642.2
62.2
42.3
0...m
ayerbie
re2
sm
ulo
v2bw384
20
122.0
615.6
415.0
20.6
21.0
40.4
01
10
0.0
00.0
014.2
814.2
88.7
714.2
814.2
814.2
82.2
32.3
1...m
ayerbie
re2
um
ulo
v1bw256
20
143.9
4119.0
5118.4
50.6
01.0
10.1
01
10
0.0
00.0
0118.0
6118.0
6116.4
2118.0
6118.0
6118.0
63.5
2.3
2RW
SExam
ple
18.txt
10
160.6
1193.3
5192.7
60.5
91.0
00.1
01
10
0.0
00.0
0192.2
8192.2
8192.1
3192.2
8192.2
8192.2
83.3
2.3
3...a
yerbie
re2
sm
ulo
v4bw0512
10
128.7
82.5
72.0
40.5
31.2
60.1
01
10
0.0
00.0
01.5
81.5
80.1
71.5
81.5
81.5
82.2
62.3
4...a
yerbie
re2
sm
ulo
v4bw0896
10
1274.5
28.2
67.7
90.4
71.0
60.5
01
10
0.0
00.0
06.9
66.9
60.5
46.9
66.9
66.9
62.2
92.3
5float
mult2.c
.50
10
109.3
940.0
239.5
70.4
51.0
10.2
01
10
0.0
00.0
039.0
839.0
838.4
239.0
839.0
839.0
83.1
12.3
6...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
33.9
63.5
30.4
31.1
20.2
01
10
0.0
00.0
02.9
02.9
00.2
72.9
02.9
02.9
02.2
72.3
7...a
yerbie
re2
sm
ulo
v4bw0768
10
563.4
25.8
05.3
70.4
31.0
80.4
01
10
0.0
00.0
04.7
14.7
10.4
04.7
14.7
14.7
12.2
82.3
8...m
ayerbie
re2
um
ulo
v1bw224
20
108.4
290.1
689.9
60.2
01.0
00.1
01
10
0.0
00.0
089.5
389.5
388.6
089.5
389.5
389.5
33.4
2.3
9lfsr
lfsr
008
159
048
20
100.5
644.3
244.1
30.1
91.0
00.1
01
10
0.0
00.0
043.6
343.6
343.2
043.6
343.6
343.6
34.9
3
123
Tab
leC
.7:
Det
aile
dre
sult
sof
PB
oole
ctor
wit
hh
igh
lim
itco
nfi
gu
rati
on
2(P
ara
llel
-In
crem
enta
l-H
igh
Lim
it2)
com
pare
dto
the
refe
ren
ceti
mes
Bas
e-In
crem
enta
l.L
ast
colu
mn
par-
inc
conta
ins
cross
refe
ren
ceto
det
ail
edre
sult
sof
Para
llel
-In
crem
enta
lin
Tab
.C.1
.
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental-Hig
hLim
it2)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
1.1
brum
mayerbie
re2
sm
ulo
v1bw32
20
1056.2
5oot
311.8
21488.1
85.7
70.3
28
77
14
00.0
11.0
5310.2
7930.6
5928.9
40.0
251.1
46.0
81.1
1.2
lfsr
lfsr
008
159
128
20
285.7
4oot
1004.4
1795.5
91.7
90.8
36
40
0.0
712.5
9989.6
11549.1
11545.6
115.4
6621.2
0140.8
35.2
61.3
log-s
licin
gbvurem
16
20
oot
oot
1092.5
7707.4
31.6
50.2
824
22
20
0.0
10.3
61091.7
72897.4
92897.3
50.0
0375.0
936.2
21.2
1.4
...st
v7
r12
vr10
c1
s15994
10
1367.7
1oot
1151.7
9648.2
11.5
61.4
26
27
20
2.1
676.1
11062.8
31091.1
21074.4
70.2
6149.6
920.5
95.1
91.5
log-s
licin
gbvudiv
18
20
oot
oot
1393.5
6406.4
41.2
90.6
928
22
21
0.0
25.1
51385.6
74238.8
14238.0
90.0
0510.3
547.6
31.7
1.6
...iffyIn
terle
avedM
odM
ult-8
20
oot
oot
1526.9
3273.0
71.1
80.3
22
142
77
00.0
20.4
41525.9
35633.4
95633.0
10.0
069.7
315.1
41.3
1.7
log-s
licin
gbvult
9988
20
555.0
3oot
1621.0
9178.9
11.1
10.6
511
60
0.0
17.2
11609.6
33588.8
13586.8
71.1
5680.6
2170.9
05.1
15
1.8
log-s
licin
gbvsm
od
15
20
oot
oot
1702.7
297.2
81.0
60.4
23
126
50
24
0.0
71.9
41699.6
95410.2
65409.5
20.0
0110.6
016.1
51.9
1.9
...e
st
v7
r12
vr5
c1
s29826
10
896.5
0oot
1737.6
562.3
51.0
41.4
41
42
20
3.2
2114.4
81603.7
81696.7
91673.3
80.0
4170.7
120.4
45.2
1
2.1
log-s
licin
gbvudiv
17
20
1270.3
61373.3
4609.8
6763.4
82.2
50.5
926
21
18
0.0
24.9
0602.3
81672.7
11672.0
10.0
1162.1
520.9
12.1
2.2
log-s
licin
gbvsm
od
14
20
1414.5
41350.5
9631.9
6718.6
32.1
40.3
19
108
39
14
0.0
51.5
4629.5
62218.7
72218.1
80.0
054.8
88.1
62.3
2.3
log-s
licin
gbvurem
15
20
oot
1147.7
1489.6
2658.0
92.3
40.1
824
22
17
0.0
10.3
6488.7
21087.9
01087.7
50.0
0123.6
514.5
12.2
2.4
...st
v5
r15
vr10
c1
s11127
10
346.0
31009.9
9563.4
2446.5
71.7
91.2
10
11
20
0.6
424.3
8534.2
0583.1
5576.7
70.0
6160.4
627.7
73.8
2.5
log-s
licin
gbvm
ul12
20
129.1
9962.3
1545.6
3416.6
81.7
60.1
22
22
0.0
00.0
2545.3
3944.0
1943.9
82.6
6534.4
6188.8
02.4
2.6
...e
st
v5
r10
vr1
c1
s13516
10
231.3
6574.5
3266.1
2308.4
12.1
60.8
56
20
0.1
88.9
4255.2
4261.1
6257.3
30.1
7127.4
223.7
44.4
12.7
log-s
licin
gbvslt
7972
20
305.2
5931.6
7626.1
6305.5
11.4
90.3
48
40
0.0
02.8
6622.0
71039.3
91038.6
02.4
1385.6
269.2
94.2
02.8
log-s
licin
gbvudiv
16
20
390.9
9521.1
2264.0
8257.0
41.9
70.2
823
20
15
0.0
10.5
0263.1
0756.0
1755.8
40.0
065.5
510.8
02.8
2.9
...test
v7
r7
vr1
c1
s4574
10
609.3
31077.2
4914.7
1162.5
31.1
81.2
29
33
40
1.1
850.2
3856.1
81015.3
91003.2
60.0
3121.7
915.6
24.1
42.1
0lo
g-s
licin
gbvsm
od
13
20
360.2
6438.2
6290.6
3147.6
31.5
10.2
15
74
29
40.0
31.0
2288.8
9952.5
2952.1
30.0
032.4
75.7
02.1
32.1
1lo
g-s
licin
gbvult
6000
20
406.3
8608.3
3460.8
5147.4
81.3
20.2
35
40
0.0
01.7
5458.2
1778.9
9778.4
40.6
3284.6
486.5
54.3
42.1
2lo
g-s
licin
gbvurem
14
20
757.1
6270.8
8158.1
7112.7
11.7
10.1
722
19
17
0.0
10.3
0157.4
7514.0
9513.9
70.0
057.5
07.5
62.1
12.1
3...e
st
v7
r7
vr10
c1
s24535
10
237.8
8246.3
0142.1
3104.1
71.7
30.5
12
20
0.0
31.8
5139.4
5141.7
8139.4
02.3
2120.4
147.2
63.1
02.1
4lo
g-s
licin
gbvult
6997
20
201.1
9669.2
9576.1
293.1
71.1
60.2
24
40
0.0
01.3
2574.1
1813.9
9813.4
925.7
5391.9
4116.2
84.3
12.1
5lo
g-s
licin
gbvudiv
15
20
162.7
0192.9
5114.5
378.4
21.6
80.1
720
18
13
0.0
10.4
1113.7
1365.2
5365.1
00.0
126.5
26.1
92.1
52.1
6lo
g-s
licin
gbvsm
od
12
20
138.5
8174.8
0118.9
055.9
01.4
70.1
945
22
00.0
20.5
6117.8
1390.2
8390.0
40.0
022.5
34.1
12.1
92.1
7brum
mayerbie
re3
isqrtadd
20
144.7
3148.8
895.5
753.3
11.5
60.1
522
16
00.0
00.1
195.0
9285.4
0283.9
20.0
318.7
96.6
43.2
82.1
8brum
mayerbie
re3
isqrt
20
145.8
1148.1
1103.7
744.3
41.4
30.1
623
16
00.0
00.1
3103.3
0308.3
8306.8
60.0
422.9
26.8
53.2
22.1
9brum
mayerbie
re2
sm
ulo
v1bw24
20
134.3
3132.0
188.4
943.5
21.4
90.1
17
32
40
0.0
10.2
587.7
9155.2
0154.7
40.0
811.4
02.4
62.1
82.2
0lo
g-s
licin
gbvurem
13
20
147.9
7116.6
775.5
741.1
01.5
40.1
721
15
11
0.0
10.2
574.9
4208.6
9208.5
80.0
023.8
33.7
92.1
72.2
1lo
g-s
licin
gbvult
7994
20
230.1
0691.4
3656.6
134.8
21.0
50.3
24
40
0.0
01.5
0654.4
0857.5
5857.0
028.4
3513.5
1122.5
14.2
92.2
2pspace
shift1add.2
2961
20
110.7
1110.8
6107.9
72.8
91.0
31.1
01
10
0.0
00.0
0107.6
0107.6
0107.1
1107.6
0107.6
0107.6
03.1
72.2
3pspace
shift1add.2
5952
20
143.2
8129.3
8127.1
02.2
81.0
21.3
01
10
0.0
00.0
0126.6
6126.6
6126.1
1126.6
6126.6
6126.6
63.2
02.2
4lfsr
lfsr
008
159
080
20
194.3
3310.3
5308.1
22.2
31.0
10.3
01
10
0.0
00.0
0307.6
7307.6
7306.9
4307.6
7307.6
7307.6
74.6
12.2
5core
ext
con
048
002
1024
20
487.7
21716.6
41714.7
51.8
91.0
00.3
01
10
0.0
00.0
01714.4
01714.4
01714.1
01714.4
01714.4
01714.4
02.5
2.2
6...m
ayerbie
re2
um
ulo
v1bw256
20
143.9
4119.0
5117.2
91.7
61.0
20.1
01
10
0.0
00.0
0116.9
6116.9
6115.3
3116.9
6116.9
6116.9
63.5
2.2
7lfsr
lfsr
008
159
048
20
100.5
644.3
242.5
81.7
41.0
40.1
01
10
0.0
00.0
042.1
742.1
741.7
342.1
742.1
742.1
74.9
32.2
8pspace
shift1add.2
9940
20
195.3
8157.2
3155.5
21.7
11.0
11.4
01
10
0.0
00.0
0155.1
6155.1
6154.5
3155.1
6155.1
6155.1
63.2
62.2
9pspace
shift1add.2
8943
20
186.8
4148.3
3146.7
11.6
21.0
11.4
01
10
0.0
00.0
0146.3
0146.3
0145.6
8146.3
0146.3
0146.3
03.2
52.3
0lfsr
lfsr
008
159
096
20
275.8
6145.8
5144.2
61.5
91.0
10.2
01
10
0.0
00.0
0143.8
0143.8
0142.9
6143.8
0143.8
0143.8
04.8
2
Contin
ued
on
next
page
124
Table
C.7
Contin
ued
from
previo
us
page
rank
benchm
ark
stat
sequ-b
tor
ref
PBoole
ctor
(Parallel-In
crem
ental-Hig
hLim
it2)
cm
p
base-inc
tot(wc)
diff
su
mem
rnds
subproble
ms
lkhd
split
search
par-inc
tot
max
fin
tot(wc)
tot(wc)
tot(wc)
tot(cpu)
sat(cpu)
min
max
mean
[][]
[][s]
[s]
[s]
[s]
[][G
B]
[][]
[][]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[s]
[]
2.3
1bm
c-b
vm
agic
10
443.8
6453.4
4452.4
41.0
01.0
00.1
01
10
0.0
00.0
00.4
60.4
60.3
90.4
60.4
60.4
62.2
12.3
2lfsr
lfsr
008
111
096
20
133.3
2238.7
2237.7
40.9
81.0
00.2
01
10
0.0
00.0
0237.2
9237.2
9236.7
0237.2
9237.2
9237.2
94.7
12.3
3pspace
shift1add.2
3958
20
114.0
8115.5
0114.6
50.8
51.0
11.2
01
10
0.0
00.0
0114.2
9114.2
9113.7
7114.2
9114.2
9114.2
93.1
82.3
4...a
yerbie
re2
sm
ulo
v4bw1024
10
1518.8
710.9
110.1
60.7
51.0
70.5
01
10
0.0
00.0
09.3
19.3
10.7
19.3
19.3
19.3
12.2
22.3
5...b
iere3
isqrtaddin
validvc
10
185.6
016.0
015.3
00.7
01.0
50.0
01
10
0.0
00.0
015.0
215.0
215.0
015.0
215.0
215.0
23.1
22.3
6pspace
shift1add.2
7946
20
167.8
1143.3
4142.6
60.6
81.0
01.3
01
10
0.0
00.0
0141.8
2141.8
2141.2
3141.8
2141.8
2141.8
23.2
32.3
7pspace
power2sum
.5699
20
347.9
21748.1
21747.5
10.6
11.0
02.0
01
10
0.0
00.0
01747.0
71747.0
71746.6
81747.0
71747.0
71747.0
74.4
2.3
8...a
yerbie
re2
sm
ulo
v4bw0640
10
141.9
33.9
63.3
70.5
91.1
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34.1
125
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Lebenslauf
ANGABEN ZUR PERSON Christian Reisenberger
Obere Donaulände 11/6, 4020 Linz (Österreich)
+43 69981452872
Geschlecht Männlich | Geburtsdatum 26/10/1980 | Staatsangehörigkeit österreichisch
SCHUL- UND BERUFSBILDUNG
10/2011–Heute
Johannes Kepler Universität, Linz
Masterstudium Informatik
10/2008–08/2011 Bachelor of ScienceJohannes Kepler Universität, Linz
Bachelorstudium Informatik
10/2006–02/2008
Fachhochschule, Hagenberg
Bachelorstudium Hard- und Software System Engineering
09/1995–06/2000 MaturaHöhere Technische Lehranstalt, Wels
Fachbereich Elektrotechnik
BERUFSERFAHRUNG
05/2013–05/2014 SoftwareentwicklerSelbstständig
04/2011–04/2013 Industriepraktikum Software TesterFujitsu Semiconductor Embedded Solutions Austria GMBH, Linz
05/2008–12/2008 SoftwareentwicklerArs Electronica FutureLab, Linz
05/2005–06/2006 Auslandsaufenthalt in Zentralamerika- Sechsmonatiges Volontariat bei einem Sozialprojekt in Nicaragua (Granada).
- 1 Monat internationaler Friedensbeobachter in Mexiko (Chiapas)
12/2001–03/2005 LichttechnikerTheater Phönix, Linz
6/10/14 © Europäische Union, 2002-2014 | http://europass.cedefop.europa.eu Seite 1 / 2
Lebenslauf Christian Reisenberger
10/2000–09/2001 ZivildienstDas Dorf, Altenhof
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Eidesstattliche Erklarung
Ich erklare an Eides statt, dass ich die vorliegende Masterarbeit selbststandigund ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilf-smittel nicht benutzt bzw. die wortlich oder sinngemaß entnommenen Stellenals solche kenntlich gemacht habe. Die vorliegende Masterarbeit ist mit demelektronisch ubermittelten Textdokument identisch.
Linz, am 20. November 2014
Christian Reisenberger, BSc