lecture 15 loop transformations

Carnegie Mellon

Lecture 15Loop Transformations

Chapter 11.10-11.11

Dror E. Maydan CS243: Loop Optimization and Array Analysis 1

Carnegie Mellon

Loop Optimization

• Domain– Loops: Change the order in which we iterate through loops

• Goals– Minimize inner loop dependences that inhibit software pipelining– Minimize loads and stores– Parallelism

• SIMD Vector today, in general multiprocessor as well– Minimize cache misses– Minimize register spilling

• Tools– Loop interchange– Fusion– Fission– Outer loop unrolling– Cache Tiling– Vectorization

• Algorithm for putting it all together

Dror E. MaydanCS243: Loop Optimization and Array Analysis 2

Carnegie Mellon

Loop Interchange

for i = 1 to n for j = 1 to n

A[j][i] = A[j-1][i-1] * b[i]

• Should I interchange the two loops?– Stride-1 accesses are better for caches– But one more load in the inner loop– But one less register needed to hold the result of the loop


for j = 1 to n for i = 1 to n A[j][i] = A[j-1][i-1] * b[i]

Carnegie Mellon

Loop Interchange

for i = 1 to n for j = 1 to n A[j][i] = A[j+1][i-1] * b[i]

Distance Vector is (deltai, deltaj) = (1, -1)Direction Vector is (>, <)

• Dependence represents that one ref, aw must happen before another ar

• To permute loops, permute direction vectors in the same manner– Permutation is legal iff all permuted direction vectors are lexicographically positive

• Special case: Fully permutable loop nest– Either dependence “carried” by a loop outside of the nest or all components > or =

• All the loops in the nest can be arbitrarily permuted– (>, >, <) Inner two loops are fully permutable– (>=, =, >) All three loops are fully permutable


j

i

Carnegie Mellon

Loop Interchange

for i = 1 to n for j = 1 to i

…

• How do I interchangefor j = 1 to n for i = j to n

– In general ugly but doable


j

i

Carnegie Mellon

Non Perfectly Nested loops


S1 for j = 1 to n

S2

• Can’t always interchange– Can be expensive when you can


Carnegie Mellon

Loop Fusion

for i = 1 to n for j = 1 to nS1 for j = 1 to nS2

• Moving S2 across “j” iterations but not any of “i” iterations• Pretend to fuse• Legal as long as there is no direction vector from S2 to S1 with “=“ in all

the outer loops and > in one of the inner (=, =, …, =, >, …),– That would imply that S2 is now before S1


for i = 1 to n

for j = 1 to n

S1

S2

Carnegie Mellon

Loop Fusion


a[i][j] = … for j = 1 to n

… = a[i][j+1] • Legal as long as there is no direction vector from the read to the

write with “=“ in all the outer loops and > in one of the inner (=, =, …, =, >, …)– (=, 1) so can’t fusion


Carnegie Mellon

Loop Fusion


a[i][j] = … for j = 1 to n

… = a[i][j+1]

If the first “+” direction is always a small literal constant, can skew the loop and allow fusion

Bonus: can get rid of a load and maybe a store


for i = 1 to n

a[i][1] = …

for j = 2 to n

a[i][j] = …

for j = 1, n-1

… = a[i][j+1]

… = a[i][n+1]

for i = 1 to n

a[i][1] = …

for j = 2 to n

a[i][j] = …

for j = 2, n

… = a[i][j]

… = a[i][n+1]

for i = 1 to n

a[i][1] = …

for j = 2 to n {

a[i][j] = …

… = a[i][j]

}

… = a[i][n-1]

Carnegie Mellon

Loop Fission


S1 for j = 1 to n

S2

• Moving S2 across all later “i” iterations– Legal as long as no dependences from S2 to S1 with > in the

fissioned outer loops


for i = 1 to n

for j = 1 to n

S1

for i = 1 to n

for j = 1 to n

S2

Carnegie Mellon

Loop Fission


= a[i-1][j] for j = 1 to n

a[i][j] =

• Moving S2 across all later “i” iterations– Legal as long as no dependences from S2 to S1 with > in the

fissioned outer loops


for i = 1 to n

for j = 1 to n

= a[i-1][j]

for i = 1 to n

for j = 1 to n

a[i][j] = Dep from write to read of (1)

Carnegie Mellon

Inner Loop Fission

for i = 1 to n for j = 1 to n… = h[i];… = h[i+1];… … = h[i+49];… = h[i+50];

• Legal as long as there is no dependence from an S2 to an S1 where the first “>” is in the “j” loop.


for i = 1 to n

for j = 1 to n

= h[i];

…

… = h[i+25];

for j = 1 to n

… = h[26];

…

… = h[50];

Carnegie Mellon

Inner Loop Fission

for j = 1 to nS1S2S3

• Looking at edges carried by the inner most loops• Strongly Connected Components can not be fissioned• Everything else can be fissoned as long as loops are emitted in topological order


S1

S2

S3

=

=>

Carnegie Mellon

Outer Loop Unrolling


for k = 1 to n c[i][j] += a[i][k] * b[k][j];

• How many loads in the inner loop? How many MACs?


Carnegie Mellon


for i = 1 to n by 2 for j = 1 to n by 2for k = 1 to n c[i][j] += a[i][k] * b[k][j]; c[i][j+1] += a[i][k] * b[k][j+1]; c[i+1][j] += a[i+1][k] * b[k][j]; c[i+1][j+1] += a[i+1][k] * b[k][j+1];

• Is it legal?


Carnegie Mellon


If n = 2– Original order was

• (1, 1, 1) (1, 1, 2) (1, 2, 1) (1, 2, 2) (2, 1, 1) (2, 1, 2) (2, 2, 1) (2, 2, 2)

– New order is • (1, 1, 1) (1, 2, 1) (2, 1, 1) (2, 2, 1) (1, 1,

2) (1, 2, 2) (2, 1, 2) (2, 2, 2)• Equivalent to permuting the loops into

for k = 1 to 2 for i = 1 to 2 for j = 1 to 2

• If loops are fully permutable can also outer loop unroll


for i = 1 to 2 by 2

for j = 1 to 2 by 2

for k = 1 to 2

c[i][j] += a[i][k] * b[k];

c[i][j+1] += a[i][k] * b[k][j+1];

c[i+1][j] += a[i+1][k] * b[k][j];

c[i+1][j+1] += a[i+1][k] * b[k][j+1];

Carnegie Mellon

Unrolling Trapezoidal Loops


j

i

• Ugly• We unroll two level trapezoidal loops but the details are very ugly

for i=1 to n by 2 for j = 1 to i

Carnegie Mellon

Trapezoidal Example

for (i=0; i<n; i++) { for (j=2*i; j<n-i; j++) {

a[i][j] += 1; }

}

D.E. MaydanCS243: Loop Optimization and Array Analysis 18

for(i = 0; i <= (n + -2); i = i + 2) { lstar = (i * 2) + 2; ustar = (n - (i + 1)) + -1; if(((i * 2) + 2) < (n - (i + 1))) { for(r2d_i = i; r2d_i <= (i + 1); r2d_i = r2d_i + 1){ for(j = r2d_i * 2; j <= ((i * 2) + 1); j = j + 1){ a[r2d_i][j] = a[r2d_i][j] + 1; } } for(j0 = lstar; ustar >= j0; j0 = j0 + 1) { a[i][j0] = a[i][j0] + 1; a[i + 1][j0] = a[i + 1][j0] + 1; }; for(r2d_i0 = i; r2d_i0 <= (i + 1); r2d_i0 = r2d_i0 + 1) { for(j1 = n - (i + 1); j1 < (n - r2d_i0); j1 = j1 + 1) { a[r2d_i0][j1] = a[r2d_i0][j1] + 1; }; } } else { for(r2d_i1 = i; r2d_i1 <= (i + 1); r2d_i1 = r2d_i1 + 1) { for(j2 = r2d_i1 * 2; j2 < (n - r2d_i1); j2 = j2 + 1) { a[r2d_i1][j2] = a[r2d_i1][j2] + 1; } } } } if(n > i) { for(j3 = i * 2; j3 < (n - i); j3 = j3 + 1) { a[i][j3] = a[i][j3] + 1; }; }

Carnegie Mellon

Cache Tiling


for k = 1 to n c[i][j] += a[i][k] * b[k][j];

• How many cache misses?


Carnegie Mellon

Cache Tiling

for jb = 1 to n by b for kb = 1 to n by b for i = 1 to nfor j = jb to jb+b for k = kb to kb + b c[i][j] += a[i][k] * b[k][j];

• How many cache misses?– Order b reuse for each array

• If loops are fully permutable can cache tile


Carnegie Mellon

Vectorization: SIMD

for i = 1 to nfor j = 1 to n a[j][i] = 0;

• N-way parallel where N is the SIMD width of the machine


for i = 1 to n by 8 for j = 1 to n a[j][i:i+7] = 0;

Carnegie Mellon

Vectorization: SIMDfor i = 1 to n

for j = 1 to n for k = 1 to n S1

….SM

• We have moved later iterations of S1 ahead of earlier iterations of S2, …, SM, etc

• Legal as long as no dependence from a latter S to an earlier S where that dependence is carried by the vector loop– E.g legal to vectorize ‘j’ above if no dependence from a latter S to an

earlier S with direction (=, >, *)


Carnegie Mellon

Putting It All Together

• Three phase algorithm1. Use fission and fusion to build perfectly nested loops

1. We prefer fusion but not obvious that that is right2. Enumerate possibilities for unrolling, interchanging, cache

tiling and vectorizing3. Use inner loop fission if necessary to minimize register

pressure


Carnegie Mellon

Phase 2

Choose a loop to vectorize All references that refer to vector loop must be stride-1For each possible inner loop

Compute best possible unrollings for each outerCompute best possible ordering and tiling

To compute best possible unrollingTry all combinations of unrolling up to a max product of 16For each possible unrolling

Estimate the machine cycles for the inner loop (ignoring cache)Estimate the register pressure

Don’t unroll more if too much register pressure

To compute best possible ordering and tilingConsider only loops with “reuse”Choose best threeIterate over all orderings of three with a binary search on cache tile sizeNote and record the total cycle time for this configuration. Pick the best

Estimating cyclesCould compile every combination, but …


Carnegie Mellon

Machine Modeling

• Recall that software pipelining had resource limits and latency limits– Map high level IR to machine resources– Unroll high level IR operations

• Remove duplicate loads and stores• Count machine resources• Build a latency graph of unrolled operations

– Iterate over inner loop cycles and find worst cycle• Assume performance is worst of two limits

• Model register pressure– Count loop invariant loads and stores– Count address streams– Count cross iteration cse’s = a[i] + a[i-2]– Add machine dependent constant


Carnegie Mellon

Cache Modeling

• Given a loop ordering and a set of tile factors• Combine array references that differ by constant, e.g. a[i][j] and

a[i+1][j+1]• Estimate capacity of all array references, multiply by fudge factor

for interference, stop increasing block sizes if capacity is larger than cache

• Estimate quantity of data that must be brought into cache


Carnegie Mellon

Phase 3: Inner loop fission

• Does inner loop use too many registers– Break down into SCCs– Pick biggest SCC

• Does it use too many registers– If yes, too bad– If no, search for other SCCs to merge in

» Pick one with most commonality » Keep merging while enough registers


Carnegie Mellon

Extra: Reductions


a[j] += b[i][j];

Can I unroll

for i = 1 to n by 2 for j = 1 to n a[j] += b[i][j]; a[j] += b[i+1][j];• Legal

– Integer: yes– Floating point: maybe


Carnegie Mellon

Extra: Outer Loop Invariants

for i for j

a[i][j] += b[i] * cos(c[j])

Can replace with

for j t[j] = cos(c[j])for i for j

a[i][j] += b[i] * t[j];

• Need to integrate with model – Model must assume that invariant computation will be replaced with loads


lecture 15 loop transformations

Documents

n s1for

interchangefor j

j iterations

loop dror

array analysis3for j

later i iterationslegal

loop transformationschapter

s1 dror