m3 – alu designbt.nitk.ac.in/c/17b/co200/notes/m3-1.multiply.divide.pdfmodule outline integer...

M3 – ALU Design

Module Outline

● Integer Arithmetic– Adder, Subtractor, Multiplier, Divider

● Arithmetic and Logical Unit Design– ALU Design in SystemC

Multiplication

● Start with long-multiplication approach

1000× 1001 1000 0000 0000 1000 1001000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001 10001000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001 10001000

100000100000 10102 10001000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001 10001000

100000100000 10102 10001000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001 10001000

100000100000 10102 10001000

10000001000000 113 10001000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001 10001000

100000100000 10102 10001000

10000001000000 113 10010001001000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000 10011001

MultiplicandMultiplicand MultiplierMultiplier

0 10001000

ProductProduct

1000010000 1001001 10001000

100000100000 10102 10001000

10000001000000 113 10010001001000

10000000100000004 10010001001000

Multiplication HardwareStartStart

Multiplication Hardware

StartStart

1. TestMultiplier0

1. TestMultiplier0

Multiplication HardwareStartStart

1. TestM10

1. TestM10

P = P + M2P = P + M2

M10 = 1 M1

0 = 0

Multiplication HardwareStartStart

1. TestM10

1. TestM10

P = P + M2P = P + M2

M10 = 1

Shift-left M2 by 1 bitShift-left M2 by 1 bit

M10 = 0

Multiplication HardwareStartStart

1. TestM10

1. TestM10

P = P + M2P = P + M2

M10 = 1

Shift-left M2 by 1 bit,

Shift-right M1 by 1 bit

Shift-left M2 by 1 bit,

Shift-right M1 by 1 bit

M10 = 0

Multiplication HardwareStartStart

1. TestM10

1. TestM10

P = P + M2P = P + M2

M10 = 1

Shift-left M2 by 1 bit,Shift-right M1 by 1 bit

Shift-left M2 by 1 bit,Shift-right M1 by 1 bit

nth

repetition?

nth

repetition?

M10 = 0

Multiplication HardwareStartStart

1. TestM10

1. TestM10

P = P + M2P = P + M2

M10 = 1

Shift-left M2 by 1 bit,Shift-right M1 by 1 bit

Shift-left M2 by 1 bit,Shift-right M1 by 1 bit

nth

repetition?

nth

repetition?

DoneDone

M10 = 0

Y

N

Multiplication Hardware

MultiplicandShift left

MultiplicandShift left

MultiplierShift right

MultiplierShift right

Multiplication Hardware

MultiplicandShift left

MultiplicandShift left

MultiplierShift right

MultiplierShift right

ProductWrite

ProductWrite

Multiplication Hardware

MultiplicandShift left

MultiplicandShift left

MultiplierShift right

MultiplierShift right

ProductWrite

ProductWrite

n-bit ALU

Multiplication Hardware

MultiplicandShift left

MultiplicandShift left

MultiplierShift right

MultiplierShift right

ProductWrite

ProductWrite

n-bit ALU

Multiplication Hardware

MultiplicandShift left

MultiplicandShift left

MultiplierShift right

MultiplierShift right

ProductWrite

ProductWrite

n-bit ALU

ControlTest

ControlTest

M10

Multiplication Hardware

MultiplicandShift left

MultiplicandShift left

MultiplierShift right

MultiplierShift right

ProductWrite

ProductWrite

n-bit ALU

ControlTest

ControlTest

M10

MIPS Multiplication● Two 32-bit registers for product

– HI: most-significant 32 bits– LO: least-significant 32-bits

MIPS Multiplication● Two 32-bit registers for product

– HI: most-significant 32 bits– LO: least-significant 32-bits

● Instructions– mult rs, rt / multu rs, rt

● 64-bit product in HI/LO

MIPS Multiplication● Two 32-bit registers for product

– HI: most-significant 32 bits– LO: least-significant 32-bits

● Instructions– mult rs, rt / multu rs, rt

● 64-bit product in HI/LO

– mfhi rd / mflo rd● Move from HI/LO to rd● Can test HI value to see if product overflows 32 bits

MIPS Multiplication● Two 32-bit registers for product

– HI: most-significant 32 bits– LO: least-significant 32-bits

● Instructions– mult rs, rt / multu rs, rt

● 64-bit product in HI/LO

– mfhi rd / mflo rd● Move from HI/LO to rd● Can test HI value to see if product overflows 32 bits

– mul rd, rs, rt● Least-significant 32 bits of product –> rd

Division

10011000 1001010 -1000 10 101 1010 -1000 10

n-bit operands yield n-bitquotient and remainder

quotient

dividend

remainder

divisor

Division● Check for 0 divisor● Long division approach

– If divisor ≤ dividend bits● 1 bit in quotient, subtract

– Otherwise● 0 bit in quotient, bring down next dividend bit

● Restoring division– Do the subtract, and if remainder goes < 0, add divisor

back● Signed division

– Divide using absolute values– Adjust sign of quotient and remainder as required

Division Hardware

Division Hardware

Initially dividend

Initially divisor in left half

MIPS Division

● Use HI/LO registers for result– HI: 32-bit remainder– LO: 32-bit quotient

● Instructions– div rs, rt / divu rs, rt– No overflow or divide-by-0 checking

● Software must perform checks if required

– Use mfhi, mflo to access result

Module Outline

● Integer Arithmetic– Adder, Subtractor, Multiplier, Divider

● Arithmetic and Logical Unit Design– ALU Design in SystemC

Extra

Multiplication

● Start with long-multiplication approach

10001000 10011001x

100010000

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1 1000 x 01000 x 0+

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000001000002

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000 x 01000 x 0+1000001000002

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000001000002

100000010000003

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000 x 11000 x 1+

1000001000002

100000010000003

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000001000002

100000010000003

100100010010004

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000001000002

100000010000003

100100010010004

MultiplicandMultiplicand MultiplierMultiplier

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000001000002

100000010000003

100100010010004

MultiplicandMultiplicand MultiplierMultiplier

Shift leftShift left Add (0 OR Multiplicand)Add (0 OR Multiplicand)

Multiplication

● Start with long-multiplication approach

10001000 10011001x

10001000

1000010000

0

1

1000001000002

100000010000003

100100010010004

MultiplicandMultiplicand MultiplierMultiplier

Shift leftShift left Add (0 OR Multiplicand)Add (0 OR Multiplicand)

Length of the product is sum of operand lengthsLength of the product is sum of operand lengths

Optimized Multiplication Hardware

● Perform steps in parallel: add/shift

Fast Multiplication

● Using multiple adders

Optimized Divider

● One cycle per partial-remainder subtraction● Looks a lot like a multiplier!

– Same hardware can be used for both

Faster Division

● Can’t use parallel hardware as in multiplier– Subtraction is conditional on sign of remainder

● Faster dividers (e.g. SRT division) generate multiple quotient bits per step– Still require multiple steps