CPSC 321 Computer Architecture ALU Design – Integer Addition, Multiplication & Division Copyright 2002 David H. Albonesi and the University of Rochester. Additional material by Rabi Mahapatra and Hank Walker

Integer multiplication Pencil and paper binary multiplication 1000 (multiplicand) 1001 (multiplier)x

Integer multiplication 1000 (multiplicand) 1001 (multiplier) 1000 x Pencil and paper binary multiplication

Integer multiplication 1000 (multiplicand) 1001 (multiplier) x Pencil and paper binary multiplication

Integer multiplication 1000 (multiplicand) 1001 (multiplier) x Pencil and paper binary multiplication

Integer multiplication 1000 (multiplicand) 1001 (multiplier) x Pencil and paper binary multiplication

Integer multiplication 1000 (multiplicand) 1001 (multiplier) (product) x (partial products) Pencil and paper binary multiplication

Integer multiplication Pencil and paper binary multiplication Key elements  Examine multiplier bits from right to left  Shift multiplicand left one position each step  Simplification: each step, add multiplicand to running product total, but only if multiplier bit = (multiplicand) 1001 (multiplier) (product) x (partial products)

Integer multiplication Initialize product register to (multiplicand) 1001 (multiplier) (running product)

Integer multiplication Multiplier bit = 1: add multiplicand to product (multiplier) (new running product) (multiplicand)

Integer multiplication Shift multiplicand left 1001 (multiplier) (multiplicand)

Integer multiplication Multiplier bit = 0: do nothing 1001 (multiplier) (multiplicand)

Integer multiplication Shift multiplicand left 1001 (multiplier) (multiplicand)

Integer multiplication Multiplier bit = 0: do nothing 1001 (multiplier) (multiplicand)

Integer multiplication Shift multiplicand left 1001 (multiplier) (multiplicand)

Integer multiplication Multiplier bit = 1: add multiplicand to product 1001 (multiplier) (multiplicand) (product)

Integer multiplication 32-bit hardware implementation  Multiplicand loaded into right half of multiplicand register  Product register initialized to all 0’s  Repeat the following 32 times If multiplier register LSB=1, add multiplicand to product Shift multiplicand one bit left Shift multiplier one bit right LSB

Integer multiplication Algorithm

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (new running product)

Integer multiplication Drawback: half of 64-bit multiplicand register are zeros  Half of 64 bit adder is adding zeros Solution: shift product right instead of multiplicand left  Only left half of product register added to multiplicand 1000 (multiplicand) 1001 (multiplier) (product)

Integer multiplication Hardware implementation

Integer multiplication Final improvement: use right half of product register for the multiplier

Integer multiplication Final algorithm

Multiplication of signed numbers Naïve approach  Convert to positive numbers  Multiply  Negate product if multiplier and multiplicand signs differ  Slow and extra hardware

Multiplication of signed numbers Booth’s algorithm  Invented for speed Shifting was faster than addition at the time Objective: reduce the number of additions required  Fortunately, it works for signed numbers as well  Basic idea: the additions from a string of 1’s in the multiplier can be converted to a single addition and a single subtraction operation  Example: is equivalent to – requires additions for each of these bit positions requires an addition for this bit position and a subtraction for this bit position

Booth’s algorithm Starting from right to left, look at two adjacent bits of the multiplier  Place a zero at the right of the LSB to start If bits = 00, do nothing If bits = 10, subtract the multiplicand from the product  Beginning of a string of 1’s If bits = 11, do nothing  Middle of a string of 1’s If bits = 01, add the multiplicand to the product  End of a string of 1’s Shift product register right one bit

Booth recoding Example 1101 (multiplier)x 0010 (multiplicand)

Booth recoding Example (product+multiplier) 0010 (multiplicand) extra bit position

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand)

Booth recoding Example (multiplicand) (product)

Integer division Pencil and paper binary division (dividend)(divisor) 1000

Integer division Pencil and paper binary division (dividend)(divisor) (partial remainder)

Integer division Pencil and paper binary division (dividend)(divisor)

Integer division Pencil and paper binary division (dividend)(divisor)

Integer division Pencil and paper binary division (dividend)(divisor)

Integer division Pencil and paper binary division (dividend)(divisor)

Integer division Pencil and paper binary division (dividend)(divisor)

Integer division Pencil and paper binary division (dividend)(divisor) (quotient) (remainder)

Integer division Pencil and paper binary division Steps in hardware  Shift the dividend left one position  Subtract the divisor from the left half of the dividend  If result positive, shift left a 1 into the quotient  Else, shift left a 0 into the quotient, and repeat from the beginning  Once the result is positive, repeat the process for the partial remainder  Do n iterations where n is the size of the divisor (dividend)(divisor) (quotient) (remainder)

Integer division Initial state (dividend)(divisor) (quotient)

Integer division Shift dividend left one position (dividend)(divisor) (quotient)

Integer division Subtract divisor from left half of dividend (dividend)(divisor) (quotient) (keep these bits)

Integer division Result positive, left shift a 1 into the quotient (dividend)(divisor) (quotient)

Integer division Shift partial remainder left one position (dividend)(divisor) (quotient)

Integer division Subtract divisor from left half of partial remainder (dividend)(divisor) (quotient)

Integer division Result negative, left shift 0 into quotient (dividend)(divisor) (quotient)

Integer division Restore original partial remainder (how?) (dividend)(divisor) (quotient)

Integer division Shift partial remainder left one position (dividend)(divisor) (quotient)

Integer division Subtract divisor from left half of partial remainder (dividend)(divisor) (quotient)

Integer division Result negative, left shift 0 into quotient (dividend)(divisor) (quotient)

Integer division Restore original partial remainder (dividend)(divisor) (quotient)

Integer division Shift partial remainder left one position (dividend)(divisor) (quotient)

Integer division Subtract divisor from left half of partial remainder (dividend)(divisor) (quotient)

Integer division Result positive, left shift 1 into quotient (dividend)(divisor) (quotient) (remainder)

Integer division Hardware implementation What operations do we do here? Load dividend here initially

Integer and floating point revisited Integer ALU handles add, subtract, logical, set less than, equality test, and effective address calculations Integer multiplier handles multiply and divide  HI and LO registers hold result of integer multiply and divide instruction memory PCPC integer register file integer ALU integer multiplier data memory flt pt register file flt pt adder flt pt multiplier HI LO

Floating point representation Floating point (fp) numbers represent reals  Example reals: , 1.23 x , x 10 6  Floats and doubles in C Fp numbers are in signed magnitude representation of the form (-1) S x M x B E where  S is the sign bit (0=positive, 1=negative)  M is the mantissa (also called the significand)  B is the base (implied)  E is the exponent  Example: x S=0 M=22.34 B=10 E=-4

Floating point representation Fp numbers are normalized in that M has only one digit to the left of the “decimal point”  Between 1.0 and … in decimal  Between 1.0 and … in binary  Simplifies fp arithmetic and comparisons  Normalized: x 10 2, 1.23 x  Not normalized: x 10 6, x 10 -4, x  In binary format, normalized numbers are of the form Leading 1 in 1.M is implied (-1) S x 1.M x B E

Floating point representation tradeoffs Representing a wide enough range of fp values with enough precision (“decimal” places) given limited bits  More E bits increases the range  More M bits increases the precision  A larger B increases the range but decreases the precision  The distance between consecutive fp numbers is not constant! BEBE B E+1 B E+2 …… S 32 bits E??M?? (-1) S x 1.M x B E

Floating point representation tradeoffs Allowing for fast arithmetic implementations  Different exponents requires lining up the significands; larger base increases the probability of equal exponents Handling very small and very large numbers 0 exponent overflow exponent underflow exponent overflow representable positive numbers (S=0) representable negative numbers (S=1)

Sorting/comparing fp numbers fp numbers can be treated as integers for sorting and comparing purposes if E is placed to the left Example  3.67 x 10 6 > 6.34 x > 1.23 x SEM (-1) S x 1.M x B E bigger E is bigger number If E’s are same, bigger M is bigger number

Biased exponent notation 111…111 represents the most positive E and 000…000 represents the most negative E for sorting/comparing purposes To get correct signed value for E, need to subtract a bias of 011…111 Biased fp numbers are of the form (-1) S x 1.M x B E-bias Example: assume 8 bits for E  Bias is = 127  Largest E represented by which is 255 – 127 = 128  Smallest E represented by which is 0 – 127 = -127

IEEE 754 floating point standard Created in 1985 in response to the wide range of fp formats used by different companies  Has greatly improved portability of scientific applications B=2 Single precision (sp) format (“float” in C) Double precision (dp) format (“double” in C) SEM 1 bit8 bits23 bits SEM 1 bit11 bits52 bits

IEEE 754 floating point standard Exponent bias is 127 for sp and 1023 for dp Fp numbers are of the form (-1) S x 1.M x 2 E-bias  1 in mantissa and base of 2 are implied  Sp form is (-1) S x 1.M 22 M 21 …M 0 x 2 E-127 and value is (-1) S x (1+(M 22 x2 -1 ) +(M 21 x2 -2 )+…+(M 0 x2 -23 )) x 2 E-127 Sp example  Number is –1.1000…000 x =-1.5 x =1.763 x …000 SEM

IEEE 754 floating point standard Denormalized numbers  Allow for representation of very small numbers  Identified by E=0 and a non-zero M  Format is (-1) S x 0.M x 2 -bias-1  Smallest positive dp denormalized number is 0.00…01 x = smallest positive dp normalized number is 1.0 x  Hardware support is complex, and so often handled by software 0 exponent overflow exponent underflow exponent overflow representable positive numbers representable negative numbers

Floating point addition Make both exponents the same  Find the number with the smaller one  Shift its mantissa to the right until the exponents match Must include the implicit 1 (1.M) Add the mantissas Choose the largest exponent Put the result in normalized form  Shift mantissa left or right until in form 1.M  Adjust exponent accordingly Handle overflow or underflow if necessary Round Renormalize if necessary if rounding produced an unnormalized result

Floating point addition Algorithm

Floating point addition example …01100 SEM …00111 SEM Initial values

Floating point addition example …01100 SEM …00111 SEM Identify smaller E and calculate E difference difference = 2

Floating point addition example …00011 SEM …00111 SEM Shift smaller M right by E difference

Floating point addition example …00011 SEM …00111 SEM Add mantissas … …00111 = … …00100 SEM

Floating point addition example …00011 SEM …00111 SEM Choose larger exponent for result …00100 SEM

Floating point addition example …00011 SEM …00111 SEM Final answer (already normalized) …00100 SEM

Floating point addition Hardware design determine smaller exponent

Floating point addition Hardware design shift mantissa of smaller number right by exponent difference

Floating point addition Hardware design add mantissas

Floating point addition Hardware design normalize result by shifting mantissa of result and adjusting larger exponent

Floating point addition Hardware design round result

Floating point addition Hardware design renormalize if necessary

Floating point multiply Add the exponents and subtract the bias from the sum  Example: (5+127) + (2+127) – 127 = Multiply the mantissas Put the result in normalized form  Shift mantissa left or right until in form 1.M  Adjust exponent accordingly Handle overflow or underflow if necessary Round Renormalize if necessary if rounding produced an unnormalized result Set S=0 if signs of both operands the same, S=1 otherwise

Floating point multiply Algorithm

Floating point multiply example Initial values …00000 SEM …00000 SEM -1.5 x x

Floating point multiply example Add exponents …00000 SEM …00000 SEM -1.5 x = (231) 1.5 x

Floating point multiply example Subtract bias …00000 SEM …00000 SEM -1.5 x SEM – = = (104) 1.5 x

Floating point multiply example Multiply the mantissas …00000 SEM …00000 SEM -1.5 x SEM … x … = … 1.5 x

Floating point multiply example Normalize by shifting 1.M right one position and adding one to E …00000 SEM …00000 SEM -1.5 x SEM … => … … 1.5 x

Floating point multiply example Set S=1 since signs are different …00000 SEM …00000 SEM -1.5 x SEM … 1.5 x x

Rounding Fp arithmetic operations may produce a result with more digits than can be represented in 1.M The result must be rounded to fit into the available number of M positions Tradeoff of hardware cost (keeping extra bits) and speed versus accumulated rounding error

Rounding Examples from decimal multiplication Renormalization is required after rounding in c)

Rounding Examples from binary multiplication (assuming two bits for M) 1.01 x 1.01 = (1.25 x 1.25 = ) 1.10 x 1.01 = 1.111(1.5 x 1.25 = 1.875) May require renormalization after rounding 1.11 x 1.01 = (1.75 x 1.25 = ) Result has twice as many bits

Rounding In binary, an extra bit of 1 is halfway in between the two possible representations (1.125) is halfway between 1.00 (1) and 1.01 (1.25) (1.625) is halfway between 1.10 (1.5) and 1.11 (1.75)

IEEE 754 rounding modes Truncate  Remove all digits beyond those supported  > 1.00 Round up to the next value  > 1.01 Round down to the previous value  > 1.00  Differs from Truncate for negative numbers Round-to-nearest-even  Rounds to the even value (the one with an LSB of 0)  > 1.00  > 1.10  Produces zero average bias  Default mode

Implementing rounding A product may have twice as many digits as the multiplier and multiplicand  1.11 x 1.01 = For round-to-nearest-even, we need to know  The value to the right of the LSB (round bit)  Whether any other digits to the right of the round digit are 1’s The sticky bit is the OR of these digits rounds to 1.01 Round bit Sticky bit = 0 OR 1 = rounds to 1.00 LSB of final rounded result

Implementing rounding The product before normalization may have 2 digits to the left of the binary point Product register format needs to be Two possible cases bb.bbbb… 1b.bbbb…r sssss… 01.bbbb…r sssss… Need this as a result bit!

Implementing rounding The guard bit (g) becomes part of the unrounded result when the MSB = 0 g, r, and s suffice for rounding addition as well

MIPS floating point registers bit FPRs  bit registers (32-bit register pairs) for dp floating point  Software conventions for their usage (as with GPRs) Control/status register  Status of compare operations, sets rounding mode, exceptions Implementation/revision register  Identifies type of CPU and its revision number f0 f1 f30 f control/status register 310 FCR implementation/revision register FCR0 310 floating point registers

MIPS floating point instruction overview Operate on single and double precision operands Computation  Add, sub, multiply, divide, sqrt, absolute value, negate  Multiply-add, multiply-subtract Added as part of MIPS-IV revision of ISA specification Load and store  Integer register read for EA calculation  Data to be loaded or stored in fp register file Move between registers Convert between different formats Comparison instructions Branch instructions

MIPS R10000 arithmetic units instruction memory PCPC integer register file integer ALU integer ALU + multiplier data memory flt pt register file flt pt adder flt pt multiplier flt pt divider flt pt sq root EA calc

MIPS R10000 arithmetic units Integer ALU + shifter  All instructions take one cycle Integer ALU + multiplier  Booth’s algorithm for multiplication (5-10 cycles)  Non-restoring division (34-67 cycles) Floating point adder  Carry propagate (2 cycles) Floating point multiplier (3 cycles)  Booth’s algorithm Floating point divider (12-19 cycles) Floating point square root unit Separate unit for EA calculations Can start up to 5 instructions in 1 cycle