Unsigned Multiplication - Shift Multiplicand Left

Csci136 Computer Architecture IILab#6 Arithmetic Operations HW #5: Due on Mar. 1, before classFeb.23, 2005

Unsigned Multiplication- Shift Multiplicand Left • Example: • 4-bit register • 0010*0011 • Multiplicand: 8 bits • Multiplier: 4 bits • Product: 8 bits

Unsigned Multiplication- Shift Multiplicand Left • Initial • Prod+Mcand •  Mcand • Mier  • Cost ??? • Time • Space

Unsigned Multiplication- Shift Product Right • Observations: • Work on one bit each time • 1/2 bits in multiplicand always 0 • least significant bits of product never changed once formed • Save Space? • Example: • 4-bit register • 0010*0011 • Multiplicand: 4 bits • Multiplier: N/A • Product: 8 bits • Save Time? M’cand=0010 Product=0000 0011

Signed Multiplication • Booth’s algorithm

Booth’s algorithm • Example: • M’cand =3 = 0011 • -m’cand=1101 • M’ier = -2 = 1110

Start: Place Dividend in Remainder 1. Subtract the Divisor register from the Remainder register, and place the result in the Remainder register. Remainder < 0 Test Remainder Remainder  0 2a. Shift the Quotient register to the left setting the new rightmost bit to 1. 2b. Restore the original value by adding the Divisor register to the Remainder register, & place the sum in the Remainder register. Also shift the Quotient register to the left, setting the new least significant bit to 0. 3. Shift the Divisor register right 1 bit. 33 repetition? No: < 33 repetitions Yes: 33 repetitions Done Unsigned Division- Restoring Division

Unsigned Division- Restoring Division • Example: • 4-bit register • Dividend=0111, divisor=0010 initial Rem = Rem-Div Rem<0+Div, sll, Q0=0 / sll, Q0=1 Div  srl

31 30 23 22 0 S Exponent Significand 1 bit 8 bits 23 bits Floating-point Representation • IEEE 754 Standard • (-1)S x (1 + Significand) x 2(Exponent-127) • Double precision identical, except with exponent bias of 1023

1 0111 1101 100 0000 0000 0000 0000 0000 Decimal  Floating-point Representation • Example: -0.375 • -0.375 = -3/8 • -11two/1000two = -0.011two • Normalized to -1.1two x 2-2 • (-1)S x (1 + Significand) x 2(Exponent-127) • (-1)1 x (1 + .100 0000 ... 0000) x 2(125-127)

1 10000001 010 0000 0000 0000 0000 0000 Floating-point Representation  Decimal • Example: • (-1)S x (1 + Significand) x 2(Exponent-127) • (-1)1 x (1 + 0.25) x 2(129-127)

Floating-point Addition • Example: 0.5ten + -0.4375ten • 0.5ten = 1/2ten = 1.000two x2-1 • -0.4375ten = -7/16ten = -1.110two x2-2 • Alignment: -1.110two x2-2 = -0.111two x2-1 • 1.000two x2-1 +(-0.111two x2-1) = 0.001two x2-1 • Normalization: 0.001two x2-1 = 1.000two x2-4 • Rounding: 4-bit already • 1.000two x2-4 = 0.0625

Floating-point Multiplication • Example: 0.5ten x -0.4375ten • 0.5ten = 1/2ten = 1.000two x2-1 • -0.4375ten = -7/16ten = -1.110two x2-2 • Exponents: (-1+127)+(-2+127)-127 = 124 • Significands: 1.000twox1.110two=1.110000twox2-3 • Normalization: -126≤exp. ≤127 • Rounding: =1.110twox2-3 • Sign: -1.110twox2-3

HW#5 • Example: • 0111/0011: quotient=0010,remainder=0001 • Difference? • Restoring Division: • (Remainder+Divisor)*2 – Divisor • Non-restoring Division: • Remainder*2 + Divisor

Unsigned Multiplication - Shift Multiplicand Left