Topics Integer Arithmetic Floating Point Representation Floating Point

Topics • Integer Arithmetic • Floating Point Representation • Floating Point Arithmetic

2’s Complement Integers

2’s Complement Addition/Subtraction A+B A What is Overflow? How is it identified?

Unsigned Integer Multiplication

Unsigned Integer Multiplication Q x M AQ

Unsigned Integer Multiplication Flow Diagram

2’s Comp Multiplication Booth’s Algorithm Add one extra bit to the Q register: Q-1

2’s Comp Multiplication Booth’s Algorithm Q-1

2’s Comp Multiplication Booth’s Algorithm

Booth : (7) x (3) A Q M 3 7 0000 0011 0 0111 ----------1001 0011 0 0111 1100 1001 1 0111 ---------- A <- (A - M) Shift 1 st 2 nd 1110 0100 1 0111 ----------0101 0100 1 0111 0010 1010 0 0111 ---------- Shift A <- (A + M) Shift 3 rd 4 th 0001 0101 0 0111 ---------- Shift

Booth : (7) x (-3) A Q M -3 7 0000 1101 0 0111 ----------1001 1101 0 0111 1100 1110 1 0111 ----------0011 1110 1 0111 0001 1111 0 0111 ----------1010 1111 0 0111 1101 0111 ---------- A <- (A - M) Shift 1 st A <- (A + M) Shift 2 nd A <- (A - M) Shift 3 rd 4 th 1110 1011 1 0111 ---------- Shift

Booth : (-7) x (3) A Q M 3 -7 0000 0011 0 1001 ----------0111 0011 0 1001 0011 1001 ---------- A <- (A - M) Shift 1 st 2 nd 0001 1100 1 1001 ----------1010 1100 1 1001 1101 0110 0 1001 ---------- Shift 1110 1011 0 1001 ---------- Shift A <- (A + M) Shift 3 rd 4 th

Booth : (-7) x (-3) A Q M -3 -7 0000 1101 0 1001 ----------0111 1101 0 1001 0011 1110 1 1001 ----------1100 1110 1 1001 1110 0111 0 1001 ----------0101 0111 0 1001 0010 1011 1 1001 ---------- A <- (A - M) Shift 1 st A <- (A + M) Shift 2 nd A <- (A - M) Shift 3 rd 4 th 0001 0101 1 1001 ---------- Shift

Unsigned Integer Division

Unsigned Integer Division

Unsigned Integer Division

Unsigned Integer Division Divisor M Dividend Q Quotient in Q Remainder in A

What about 2’s Comp Division ? • Can we do something like Booth’s Algorithm? • Why might we use Booth’s Algorithm for multiplication but not for Division?

Single Precision Floating Point Numbers IEEE Standard 32 bit Single Precision Floating Point Numbers are stored as: S EEEE FFFFFFFFFFFF S: Sign – 1 bit E: Exponent – 8 bits F: Fraction – 23 bits Significand The value V: • If E=255 and F is nonzero, then V= Na. N ("Not a Number") • If E=255 and F is zero and S is 1, then V= - Infinity • If E=255 and F is zero and S is 0, then V= Infinity • If 0<E<255 then V= (-1)**S * 2 ** (E-127) * (1. F) (exponent range = -127 to +128) • If E=0 and F is nonzero, then V= (-1)**S * 2 ** (-126) * (0. F) ("unnormalized" values”) • If E=0 and F is zero and S is 1, then V= - 0 • If E=0 and F is zero and S is 0, then V = 0 Note: 255 decimal = 1111 in binary (8 bits)

FP Examples

Double Precision Floating Point Numbers IEEE Standard 64 bit Double Precision Floating Point Numbers are stored as: S EEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFF S: Sign – 1 bit E: Exponent – 11 bits F: Fraction – 52 bits Significand The value V: • If E=2047 and F is nonzero, then V= Na. N ("Not a Number") • If E=2047 and F is zero and S is 1, then V= - Infinity • If E=2047 and F is zero and S is 0, then V= Infinity • If 0<E<2047 then V= (-1)**S * 2 ** (E-1023) * (1. F) (exponent range = -1023 to +1024) • If E=0 and F is nonzero, then V= (-1)**S * 2 ** (-1022) * (0. F) ("unnormalized" values) • If E=0 and F is zero and S is 1, then V= - 0 • If E=0 and F is zero and S is 0, then V= 0 Note: 2047 decimal = 111111 in binary (11 bits)

32 bit 2’s Complement Integer Numbers All the Integers from -2, 147, 483, 648 to + 2, 147, 483, 647, i. e. - 2 Gig to + 2 Gig-1

32 bit FP Numbers

“Density” of 32 bit FP Numbers Note: ONLY 232 FP numbers are representable There are only 232 distinct combinations of bits in 32 bits !

The Added Denormalized FP Numbers

Floating Point Addition / Subtraction Steps: • Check for zero • Align the significands (fractions) • Add or Subtract the significands • Normalize the Result Bad Results: • • Exponent Overflow Exponent Underflow Significand Overflow Significand Underflow

Floating Point Addition/Subraction

Floating Point Multiplication 1) Check for zero 2) Multiply significands 3) Add exponents 4) Normalize Overflow/underflow?

Floating Point Multiplication

Floating Point Division 1) Check for zero 2) Divide significands 3) Subract exponents 4) Normalize Overflow/underflow?

Floating Point Division