Binary Arithmetic

• Quiz
• Division
• Extended Precision Arithmetic
• Project 1 (Homework 5)

Binary Division

• a lot simpler than decimal division
• terms: ((dividend - remainder) / divisor) = quotient (quotient * divisor) + remainder = dividend
• example: dividend 13, divisor 4, quotient 3, remainder 1
• division operation has 2 inputs (dividend and divisor) and two outputs (quotient and remainder)

Unsigned Binary Division: algorithm

1. Q = R = 0
2. compute B' = B shifted to the left until the most significant bit is 1.
3. looplab: if A < B then R = A and we are done.
4. Q = Q * 2
5. if A >= B' then:
   • Q = Q + 1
   • A = A - B'.
6. shift B' to the right (logical shift).
7. return to step "looplab".

Unsigned Binary Division: example

Nonrestoring division

• Note conditional subtraction: subtract only if A >= B
• Implementation: always do A - B, and add B back again if the result is negative.
• Note that on the next step we will subtract B/2.
• Mathematically, the computation if A - B < 0 is A - B + B - B2^{-1} = A - B + B2^{-1}
• Optimization: instead of adding B back, add B/2 (the shifted B) if the dividend A is negative.
• This is called non-restoring division
• The quotient gets a "1" bit when the result is positive, a "0" bit otherwise.

Nonrestoring Division: example

SPARC Division

• .div, .udiv, .rem, .urem,
• dividend in %o0 , divisor in %o1, result in %o0
• no special instruction such as mulscc
• assumption: division is rare, no need to have it in hardware

Intel Division

• DIV instruction for unsigned division
• DIV reg/mem: only source (divisor) is explicit, destination (dividend and quotient) is implicit:
  • 8 bits: AX = dividend, AL = quotient, AH = remainder
  • 16 bits: DX:AX = dividend, AX = quotient, DX = remainder
  • 16 bits: EDX:EAX = dividend, EAX = quotient, EDX = remainder

Extended Precision Arithmetic

• For arithmetic over more than 32/64 bits
• Example: cryptographic code (RSA)
• Example: scientific computations
• Do the arithmetic one word at a time
• Similar to decimal arithmetic (one digit at a time)

Extended Precision Addition

• Start with low-order word
• Add with carry for all but the first word (or clear carry flag, and add with carry for all words)
• SPARC: addx or addxcc
• Intel: ADC

Extended Precision Subtraction

• Start with low-order word
• Subtract with carry (borrow) for all but the first word (or clear carry flag, and subtract with carry/borrow for all words)
• SPARC: subx or subxcc
• Intel: SBB

Extended Precision Multiplication

• A * B
• Multiply A by low-order word (or bit) of B
• Shift A, then multiply by next low-order word (or bit) of B
• Add the two products, repeat
• Use extended-precision addition as subroutine

Extended Precision Division

• Produce one bit of quotient at a time
• Use restoring or non-restoring division
• Use extended-precision addition and subtraction as subroutines
• Need a test for positive/negative result or for magnitude comparison

Project

• build a four-function calculator
• adding, subtracting, multiplying, and dividing
• unsigned binary numbers up to 4000 bits (500 bytes) long
• two-week, double-credit homework (due October 7th)
• input-output: /home/1/esb/312/inout.c
• full description: http://www.ics.hawaii.edu/~esb/1998fall.ics312/hw5.html