Computer Arithmetic

1
Computer Arithmetic

• Readings
• Appendix A (A.1 A.7) of Hennessy and
  Pattersons Computer Architecture A
  Quantitative Approach

2
Computer Arithmetic

• Number storage (like memory and registers) is
  generally accomplished with bits, which are
  ON-OFF switches that represent binary digits
• A byte is 8 bits
• A word is 4 bytes (a doubleword is 8 bytes and a
  half-word is 2 bytes)
• The bits represent 1s and 0s in the binary
  (base-2) number system

3
Integer Representation

• Unsigned integers are represented as sequences of
  bits (e.g. 10010101 149 base 10)
• Signed integers reserve the most significant bit
  as the sign bit -- if it is 1 it is negative
• Three ways to represent negatives
• Sign bit only
• Ones complement (toggle all the bits)
• Twos complement (toggle all the bits and add one
  to the number)
• We prefer twos complement because it behaves
  nicely, especially with respect to zero (ones
  complement has two representations of zero, but
  not twos complement)

4
Integer Addition/Subtraction

• String together a chain of full adders (one for
  each bit in the integers), linking the carry-out
  of one adder to the carry-in of the next
• For addition, feed a 0 to the first adders
  carry-in
• In a twos complement system, to compute a-b,
  simply add a to the ones complement of b and use
  a 1 as the carry-in digit
• Note that with twos complement, the result of
  addition or subtraction is correct for the stored
  bits regardless of the carry-out from the highest
  order bit
• Overflow occurs when the carry-out is different
  from the carry-in to the last adder

5
Integer Multiplication (unsigned)

• Requires extra register P that is initially zero
• Assume operands are in registers A and B
• Repeat the following for each bit
• If the least significant digit of A is 1, then
  add B to P otherwise, add 0 to P
• Shift A one bit right, discarding the bit that
  falls off the end
• Shift P one bit right, moving the bit that falls
  off the end to the high-order position of A
• Product is in P and A, where A has the low-order
  bits and P the high-order bits

6
Integer Division (unsigned)

• Again, assume that A and B contain the operands
  and P is an extra zero register
• Repeat the following for each bit
• Shift P one bit left, discarding anything that
  falls off the left end
• Shift A one bit left, moving the bit that falls
  off the end to the low-order position in P
• Subtract B from P
• Set the low-order bit of A to the complement of
  the sign bit of P
• If P is negative, add B back to P
• Register A has quotient, P has remainder

7
Signed Multiplication?

• A technique called Booth recoding allows precise
  treatment of negative multipliers in hardware
  see handout for details
• Since division is slow anyway, signed division is
  implemented by (initially) ignoring the sign bit,
  doing an unsigned divide, and then calculating
  new sign bit

8
Brief Introduction to Floating Point Arithmetic

• FP standard is IEEE 754
• A single-precision (32-bit) number x is
  represented with a sign bit, a 23-bit fractional
  part f and an 8-bit biased exponent e such
  that x 1.f x 2e-127
• Question Why would anyone want to write a biased
  exponent instead of twos complement?
• Some numbers do not have exact floating point
  representations because of the base-2 exponent
• x0.5 f 0 and e 126 (in binary)
• What about x 0.3? 0.9?

9
Special FP Numbers

• Infinity e 255, f 0 (sign bit applies)
• NaN e 255, f gt 0
• Result for things like sqrt(-1), 0/0, 0
  infinity
• Zero e 0, f 0
• Trick Question If zero is represented in this
  way with e0, then what is the representation of
  1 in IEEE 754?
• Denormal numbers are used to represent very small
  quantities (i.e. smaller than 2-127)
• A denormal number is indicated by e 0, f gt 0
• Decimal equivalent is 0.f x 2-127

10
FP Multiplication

• Simplest operation
• Use an integer multiply on the fractional parts
  to obtain the new fractional part (take the first
  23 bits of product, round up if the discarded
  bits exceed 1000)
• The XOR of the sign bits is the new sign
• Add the biased exponents together and subtract
  127 to get the new exponent
• Overflow occurs when the exponent addition
  overflows however, underflow does not always
  occur when the exponent addition underflows (the
  result may be a denormalized number)
• Some machines use software to trap a denormalized
  result rather than complicate hardware

11
Other FP Operations

• See the reading for details if you are interested
  I do not expect you to know them in great
  detail
• Some highlights
• In addition/subtraction, precision is maintained
  by shifting the operand with the smaller exponent
  to the right until its exponent is the same as
  the larger one
• Rounding is similar to the case in multiplication
• In division, the general algorithm to calculate
  a/b is to calculate 1/b using Newtons method,
  and then multiply by a to get a/b