COE 308: Computer Architecture T032 Dr' Marwan AbuAmara

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COE 308 Computer Architecture (T032)Dr. Marwan
Abu-Amara

• Integer Floating-Point Arithmetic (cont.)
• (Appendix A, Computer Architecture A
  Quantitative Approach, J. Hennessy D.
  Patterson, 1st Edition, 1990)

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Radix-2 Multiplication Division

• Signed Numbers Multiplication To perform 2s
  complement multiplication, use same algorithm as
  before but with the following modifications
• At the ith multiply step, LSB of A is ai, and
• for 1st step (i.e. when i 0), take ai-1 to be
  0
• Shift P arithmetically (i.e. copy sign bit) 1 bit
  to right

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Example 1 of Multiplication

• Multiply -6 by -5 ? A -6 a3a2a1a0 10102 B
  -5 10112
• Iteration Step P A ai-1
• 0000 1010 0
• 0 1 0000
• 0 0000 1010
• 2 0000 0101 0
• -B 1 0101
• 1 0101 0101
• 2 0010 1010 1
• B 1 1011
• 2 1101 1010
• 2 1110 1101 0
• -B 1 0101
• 3 0011 1101
• 2 0001 1110 1 ? Product P A 30

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Example 2 of Multiplication

• Multiply -6 by 5 ? A -6 a3a2a1a0 10102 B
  5 01012
• Iteration Step P A ai-1
• 0000 1010 0
• 0 1 0000
• 0 0000 1010
• 2 0000 0101 0
• -B 1 1011
• 1 1011 0101
• 2 1101 1010 1
• B 1 0101
• 2 0010 1010
• 2 0001 0101 0
• -B 1 1011
• 3 1100 0101
• 2 1110 0010 1 ? Product P A -30

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Floating Point

• A floating-point number (FP ) is divided into 2
  parts
• Exponent
• Significand (or Mantissa)
• FP significand ? baseexponent (e.g. exponent
  -2 significand 1.5 ? FP 1.5 ? 2-2
  0.375)
• Single-precision is represented using 32 bits
• 1 for sign
• 8 for exponent
• 23 for fraction
• Exponent is a signed represented using the bias
  method with a bias of 127
• Significand Mantissa 1 fraction
• Thus, if e value of exponent field, and f
  value of fraction field, then FP represented is
  1.f ? 2e127

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Floating Point (cont.)

• Example What single-precision FP does the
  following 32-bit word represent? 110000001010000
• 1 10000001 010000 ?
• sign 1 ve
• exponent field e 100000012 129 (?
  exponent 129127 2)
• fraction field f .0100002 0.012 0.25
• ? FP 1.f ? 2e127 1.25 ? 2129127 1.25
  ? 4 5
• Range of exponent field (i.e. e) is from 1 to 254
  (i.e. exponent is from 126 to 127)
• e 0 or 255 are used to represent special values

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Floating Point (cont.)
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Floating Point Addition

• What is the sum of 1,234,823.333 .0011?
• Need to line up the decimal points first
• This is the same as shifting the significand
  while changing the exponents
• 1,234,823.333 1.234823333 ? 106
• .0011 1.1 ? 10-3 0.0000000011 ? 106
• Add significands (using integer addition)
• Significand sum 1.234823333 0.0000000011
• 1.2348233341
• Normalize the result, if needed
• Result 1.2348233341 ? 106

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Floating Point Addition (cont.)

• Binary FP Addition Algorithm
• Similar to decimal FP addition method
• Let ei exponent, si significand (i.e. 1 fi
  24 bits), then steps of algorithm are
• If e1 lt e2, swap the operands, calculate d e1
  e2 (note that d ? 0), and set exponent of result
  to e1
• Shift s2 by d places to the right
• Add s1 result of step 2, and store result in s1
• Normalize
• If result of step 3 (i.e. s1) ? 2 ? Shift s1 by 1
  place right add 1 to exponent
• If s1 lt 1 ? Shift s1 to left until leftmost
  binary digit is 1 subtract of shifts from
  exponent
• If s1 0 ? Load special zero pattern into
  exponent
• Otherwise, do nothing (i.e. done)