COE 308: Computer Architecture (T041) Dr. Marwan Abu-Amara

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COE 308 Computer Architecture (T041)Dr. Marwan
Abu-Amara

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

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Basic Techniques of Integer Arithmetic

• Ripple-Carry Addition
• Half Adder Takes 2 inputs ai bi and produces a
  sum bit, si, and a carry bit, ci1, as output
• Mathematically si (ai bi) mod 2
• ci1 ?(ai bi) / 2?
• Logically
• ci1 aibi

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Basic Techniques of Integer Arithmetic (cont.)

• Ripple-Carry Addition
• Full Adder Takes 3 inputs ai, bi, ci and
  produces a sum bit, si, and a carry bit, ci1, as
  output
• Logically
• ci1 aibi aici bici
• Problem in building an n-bit adder is propagating
  the carries
• Ripple-carry adder uses n full adders to build an
  n-bit adder
• Delay of ci1 is 2 levels of logic ? 2n logic
  levels in an n-bit adder (i.e. O(n) to generate
  final result)
• Can use n-bit adder to get AB by feeding A into
  A input inverse of B into B input, and set
  c0 to 1

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Basic Techniques of Integer Arithmetic (cont.)
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Radix-2 Multiplication Division

• Multiplication 2 unsigned numbers, A B
  (an-1an-2 a1a0 bn-1bn-2 b1b0). Register P
  (i.e. product) is initially set to 0. Algorithm
• Repeat n times
• If least significant bit of A is 1 ? P P B
• Otherwise ? P P 0
• Shift the register pair P A to right by 1 bit
  such that low-order bit of P is moved into As
  high-order bit, and low-order bit of A is shifted
  out
• Final result is in the register pair P A with P
  having the high order bits

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Radix-2 Multiplication Division

• Restoring Division 2 unsigned numbers, A B
  (an-1an-2 a1a0 bn-1bn-2 b1b0). Register P
  is initially set to 0. Algorithm
• Repeat n times
• Shift register pair P A one bit left (with
  high-order bit of A moved into Ps low-order bit)
• Subtract B from P (i.e. P P B)
• If result of step 2 lt 0 (i.e. ()ve) ? Set
  low-order bit of A to 0
• Otherwise ? Set low-order bit of A to 1
• If result of step 2 lt 0 (i.e. ()ve) ? P P B
• Final result A quotient, P remainder

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Radix-2 Multiplication Division
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Example of Division

• Divide 14 by 3 ? A 14 11102 B 3 00112
• Iteration Step P A Iteration Step
  P A .
• Initial. 0 0000 1110
• 1 1 0 0001 110_ 4 1 0 0010 010_
• 2 -0 0011 2 -0 0011
• 3 -0 0010 1100 3 -0 0001 0100
• (add B to P) 4 0 0001 1100 (add B to P) 4
  0 0010 0100
• 2 1 0 0011 100_
• 2 -0 0011 ? Quotient A 01002
• 3 0 0000 1001 4
• 4 0 0000 1001 Remainder P 00102
• 3 1 0 0001 001_ 2
• 2 -0 0011
• 3 -0 0010 0010
• (add B to P) 4 0 0001 0010

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Radix-2 Multiplication Division

• Non-Restoring Division 2 unsigned numbers, A B
  (an-1an-2 a1a0 bn-1bn-2 b1b0). Register P
  is initially set to 0. Algorithm
• Repeat n times
• If P is ()ve
• Shift register pair P A one bit left
• P P B
• Else (i.e. If P is not ()ve)
• Shift register pair P A one bit left
• P P B
• If P is ()ve ? Set low-order bit of A to 0
• Otherwise ? Set low-order bit of A to 1
• Final result A quotient, P remainder

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Signed Numbers

• 4 methods
• Sign magnitude
• 1s complement
• 2s complement (most widely used)
• Biased A fixed bias is picked so that sum of
  bias represented is always gt 0 (used in
  floating-point)
• Example Represent 3 in (1) sign magnitude, (2)
  1s complement, (3) 2s complement
• 3 00112 ? (1) sign magnitude 00112 10112
• ? (2) 1s complement 11002
• ? (3) 2s complement 11012

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Signed Numbers (cont.)

• Overflow in unsigned s when theres a carry-out
  of MSB
• Overflow in 2s complement when carry-in to MSB
  is different from carry-out of MSB
• Example
• 5 (7) ? 10 ? different
• 1011 (5) ? overflow
• 1001 (7)
• 0100
• 5 (7) ? 00 ? same
• 0101 (5) ? no overflow
• 1001 (7)
• 1110

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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

If ai ai-1 then
0 0 Add 0 to P
0 1 Add B to P
1 0 Sub B from P
1 1 Add 0 to P
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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.)
e f FP Represented Comment
255 0 ?
255 ? 0 NaN Not a Number
0 0 0
0 ? 0 denormal (or subnormal ) Deals with very small values
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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)