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Arithmetic

Arithmetic. By Li Wen Professor Sin-Min Lee SJSU CS 147 Fall 2007. Table of Contents. Floating-point arithmetic Floating-point addition and subtraction Floating-point multiplication and division High-performance arithmetic High-performance addition High-performance multiplication

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Arithmetic

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  1. Arithmetic By Li Wen Professor Sin-Min Lee SJSU CS 147 Fall 2007

  2. Table of Contents • Floating-point arithmetic • Floating-point addition and subtraction • Floating-point multiplication and division • High-performance arithmetic • High-performance addition • High-performance multiplication • High-performance division • Residue Arithmetic • Questions and comments • Reference

  3. Floating-point Arithmetic • One way to add 101 + 1110 = .101 x 23 + .111 x 24 = .010 x 24 + .111 x 24 = (.010 + .111) x 24 = 1.001 x 24 = .1001 x 25 = .100 x 25 Adjusting the significant • Make the exponent equal to the larger exponent • 2. Lost .001 x 23 precision Lost total .001x23 +.0001x24 =.0011x24 precision • Round to three significant • Lost .0001 x 24

  4. Floating-point Arithmetic (cont.) • The other way to add 101 + 1110 = .101 x 23 + .111 x 24 = .0101 x 24 + .111 x 24 = (.0101 + .111) x 24 = 1.0011 x 24 = .10011 x 25 = .101 x 25 Adjusting the significant Make the exponent equal to the larger exponent • Round to nearest even method • got .00001 x 25 more Got total .00001x25 more

  5. Floating-point Arithmetic (cont.) • Both are incorrect • According to IEEE 754: the final result should be the same as if the maximum precision needed is used before applying the rounding method. • The correct result is .101 x 25 , which is the second example above

  6. Floating-point Arithmetic (cont.) • Using guard (g), round (r) and sticky (s) bit will make the correct result without using too much hardware. • Guard (g): one additional digit to the right • Round (r): one additional digit to the right • Sticky (s): any nonzero digits to the right of the round bit • Round method used: round to nearest even number on ties

  7. Floating-point Arithmetic (cont.) • Example: 101 .101 x 23 .0101 x 24 1110 .111 x 24 + .111 x 24 1.00110x24 grs=100 No 1 after r, so s = 0, stick 100 to .100110 .100110 x 25≈ .101 x 25 Reason of Complication PP 75

  8. Floating-point Arithmetic (cont.) • Addition example .1011 x 27 .1011 x 27 .0111 x 24 +.0000111 x 27 .1011111 x 27 grs=111 result .1100 x 27

  9. Floating-point Arithmetic (cont.) • Subtraction example .1011 x 27 .1011 x 27 .0111 x 24 - .0000111 x 27 .1010001 x 27 grs=001 result .1011 x 27

  10. Floating-point Arithmetic (cont.) • Multiplication example +.101 x 22 x - .110 x 2-3 - . 01111 x 2-1 Result: -.100 x 2-1 2+(-3) = -1

  11. Floating-point Arithmetic (cont.) • Division example + .110 x 25 /+.100 x 24 1.1 x 21 Result: .110 x 22 5 - 4 = 1

  12. High-performance arithmetic • The goal is to improve the speed of arithmetic operation. • High-Performance Addition, Multiplication, Division, and Residue Arithmetic

  13. High-Performance Addition • Method used in addition: carry lookahead adder • Example: si = aibici+aibici+aibici+aibici ci+1= bici+aici+aibi ci+1 = aibi+(ai+bi)ci ci+1 = Gi+Pici Where Gi=aibi and Pi = ai+bi

  14. High-Performance Addition (cont.) • Gi = aibi Gi is called generate, created by AND gate When Gi = 1, a carry is generated at stage i. • Pi = ai+bi Pi is called propagate, created by OR gate When Pi = 1, a carry is propagate at stage i if either ai or bi is a 1.

  15. High-Performance Addition (cont.)

  16. High-Performance Addition (cont.) • Example 3-2 Carry lookahead adder (CLA) Group carry lookahead adder (GCLA) Group Generate (GG) Group Propagate (GP)

  17. High-Performance Multiplication • Two methods are used for multiplication: • Skipping over blocks of 1s • Cross product in parallel multiplier • The Booth Algorithm • No + or – number • No addition, but shifting

  18. High-Performance Multiplication (cont.) • For example, for multiplier 001110 • From 0 to 1, -21; from 1 to 0 +24; 24-21 = (14)10 • Recorded as 0 +1 0 0 -1 0 by Booth Algorithm • Multiplicand 010101 (21)10 • Operation 1: (21)10 x (-21) • Operation 2: (21)10 x (24) • Operation 3: (294)10

  19. High-Performance Multiplication (cont.) • Problem with Booth Algorithm • Saving in the previous example: two operations needed -1 and +1 are recorded, but consider the following example: 010101 (21)10 • Will be recorded as +1 -1 +1 -1 +1 -1 by the same algorithm, where six operation will be needed. • Modified Booth Algorithm (bit-pair recording): group each (+1,-1) as +1, each (-1,+1) as -1, and (+1,+1) (-1,-1) cannot occur. Bit pairs table PP 83

  20. High-Performance Multiplication (cont.) • Method used in Array Multipliers: do each process in parallel. • Solve problem: time reduce from w2 to 2w for the length of w • Parallel pipelined array multiplier

  21. High-Performance Division • Method used: Newton’s iteration • Goal: find where the function f(x) crossed the x axis by starting guess xi then using the error between f(xi) and zero to refine the guess. • Result: until the last two guessed results are close to each other.

  22. Residue Arithmetic • Residue arithmetic can solve addition, subtraction, and multiplication problems. • The first 20 residue number for given moduli 5,7,9,4, PP 87

  23. References • http://en.wikipedia.org/wiki/Adder_(electronics) obtained by 9/23/07 • http://vlsi.uwindsor.ca/presentations/biswas4_seminar_1.pdf obtained by 9/23/07 • Computer Architecture and Organization, Miles Murdocca and Vincent Heuring

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