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Enhancing Arithmetic Operations: Fast Adders, Booth Multiplier, and Floating Point Numbers

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Learn about efficient arithmetic techniques like Fast Adders, Booth Multiplier, and Floating Point Numbers in computer science. Explore advanced algorithms for addition, multiplication, and real number representation.

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Enhancing Arithmetic Operations: Fast Adders, Booth Multiplier, and Floating Point Numbers

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  1. Arithmetic IIICPSC 321 Andreas Klappenecker

  2. Any Questions?

  3. Today’s Menu Addition Multiplication Floating Point Numbers

  4. Recall: Full Adder cin s a b cout 3 gates delay for first adder, 2(n-1) for remaining adders

  5. Ripple Carry Adders • Each gates causes a delay • our example: 3 gates for carry generation • book has example with 2 gates • Carry might ripple through all n adders • O(n) gates causing delay • intolerable delay if n is large • Carry lookahead adders

  6. Faster Adders Why are they called like that? cout=ab+cin(a xor b) =ab+acin+bcin =ab+(a+b)cin = g + p cin Generate g = ab Propagate p = a+b

  7. Fast Adders Iterate the idea, generate and propagate ci+1 = gi + pici = gi + pi(gi-1 + pi-1 ci-1) = gi + pigi-1+ pipi-1ci-1 = gi + pigi-1+ pipi-1gi-2 +…+ pipi-1 …p1g0 +pipi-1 …p1p0c0 Two level AND-OR circuit Carry is known early!

  8. A Simple ALU for MIPS • Need to support the set-on-less-than instruction (slt) • remember: slt is an arithmetic instruction • produces 1 if rs < rt and 0 otherwise • use subtraction: (a-b) < 0 implies a < b • Need to support test for equality (beq $t5, $t6, $t7) • use subtraction: (a-b) = 0 implies a = b

  9. ALU 000 = and001 = or010 = add110 = subtract111 = slt • Note: zero is a 1 when the result is zero!

  10. Multipliers

  11. Multiplication • More complicated than addition • accomplished via shifting and addition • Let's look at 3 versions based on the grade school algorithm0010 (multiplicand)__ x_1011 (multiplier) 0010 x 1 00100 x 1 001000 x 0 0010000 x 1 00010110 • Shift and add if multiplier bit equals 1

  12. Multiplication 0010 (multiplicand) __ x_1011 (multiplier) 0010 x 1 00100 x 1 001000 x 0 0010000 x 1 0010110

  13. Multiplication • If each step took a clock cycle, this algorithm would use almost 100 clock cycles to multiply two 32-bit numbers. • Requires 64-bit wide adder • Multiplicand register 64-bit wide

  14. Variations on a Theme • Product register has to be 64-bit • Nothing we can do about that! • Can we take advantage of that fact? • Yes! Add multiplicand to 32 MSBs • product = product >> 1 • Repeat last steps 0010 (multiplicand) __ x_1011 (multiplier) 0010 x 1 00100 x 1 001000 x 0 0010000 x 1 0010110

  15. Second Version

  16. Version 1 versus Version 2

  17. Critique • Registers needed for • multiplicand • multiplier • product • Use lower 32 bits of product register: • place multiplier in lower 32 bits • add multiplicand to higher 32 bits • product = product >> 1 • repeat

  18. Final Version Multiplier (shifts right)

  19. Summary It was possible to improve upon the well-known grade school algorithm by • reducing the adder from 64 to 32 bits • keeping the multiplicand fixed • shifting the product register • omitting the multiplier register

  20. The Booth Multiplier Let’s kick it up a notch!

  21. Runs of 1’s • 011102 = 14 = 8+4+2 = 16 – 2 • Runs of 1s (current bit, bit to the right): • 10 beginning of run • 11 middle of a run • 01 end of a run of 1s • 00 middle of a run of 0s

  22. Run’s of 1’s • 0111 1111 11002 = 2044 • How do you get this conversion quickly? • 0111 11112 = 128 – 1 = 127 • 0111 1111 11112 = 2048 – 1 • 0111 1111 11002 = 2048 – 1 – 3 = 2048 – 4

  23. Example 0010 0110 0000 shift -0010 sub 0000 shift 0010 add 00001100 0010 0110 0000 shift 0010 add 0010 add 0000 shift 00001100

  24. Booth Multiplication Current and previous bit 00: middle of run of 0s, no action 01: end of a run of 1s, add multiplicand 10: beginning of a run of 1s, subtract mcnd 11: middle of string of 1s, no action

  25. Example: 0010 x 0110

  26. Negative numbers Booth’s multiplication works also with negative numbers: 2 x -3 = -6 00102 x 11012 = 1111 10102

  27. Negative Numbers 00102 x 11012 = 1111 10102 0) Mcnd 0010 Prod 0000 1101,0 1) Mcnd 0010 Prod 1110 1101,1 sub 1) Mcnd 0010 Prod 1111 0110,1 >> 2) Mcnd 0010 Prod 0001 0110,1 add 2) Mcnd 0010 Prod 0000 1011,0 >> 3) Mcnd 0010 Prod 1110 1011,0 sub 3) Mcnd 0010 Prod 1111 0101,1 >> 4) Mcnd 0010 Prod 1111 0101,1 nop 4) Mcnd 0010 Prod 1111 1010,1 >>

  28. Summary • Extends the final version of the grade school algorithm • Simple change: add, subtract, or do nothing if last and previous bit respectively satisfy 0,1; 1,0 or 0,0; 1,1 • 0111 11002 = 128 – 4 = 1000 0002 – 0000 01002

  29. Floating Point Numbers

  30. Floating Point Numbers We often use calculations based on real numbers, such as • e = 2.71828… • Pi = 3.14592… We represent approximations to such numbers by floating point numbers • 1.xxxxxxxxxx2 x 2yyyy

  31. Floating-Point Representation: float We need to distribute the 32 bits among sign, exponent, and significand • seeeeeeeexxxxxxxxxxxxxxxxxxxxxxx The general form of such a number is • (-1)s x F x 2E • s is the sign, F is derived from the significand field, and E is derived from the exponent field

  32. Floating Point Representation: double • 1 bit sign, 11 bits for exponent, 52 bits for significand • seeeeeeeeeeexxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Range of float: 2.0 x 10-38 … 2.0 x 1038 Range of double: 2.0 x 10-308 … 2.0 x 10308

  33. IEEE 754 Floating-Point Standard • Makes leading bit of normalized binary number implicit 1 + significand • If significand is s1 s2 s3 s4 s5 s6 … then the value is (-1)s x (1 + s1/2 + s2/4 + s3/8 + … ) 2E • Design goal of IEEE 754: Integer comparisons should yield meaningful comparisons for floating point numbers

  34. IEEE 754 Standard • Negative exponents are a difficulty for sorting • Idea: most positive … most negative 1111 1111 … 0000 0000 • IEEE 754 uses a bias of 127 for single precision. • Exponent -1 is represented by -1 + 127 = 126

  35. IEEE 754 Example Represent -0.75 in single precision format. -0.75 = -3/4 = -112 / 4 = -0.112 In scientific notation: -0.11 x 20 = -1.1 x 2-1 the latter form is normalized sc. notation Value: (-1)s x (1+ significand) x 2(Expnt – 127)

  36. Example (cont’d) • -1.1 x 2-1 = (-1)1 x (1 + .1000 0000 0000 0000 0000 000) x 2(126 – 127) The single precision representation is 1 0111 11101000 0000 0000 0000 0000 000 BAM!

  37. Conclusion • We learned how to multiply • Three variations on the grade school algorithm • Booth multiplication • Floating point representation a la IEEE 754 (Photo’s are courtesy of www.emerils.com, some graphs are due to Patterson and Hennessy)

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