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CSC 4250 Computer Architectures

CSC 4250 Computer Architectures. September 5, 2006 Appendix H. Computer Arithmetic. Integers. Using an 8-bit pattern, how many different integers can we represent? 2 8 Correct? How can we represent negative integers? Use a sign bit How many different integers are there?. Small Numbers.

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CSC 4250 Computer Architectures

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  1. CSC 4250Computer Architectures September 5, 2006Appendix H. Computer Arithmetic

  2. Integers • Using an 8-bit pattern, how many different integers can we represent? • 28 • Correct? • How can we represent negative integers? • Use a sign bit • How many different integers are there?

  3. Small Numbers • How can we represent numbers that are smaller than 1?

  4. Fixed Point Data Type • Bit pattern: 0100 0010 • Binary point just to the right of sign bit (in text) • Can be anywhere inside bit pattern • Binary point to far right of bit pattern • What do we get? • Integer • Value of bit pattern as integer: 0100 0010 • Value of bit patt. as fixed point: 0100 0010

  5. Blocked Floating Point • Fixed point → Low cost floating point • Exponent kept in separate variable • Exponent shared by a set of fixed point variables • Applications? • Overflow?

  6. Saturating Arithmetic • In DSP, if result is too large to be represented, it is set to the largest representable number with the appropriate sign • What happens when we get a floating-point overflow?

  7. IEEE 754 Floating-Point (32 bit) • Value = (–1)s (1+f) 2(e–127) • Significand = 1+f • Compare : scientific notation, say 3.45 × 106 • Base = 2 • So, 13 is represented by 1.101 × 23 with s=0, e=10000010, f=10100…00 • Can you give one advantage of the implicit 1? • Can you give one disadvantage of the implicit 1?

  8. IEEE 754 Floating-Point (64 bit) 1 bit 11 bits 52 bits • Value = (–1)s (1+f) 2(e–1023) • Significand = 1+f

  9. Base • IEEE: 2 • Other bases: 16 IBM 8 Burroughs • Normalize such that the first digit is nonzero • For base 16, the first digit could be 1, 2, …, 14, 15 • Need new symbols to represent 10,11, 12, 13, 14, 15 • What is base 16 called?

  10. Hexadecimal • What is hexa? • Greek for six • What is decimal? • Latin for ten • What is Latin prefix for six? • Sexa • What is old name for base 16? • Sexidecimal • Which company changed the name?

  11. Hexadecimal System • For base 16, we have 1 = 1/16 × 161 with f = 0001 00…00, 2 = 2/16 × 161 with f = 0010 00…00, 4 = 4/16 × 161 with f = 0100 00…00, 8 = 8/16 × 161 with f = 1000 00…00, 16 = 1/16 × 162 with f = 0001 00…00, • Disadvantage: As many as three leading zero bits • Advantage: Base larger → Range larger • 32 bit FP rep.: 1 sign bit, 7 exp. bits, 24 fraction bits (exponent bias = 64).

  12. Floating Point Representation of 1 • IEEE: s = 0, e = 01111111, f = 00000…00; Value = (1) 2(127–127) = 1. • IBM Hexadecimal: s = 0, e = 1000001, f = 00010000…00; Value = (1/16) 16(65–64) = 1.

  13. IEEE FP Representation of One to Sixteen

  14. Integer Comparisons of FP Number • Ease to use • Sorting • Sign bit is most significant: • Easy to check if number is greater than, less than, or equal to zero • Exponent before fraction: • Larger exponent → larger number • Use bias such that exponent values ≥ 0 • 1 ≤ e ≤ 254 → –126 ≤ e – bias ≤ 127 • e = 0 and e = 255?

  15. Representation of Zero • All bits (except sign bit) equal zero • e = 0 → zero exponent field • Zero fraction field (no implicit 1) • Plus and minus zero

  16. Denormal Numbers • e = 0 → zero exponent field • f ≠ 0 → no implicit 1 • Value = (–1)s *f *2–126 • Gradual underflow

  17. IEEE Representation • Fill in the blanks

  18. Underflow to Zero • In old days, a FP number that underflowed could be set to zero • An old “fast” test for equality: If a − b = 0, then a = b • Test would fail if a − b underflowed

  19. Infinity • e = 11…1 → maximum exponent field • Zero fraction field • Plus and minus infinity • 1/0 = ∞; 1+∞ = ∞ • Useful in trigonometry: sin(tan–1∞) = sin π/2 = 1

  20. NaN • Not a Number • e = 11…1 → maximum exponent field • Nonzero fraction field • Can you give an operation that generates NaN? • What is 1+NaN?

  21. Examples • Largest positive number: s = 0, e = 11111110, f = 11111…11; Value = (1+2–1+…+2–23) 2(254–127) = (2−2–23) 2127 ≈ 2128 ≈ 2.56×1038 • One: s = 0, e = 01111111, f = 00000…00; Value = (1) 2(127–127) = 1. • Smallest positive normal number: s = 0, e = 00000001, f = 00000…00; Value = (1) 2(1–127) = 2–126 ≈ 1.6×10–38

  22. Examples (2) • Largest positive denormal number: s = 0, e = 00000000, f = 11111…11; Value = (2–1+…+2–23) 2–126 = (1−2–23) 2–126 ≈ 2–126 • Differs from smallest normal by 2–149 • Smallest positive denormal number: s = 0, e = 00000000, f = 00000…01; Value = (2–23) 2–126 = 2–149 ≈ 2×10–45 • Plus zero: s = 0, e = 00000000, f = 00000…00.

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