1 / 22

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.

ahmed-carson
Télécharger la présentation

CSC 4250 Computer Architectures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related