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CSE 246: Computer Arithmetic Algorithms and Hardware Design

CSE 246: Computer Arithmetic Algorithms and Hardware Design. Winter 2004 Lecture 9. Instructor: Prof. Chung-Kuan Cheng. Topics:. Floating Point Numbers (IEEE P754) Standard Operations Exceptional Situations Rounding Modes. Standard. 2 32  Typically. Goal: Dynamic Range:

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CSE 246: Computer Arithmetic Algorithms and Hardware Design

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  1. CSE 246: Computer Arithmetic Algorithms and Hardware Design Winter 2004 Lecture 9 Instructor: Prof. Chung-Kuan Cheng

  2. Topics: • Floating Point Numbers (IEEE P754) • Standard • Operations • Exceptional Situations • Rounding Modes

  3. Standard 232  Typically • Goal: Dynamic Range: largest #/ smallest # • If too large, holes between #’s

  4. Standard • ulp (unit in the last place) • Difference between two consecutive values of the significand. 3 Parts  x =  s  be Sign Bit Significand 8-bit exponent

  5. Standard • a1a2a3a4a5a6a7a8b1b2b3b22b23 • 1.* normalized number • 0.* denormalized number 0 0.b1b2b3b22b23  2-126 1 --------------------------------- 1. b1b2b3b22b23  2-126 2 . . . 253 254 ------------------------------- 1.b1b2b3b22b23  2127 •   if bi = 0 for all i = 1,2,…,23, NaN otherwise NaN  Not a Number

  6. Standard 0.01x2-3 = 0.00x2-2 • Same number, so normalize to remove redundancy • Smallest Number 0.00…01x2-126 = 1.0x2-23x2-126 = 1x2-149 • 1.1101111001110011100101 • Difference between 2 #’s small for normalized 0.0001 2 times compared to magnitudes 0.0010

  7. Standard - Example s. eeeeeeee nnnnnnnnnnnnnnnnnnnnnnn 0.00000000 00000000000000000000000 = 0.000…0x2-126 1.00000000 00000000000000000000000 = 0 0.00000001 00000000000000000000000 = 1.000…0x2-126 - minimal normalized # 0.00000001 00000000000000000000001 = 1.000…1x2-126 . . . 0.01111111 00000000000000000000001 = 1.000…1x20 0.10000000 00000000000000000000001 = 1.000…0x21

  8. Standard – Example Cont. 0.11111110 00000000000000000000001 = 1.000…1x2127 0.11111110 11111111111111111111111 = 1.111…1x2127 - Normalized Maximum 0.11111111 00000000000000000000000 =  Nmin = 1.0 x 2-126 Nmax = (2 – 2-23)2127

  9. Double Floating Point • a1a2…a11b1b2…b52 000…00 0. b1b2…b52 x 2-1023 000…01 1. b1b2…b52 x 2-1022 . . . 011…11 1. b1b2…b52 x 20 100…00 1. b1b2…b52 x 21 . . . 111…10 1. b1b2…b52 x 21023 111…11 =  if bi = 0 for all i = 1,2,…,52

  10. Overflow/Underflow Underflow Denser Sparser Nmin Nmax Overflow

  11. Addition/Multiplication • s1xbe1 + (s2xbe2) = sxbe = s1xbe1 + s2/be1-e2 x be1 = (s1  s2/be1-e2) x be1 • (s1xbe1) x (s2xbe2) = (s1xs2)be1+e2

  12. Exceptions a/0 =  if a > 0 a/ = 0 if a != 0 a·0 = 0 a· =  if a > 0 0· = invalid operation (NaN) 0/0 = invalid operation (NaN) NaP op a = NaN a +  =   -  = NaN

  13. Rounding Mode • Adder Output = Cout z1z0.z-1z-2…z-l GRS Guard Bit Round Bit Sticky Bit, OR of all bits below bit R 1.101 x 23 +1.110 x 23 11.011 x 23 1.1011x24 Normalize – need to round or

  14. Rouding 1.110 x 23 - 1.101 x 23 0.001 x 23 1.000 x 20 normalize 1.101 x 23 - 1.111 x 22 1.101 x 23 - 0.1111 x 23 0.1101 x 23 1.101 x 22 Guard bit

  15. Rounding • Round to the nearest even • toward 0 1.1011 • Toward + 1.1100 • Toward - 1.1011

  16. Conventional Rounding Error Rounding Error 1.10100  1.101 = 0 1.10101  1.101 = -0.25 1.10110  1.110 = +0.5 1.10111  1.110 = +0.25 Average Error = 0.5/4 = 0.125

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