CSE 246: Computer Arithmetic Algorithms and Hardware Design

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CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Winter 2004 Lecture 9

• Instructor
• Prof. Chung-Kuan Cheng

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Topics

• Floating Point Numbers (IEEE P754)
• Standard
• Operations
• Exceptional Situations
• Rounding Modes

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Standard
232 ? Typically

• Goal Dynamic Range
• largest / smallest
• If too large, holes between s

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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
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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

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Standard

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

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Standard - Example

• s. eeeeeeee nnnnnnnnnnnnnnnnnnnnnnn
• 0.00000000 00000000000000000000000
  0.0000x2-126
• 1.00000000 00000000000000000000000 0
• 0.00000001 00000000000000000000000
  1.0000x2-126
• - minimal normalized
• 0.00000001 00000000000000000000001
  1.0001x2-126
• .
• .
• .
• 0.01111111 00000000000000000000001 1.0001x20
• 0.10000000 00000000000000000000001 1.0000x21

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Standard Example Cont.

• 0.11111110 00000000000000000000001 1.0001x2127
• 0.11111110 11111111111111111111111 1.1111x2127
• - Normalized Maximum
• 0.11111111 00000000000000000000000 ?
• Nmin 1.0 x 2-126
• Nmax (2 2-23)2127

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Double Floating Point

• ?a1a2a11b1b2b52
• 00000 0. b1b2b52 x 2-1023
• 00001 1. b1b2b52 x 2-1022
• .
• .
• .
• 01111 1. b1b2b52 x 20
• 10000 1. b1b2b52 x 21
• .
• .
• .
• 11110 1. b1b2b52 x 21023
• 11111 ?? if bi 0 for all i 1,2,,52

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Overflow/Underflow
Underflow
Sparser
Denser
Nmax
Nmin
Overflow
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Addition/Multiplication

• ?s1xbe1 (?s2xbe2) ?sxbe
• ?s1xbe1 ?s2/be1-e2 x be1
• (?s1 ? s2/be1-e2) x be1
• (?s1xbe1) x (?s2xbe2) ?(s1xs2)be1e2

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Exceptions

• a/0 ? if a gt 0
• a/? 0 if a ! 0
• a0 0
• a? ? if a gt 0
• 0? invalid operation (NaN)
• 0/0 invalid operation (NaN)
• NaP op a NaN
• a ? ?
• ? - ? NaN

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Rounding Mode

• Adder Output Cout z1z0.z-1z-2z-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

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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
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Rounding

• Round to the nearest even
• toward 0 1.1011
• Toward ? 1.1100
• Toward -? 1.1011

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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