Floating Point Numbers

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Floating Point Numbers

• Muddsar JamilCS 147

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Introduction Representation

• Provides the ability to represent very large
  numbers, as well as very small numbers.Example
  1 Trillion 40 bits to left of radix pt.
• Retaining as much precision as needed increase
  calculation efficiency.
• A great deal of extra hardware is required in
  order to store/manipulate numbers with 80 bits or
  more.

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Computer Representation of Floating Point Numbers

• _ _ _ _ . _ _ _ _ _ _ _ _
  _ _ _ _
• This format makes for easier comparison , /,
  lt,
• Example Convert (358)10 in to the above format
  to be used as a floating point number.Java
  Float x new Float(358.0f)?

Sign Bit0 1 -
Three base 16 digits
3-bitexponent
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Example Continued

• First step is to convert 358 from base 10 to 16.
• Using Horner's method
• 358/16 22 --- R 6
• 22/16 1 --- R 6
• 35810 16616
• Next, convert to floating-point and Normalize
• (166)10 (166.)16 x 160
• Normalize ( .166 )16 x 163
• The exponent is 3, but we represent it in excess
  4
• 0 1 1 (3)10
• Excess 4 1 0 0 (4)10
• 1 1 1
• 0 1 1 1 . 0 0 0 1 0 1 1 0 0
  1 1 0
• 3 1 6
  6 Sign Expon.
  Fraction

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Fractional -gt Fixed Point Conversion

• Convert (XYZ.375)10 to Binary
• First, convert XYZ using Horner's method.
• Next, Convert the .37510 as following
• .375 x 2 0.75
• .75 x 2 1.5
• .5 x 2 1.0
• So (.375)10 (.011)2

Most Significant BitLeast Significant Bit
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IEEE 754 Floating Point Standard

• Created in 1985 to ensure standard representation
  among different systems.
• Most new architectures support IEEE 754.
• Two Formats
• Single Precision 1 8 23
• Double Precision

32 bits total
Sign Expon. Fraction
64 bits
1
11
52
Sign Expon. Fraction
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IEEE 754 Representations

• Can Represent (among others)?
• Non-zero, normalized numbers
• Clean zero
• All 0s in exponent and fraction
• Sign bit can be 0, or 1, to represent 0 or -0
• Infinity / Overflow / NaN
• Exponent contains all 1s, Fraction is all 0s
• Sign bit can be 0, or 1
• 0 / 0
• Sqrt(-1)