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Floating point is a number representation consisting of a mantissa, M, an exponent, E, and an (assumed) radix (or "base") . The number represented is M*R^E where R is the
radix - usually ten but sometimes 2.

Many different representations are used for the mantissa and exponent themselves. The [IEEE] specify a standard representation which is used by many hardware floating-point
systems.  This is [IEEE 754|http://grouper.ieee.org/groups/754/].  There is also lots of documentation at http://cch.loria.fr/documentation/IEEE754/.

The opposite is fixed-point.

!Single Precision

The IEEE single precision floating point standard representation requires a 32 bit word, which may be represented as numbered from 0 to 31, left to right. The first bit is the sign bit, S, the next eight bits are the exponent bits, 'E', and the final 23 bits are the fraction 'F':

  S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF
  0 1      8 9                    31

The value V represented by the word may be determined as follows:

* If E=255 and F is nonzero, then V=NaN ("Not a number")
* If E=255 and F is zero and S is 1, then V=-Infinity
* If E=255 and F is zero and S is 0, then V=Infinity
* If 0<E<255 then V=(-1)**S * 2 ** (E-127) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point.
* If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-126) * (0.F) These are "unnormalized" values.
* If E=0 and F is zero and S is 1, then V=-0
* If E=0 and F is zero and S is 0, then V=0

In particular,

  0 00000000 00000000000000000000000 = 0
  1 00000000 00000000000000000000000 = -0

  0 11111111 00000000000000000000000 = Infinity
  1 11111111 00000000000000000000000 = -Infinity

  0 11111111 00000100000000000000000 = NaN
  1 11111111 00100010001001010101010 = NaN

  0 10000000 00000000000000000000000 = +1 * 2**(128-127) * 1.0 = 2
  0 10000001 10100000000000000000000 = +1 * 2**(129-127) * 1.101 = 6.5
  1 10000001 10100000000000000000000 = -1 * 2**(129-127) * 1.101 = -6.5

  0 00000001 00000000000000000000000 = +1 * 2**(1-127) * 1.0 = 2**(-126)
  0 00000000 10000000000000000000000 = +1 * 2**(-126) * 0.1 = 2**(-127)
  0 00000000 00000000000000000000001 = +1 * 2**(-126) *
                                       0.00000000000000000000001 =
                                       2**(-149)  (Smallest positive value)

!Double Precision

The IEEE double precision floating point standard representation requires a 64 bit word, which may be represented as numbered from 0 to 63, left to right. The first bit is the sign bit, S, the next eleven bits are the exponent bits, 'E', and the final 52 bits are the fraction 'F':

  S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
  0 1        11 12                                                63

The value V represented by the word may be determined as follows:

* If E=2047 and F is nonzero, then V=NaN ("Not a number")
* If E=2047 and F is zero and S is 1, then V=-Infinity
* If E=2047 and F is zero and S is 0, then V=Infinity
* If 0<E<2047 then V=(-1)**S * 2 ** (E-1023) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point.
* If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-1022) * (0.F) These are "unnormalized" values.
* If E=0 and F is zero and S is 1, then V=-0
* If E=0 and F is zero and S is 0, then V=0
4 pages link to FloatingPoint:
Last edited on Sunday, August 10, 2003 5:42:22 pm by CraigBox
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