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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. 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