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CraigBox |
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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 |
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radix - usually ten but sometimes 2. |
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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 |
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systems. This is [IEEE 754|http://grouper.ieee.org/groups/754/]. There is also lots of documentation at http://cch.loria.fr/documentation/IEEE754/. |
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The opposite is fixed-point. |
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!Single Precision |
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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': |
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S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF |
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0 1 8 9 31 |
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The value V represented by the word may be determined as follows: |
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* If E=255 and F is nonzero, then V=NaN ("Not a number") |
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* If E=255 and F is zero and S is 1, then V=-Infinity |
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* If E=255 and F is zero and S is 0, then V=Infinity |
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* 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. |
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* If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-126) * (0.F) These are "unnormalized" values. |
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* If E=0 and F is zero and S is 1, then V=-0 |
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* If E=0 and F is zero and S is 0, then V=0 |
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In particular, |
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0 00000000 00000000000000000000000 = 0 |
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1 00000000 00000000000000000000000 = -0 |
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0 11111111 00000000000000000000000 = Infinity |
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1 11111111 00000000000000000000000 = -Infinity |
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0 11111111 00000100000000000000000 = NaN |
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1 11111111 00100010001001010101010 = NaN |
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0 10000000 00000000000000000000000 = +1 * 2**(128-127) * 1.0 = 2 |
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0 10000001 10100000000000000000000 = +1 * 2**(129-127) * 1.101 = 6.5 |
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1 10000001 10100000000000000000000 = -1 * 2**(129-127) * 1.101 = -6.5 |
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0 00000001 00000000000000000000000 = +1 * 2**(1-127) * 1.0 = 2**(-126) |
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0 00000000 10000000000000000000000 = +1 * 2**(-126) * 0.1 = 2**(-127) |
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0 00000000 00000000000000000000001 = +1 * 2**(-126) * |
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0.00000000000000000000001 = |
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2**(-149) (Smallest positive value) |
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!Double Precision |
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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': |
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S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF |
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0 1 11 12 63 |
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The value V represented by the word may be determined as follows: |
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* If E=2047 and F is nonzero, then V=NaN ("Not a number") |
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* If E=2047 and F is zero and S is 1, then V=-Infinity |
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* If E=2047 and F is zero and S is 0, then V=Infinity |
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* 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. |
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* If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-1022) * (0.F) These are "unnormalized" values. |
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* If E=0 and F is zero and S is 1, then V=-0 |
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* If E=0 and F is zero and S is 0, then V=0 |
