FloatingPointRepresentation

This e ectively provides many guard bits for single

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Unformatted text preview: guard bits for single and double precision operations, but if an extended precision operation on extended precision operands is desired, at least one additional guard bit is needed. In fact, the following example (given in single precision for convenience) shows that one, two or even 24 guard bits are not enough to guarantee correctly rounded addition with 24-bit signi cands when the rounding mode is round to nearest. Consider computing x ; y where x = 1:0 and y = (1:000 : : : 01)2 2;25 , where y has 22 zero bits between the binary point and the nal one bit. In exact arithmetic, which requires 25 guard bits in this case, we get: (1:00000000000000000000000j )2 20 ; (0:00000000000000000000000j0100000000000000000000001 )2 20 = (0:11111111111111111111111j1011111111111111111111111 )2 20 Normalizing the result, and then rounding this to the nearest oating point number, we get (1:111 : : : 1)2 2;1 , which is the correctly rounded value of the exact sum of the numbers. However, if we were to use on...
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