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Unformatted text preview: t. When one of the arguments is a special value such as ¼, ½ or Æ Æ , the standard speciﬁes conventions that attempt to be reasonable. For example ½ ¼ is deﬁned to yield ½, while ½ · ¼ is deﬁned to yield ·½. One strength of the IEEE standard’s method of specifying the behavior of ﬂoatingpoint operations is that it is independent of any particular hardware or software realization. Thus, we can examine its abstract mathematical properties without considering how it is actually implemented. We saw earlier that integer addition, both unsigned and two’s complement, forms an Abelian group. Addition over real numbers also forms an Abelian group, but we must consider what effect rounding has on these properties. Let us deﬁne Ü +f Ý to be ÊÓÙÒ ´Ü · Ý µ. This operation is deﬁned for all values of Ü and Ý , although it may yield inﬁnity even when both Ü and Ý are real numbers due to overﬂow. The operation is commutative, with Ü +f Ý Ý +f Ü for all values of Ü and Ý. On the other hand, the operation is not associative. For example, with singleprecision ﬂoating point the expression (3.14+1e10)1e10 would evalu...
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 Spring '10
 Schultz
 The American

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