For example consider computing x y with x 102 20 and

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Unformatted text preview: mputing x ; y with x = (1:0)2 20 and y = (1:1111 : : : 1)2 2;1 , where the fraction eld for y contains 23 ones after the binary point. (Notice that y is only slightly smaller than x in fact it is the next oating point number smaller than x.) Aligning the signi cands, we obtain: ( 1:00000000000000000000000j )2 20 ; ( 0:11111111111111111111111j1 )2 20 = ( 0:00000000000000000000000j1 )2 20: This is an example of cancellation, since almost all the bits in the two numbers cancel each other. The result is (1:0)2 2;24 , which is a oating point number, but in order to obtain this correct result we must be sure to carry out the subtraction using an extra bit, called a guard bit, which is shown after the vertical line following the b23 position. When the IBM 360 was rst released, it did not have a guard bit, and it was only after the strenuous objections of certain computer scientists that later versions of the machine incorporated a guard bit. Twenty- ve years later, the Cray supercomputer still does not have a guard bit. When the operation just...
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