FloatingPointRepresentation

Now consider adding 3 to 3 223 we get

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Unformatted text preview: to 3 2;23. We get ( 1:1000000000000000000000 )2 21 + ( 0:0000000000000000000001j1 )2 21 = ( 1:1000000000000000000001j1 )2 21 : This time, the result is not an IEEE single precision oating point number, since its signi cand has 24 bits after the binary point: the 24th is shown beyond the vertical bar. Therefore, the result must be correctly rounded. Rounding down gives the result (1:10000000000000000000001)2 21, while rounding up gives (1:10000000000000000000010)2 21. In the case of rounding to nearest, there is a tie, so the latter result, with the even nal bit, is obtained. For another example, consider the numbers 1, 2;15 and 215, which are all oating point numbers. The result of adding the rst to the second is (1:000000000000001)2, which is a oating point number the result of adding the rst to the third is (1:000000000000001)2 215, which is also a oating point number. However, the result of adding the second number to the third is (1:000000000000000000000000000001)2 215 which is not a oating point number in the IEEE single precision format, since the fraction eld would need 30 bits to represent this number exactly. In this example, usin...
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