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FloatingPointRepresentation

Now we ask the question how is correctly rounded

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Unformatted text preview: ion: how is correctly rounded arithmetic implemented? Let us consider the addition of two oating point numbers x = m 2E and y = p 2F , using IEEE single precision. If the two exponents E and F are the same, it is necessary only to add the signi cands m and p. The nal result is (m + p) 2E, which then needs further normalization if m + p is 2 or larger, or less than 1. For example, the result of adding 3 = (1:100)2 21 to 2 = (1:000)2 21 is: ( 1:10000000000000000000000 + ( 1:00000000000000000000000 = ( 10:10000000000000000000000 )2 21 ) 2 21 )2 21 and the signi cand of the sum is shifted right 1 bit to obtain the normalized format (1:0100 : : : 0)2 22 . However, if the two exponents E and F are 17 di erent, say with E > F , the rst step in adding the two numbers is to align the signi cands, shifting p right E ; F positions so that the second number is no longer normalized and both numbers have the same exponent E . The signi cands are then added as before. For example, adding 3 = (1:100)2 21 to 3=4 = (1:100)2 2;1 gives: ( 1:1000000000000000000000 )2 21 + ( 0:0110000000000000000000 )2 21 = ( 1:1110000000000000000000 )2 21: In this case, the result does not need further normalization. Now consider adding 3...
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