FloatingPointRepresentation

However if we were to use only two guard bits or

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Unformatted text preview: ly two guard bits (or indeed any number from 2 to 24), we would get the result: (1:00000000000000000000000j )2 20 ; (0:00000000000000000000000j01 )2 20 = (0:11111111111111111111111j11 )2 20 Normalizing and rounding then results in rounding up instead of down, giving the nal result 1:0, which is not the correctly rounded value of the exact sum. Machines that implement correctly rounded arithmetic take such possibilities into account, and it turns out that correctly rounded results can be achieved in all cases using only two guard bits together with an extra bit, called a sticky bit, which is used to ag a rounding problem of this kind. Floating point multiplication, unlike addition and subtraction, does not require signi cands to be aligned. If x = m 2E and y = p 2F , then x y = (m p) 2E+F so there are three steps to oating point multiplication: multiply the signi cands, add the exponents, and normalize and correctly round the result. 20 Single precision signi cands are easily multiplied in an extended precision register, since the product of two 24-bit signi cand bitstrings is a 48-bit bitstring which is then correctly rounded to 24 bits after normalization. Multiplication of double precis...
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