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 24bit signi cand bitstrings is a 48bit bitstring
which is then correctly rounded to 24 bits after normalization. Multiplication
of double precis...
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 Spring '09
 LIE
 oating point

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