# Download Code optimizers and register organizations for vector by Lee C.G. PDF

By Lee C.G.

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Spring Jo,nt Computer Conference, pp.

There is a much more efficient method, however, that dramatically improves the accuracy of sums, namely Theorem 8. xrjl-c T=S+Y C=(T– S=T } Summation using the N{ S)– Y Then the Exj(l s 2f. + dj) + 0(iVe2)X computed sum S is equal I xi 1, where to I 6, I Using the naive formula ~xl, the computed sum is equal to Xx~(l + 6J) where I 6, I < (n - j)e. Comparing this with the error in the Kahan summation formula shows a dramatic improvement. Each summand is perturbed by only 2 e instead of perturbations as large as n e in the simple formula.

10 I p I) will be correct down to the last bit. If ixflag is 1, then some digits were truncated, since round to zero always truncates. The significant of the product will look like 1. bl “ o“ bzz bz~ error may oc“ “ “ b~l. A double-rounding cur if bz~ “ . “ b~l = 10 “ “ “ O. A simple way to account for both cases is to perform a logical OR of ixflag with b31. Then round (N “ 10 I p I) will be computed correctly in all cases. 3 Errors . Arithmetic where ~6, ~ < c, and ignoring order terms in 6 i gives 43 second- The first eaualitv of (31) shows that the computed ~alue”of EXJ is the same as if an exact summation was performed on perturbed values of x,.