On 07/27/2012 09:26 AM, Bruce Evans wrote:
% hm1 = -1;
% for (i=0;i<12;i++) hm1 += val[i];
% return (cpack(0.5 * log1p(hm1), atan2(y, x)));
It is the trailing terms that I think don't work right here. You sort
them and add from high to low, but normally it is necessary to add
from low to high (consider terms [1, DBL_EPSILON/2, DBL_EPSILON/4]).
Adding from high to low cancels with the -1 term, but then only
particular values work right. Also, I don't see how adding the low
terms without extra precision preserves enough precision.
I understand what you are saying. But in this case adding in order of
smallest to largest (adding -1 last) won't work. If all the signs in
the same direction, it would work. But -1 has the wrong sign.
But I can also tell you that I haven't thought my algorithm through
every special case. I can tell you it seems to work in all the examples
I tried. But I don't have a mathematical proof.
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