On Wed, 25 Jul 2012, Stephen Montgomery-Smith wrote:
This function seems to be able to compute clog with a worst case relative
error of 4 or 5 ULP.
...
I lost your previous reply about this after reading just the first part.
Please resend if interested.
First part recovered by vidcontrol:
VC> > I'm still working on testing and fixing clog. Haven't got near the more
VC> > complex functions.
VC> >
VC> > For clog, the worst case that I've found so far has x^2+y^2-1 ~= 1e-47:
VC> >
VC> > x = 0.999999999999999555910790149937383830547332763671875000000000
VC> > y =
VC> > 0.0000000298023223876953091912775497878893005143652317201485857367516
VC> > (need high precision decimal or these rounded to 53 bits binary)
VC> > x^2+y^2-1 = 1.0947644252537633366591637369e-47
VC>
VC> That is exactly 2^(-156). So maybe triple quad precision really is enough.
Hmm. But you need 53 more value bits after the 156. Quadruple precision
gives 3 to spare. I didn't notice that this number was exactly a power
of 2, but just added 15-17 for the value bits in decimal to 47 to get over
60.
VC> > so it needs more than tripled double precision for a brute force
VC> > evaluation, and the general case is probably worse. I'm working
VC> > on a rearrangement so that doubled double precision works in the
VC> > general case. Both your version and my version get this case right,
VC> > but mess up different much easier cases. It takes insanely great
VC> > accuracy to get even 1 bit in this case right, but now that we
Tripled double precision is enough for this because -1 cancels with
leading terms, giving almost quadrupled double precision:
% 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.
Bruce
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