Date: Fri, 27 Jul 2012 14:30:11 GMT From: Bruce Evans <brde@optusnet.com.au> To: freebsd-bugs@FreeBSD.org Subject: Re: bin/170206: complex arcsinh, log, etc. Message-ID: <201207271430.q6REUBem059502@freefall.freebsd.org>
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The following reply was made to PR bin/170206; it has been noted by GNATS. From: Bruce Evans <brde@optusnet.com.au> To: Stephen Montgomery-Smith <stephen@freebsd.org> Cc: FreeBSD-gnats-submit@freebsd.org, freebsd-bugs@freebsd.org Subject: Re: bin/170206: complex arcsinh, log, etc. Date: Sat, 28 Jul 2012 00:26:56 +1000 (EST) 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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