On Thu, Dec 24, 2015 at 06:29:09PM -0800, Ganesh Ajjanagadde wrote: > On Thu, Dec 24, 2015 at 6:18 PM, Michael Niedermayer > <mich...@niedermayer.cc> wrote: > > On Thu, Dec 24, 2015 at 06:07:17PM -0800, Ganesh Ajjanagadde wrote: > >> On Thu, Dec 24, 2015 at 3:55 PM, Ganesh Ajjanagadde <gajja...@mit.edu> > >> wrote: > >> [...] > >> > 2. accuracy - yes, I am the only one who seems to care about it enough > >> > to bring it up everytime. On the other hand, I have documented the > >> > caveat and will transfer relevant information to avpriv_exp10 if we go > >> > that route, so I am fine with it. > >> > >> My long standing faith in GNU libm has been shattered, and I am > >> perfectly alright with this accuracy wise. BTW, I can reduce the error > >> by ~ 30% with 2 extra multiplications and an addition (a negligible > >> cost in front of the exp) in a very easy to understand way (no "magic" > >> numbers). Belongs in separate patch IMHO. > >> For those curious, here is the sequence: > >> 1. GNU libm makes a huge fuss about correct rounding (even 0.5 ulp), > >> refusing to take in slightly less accurate, but much faster functions: > >> https://news.ycombinator.com/item?id=8828936, particularly > >> https://news.ycombinator.com/item?id=8830486. Ok, I respect that > >> sentiment as long as they actually live by that. Experiments with sin, > >> cos, and other relatively simple libm functions confirmed that their > >> implementations are very accurate. > >> 2. Beginning of suspicion: while working on swr/resample (and merging > >> in Boost's code for bessel), I noticed GNU libm actually implements j0 > >> and other Bessel functions (man j0). They have a nice BUGS section > >> detailing errors up to 2e-16 on -8 to 8. > >> 3. Work on erf - I noticed that even here, GNU's implementation is not > >> correctly rounded in all cases, and Boost's is ~30% faster at similar > >> levels of accuracy: Boost's math function implementers seem to be > >> pragmatists wrt such rounding, > >> http://www.boost.org/doc/libs/1_48_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_function.html, > >> and come clean on how/to what degree things are correct. I do a man > >> erf, no BUGS section, nothing telling me anything regarding its > >> quality. I have to dig into the source to see that the claim is 1ulp, > >> which seems correct from some simple testing. BTW, this increased > >> speed, up front discussions of accuracy, readable and clean > >> implementations, and licensing issues are why I pull stuff from Boost > >> in case some of you wondered. > >> 4. Work on exp10 - turns out their initial implementation was an > >> exp(log(10)*x), which suffers from accuracy loss at large/small > >> numbers. Old bug report: > >> https://sourceware.org/bugzilla/show_bug.cgi?id=13884, and apparently > >> "fixed" by computing 2 exps (one being a small correction term, the > >> other the main term), > >> https://github.com/andikleen/glibc/blob/rtm-devel9/sysdeps/ieee754/dbl-64/e_exp10.c. > >> I assumed with all that effort and "magic" constants log10_high, > >> log10_low (what are they?), this would actually solve the rounding > >> issue: there is essentially no excuse for slowing down clients 2x > >> unless it actually achieves GNU libm's goal of correct rounding. > >> The beauty is, it does not. Illustration: > >> arg : -303.137207600000010643270798027515 > >> exp10 : 7.2910890073523505e-304, 2 ulp > >> exp10l: 7.2910890073523489e-304, 0 ulp > >> simple: 7.2910890073526541e-304, 377 ulp > >> corr : 7.2910890073524274e-304, 97 ulp > >> real : 7.2910890073523489e-304, 0 ulp > > > > how many ulps apart are exp10(x) and exp10(x + epsilon) > > that is the double and immedeatly next representable double arguments? > > More precisely I think you mean exp10(nextafter(x, INFINITY)). Here > are the answers (with incorrectly rounded exp): > next : 7.2910890073533049e-304, 1179 > prev : 7.2910890073513962e-304, 1179 > > i.e > exp10(nextafter(x, INFINITY)) > exp10(nextafter(x, -INFINITY)) > > or with the correct exp10l: > next : 7.2910890073533033e-304, 1178 > prev : 7.2910890073513954e-304, 1178 > > i.e > exp10l(nextafter(x, INFINITY)) > exp10l(nextafter(x, -INFINITY))
377 ulp looks rather good to me if the closest representable arguments are 1178 ulp apart [...] -- Michael GnuPG fingerprint: 9FF2128B147EF6730BADF133611EC787040B0FAB Concerning the gods, I have no means of knowing whether they exist or not or of what sort they may be, because of the obscurity of the subject, and the brevity of human life -- Protagoras
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