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)) > > [...] > > -- > Michael GnuPG fingerprint: 9FF2128B147EF6730BADF133611EC787040B0FAB > > The educated differ from the uneducated as much as the living from the > dead. -- Aristotle > > _______________________________________________ > ffmpeg-devel mailing list > ffmpeg-devel@ffmpeg.org > http://ffmpeg.org/mailman/listinfo/ffmpeg-devel > _______________________________________________ ffmpeg-devel mailing list ffmpeg-devel@ffmpeg.org http://ffmpeg.org/mailman/listinfo/ffmpeg-devel
