[FFmpeg-devel] [PATCH 00/15] replace pow(10, x) by exp10(x) across FFmpeg

[Michael Niedermayer]

On Thu, Dec 24, 2015 at 06:29:09PM -0800, Ganesh Ajjanagadde wrote:
> On Thu, Dec 24, 2015 at 6:18 PM, Michael Niedermayer
> <michael at 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 <gajjanag at 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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