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? [...] -- Michael GnuPG fingerprint: 9FF2128B147EF6730BADF133611EC787040B0FAB The educated differ from the uneducated as much as the living from the dead. -- Aristotle
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