[FFmpeg-devel] [PATCHv2] lavu/libm: add erf hack and make dynaudnorm available everywhere
Ganesh Ajjanagadde
gajjanagadde at gmail.com
Mon Dec 21 18:08:56 CET 2015
On Sun, Dec 20, 2015 at 4:04 PM, Ganesh Ajjanagadde
<gajjanagadde at gmail.com> wrote:
> On Sun, Dec 20, 2015 at 12:04 PM, Ganesh Ajjanagadde
> <gajjanagadde at gmail.com> wrote:
>> Source code is from Boost:
>> http://www.boost.org/doc/libs/1_46_1/boost/math/special_functions/erf.hpp
>> with appropriate modifications for FFmpeg.
>>
>> Tested on interval -6 to 6 (beyond which it saturates), +/-NAN, +/-INFINITY
>> under -fsanitize=undefined on clang to test for possible undefined behavior.
>>
>> (Below will be trimmed for actual commit, given for completeness here)
>>
>> Note that this will remove the erf dependency for dynaudnorm, making it
>> universally available for clients. I have not done this in this patch;
>> may be easily done once this is tested on a broken libm and reviewed.
>>
>> This function turns out to actually be more accurate and faster than the
>> libm (GNU/BSD's/Mac OS X), and I can think of 3 reasons why upstream
>> does not use this:
>> 1. They are not aware of it.
>> 2. They are concerned about licensing - this applies especially to GNU
>> libm.
>> 3. They do not know and/or appreciate the benefits of rational
>> approximations over polynomial approximations. Boost uses them to great
>> effect, see e.g swr/resample for bessel derived from them, which is also
>> similarly superior to libm variants.
>>
>> First, performance.
>> sample benchmark (clang -O3, Haswell, GNU/Linux):
>>
>> 3e8 values evenly spaced from 0 to 6
>> time (libm):
>> ./test 13.39s user 0.00s system 100% cpu 13.376 total
>> time (boost based):
>> ./test 9.20s user 0.00s system 100% cpu 9.190 total
>>
>> Second, accuracy.
>> 1e8 eval pts from 0 to 6
>> maxdiff (absolute): 2.2204460492503131e-16
>> illustration:
>> erf(0.6):
>> libm : 0.60385609084792602
>> boost : 0.60385609084792591
>> real : 0.60385609084792591
>>
>> i.e libm is actually incorrectly rounded. Note that this is clear from:
>> https://github.com/JuliaLang/openlibm/blob/master/src/s_erf.c (the Sun
>> implementation used by both BSD and GNU libm's), where only 1 ulp is
>> guaranteed.
>> I suspect Boost's is correctly rounded (0.5 ulp) based on their error
>> analysis but have not checked or verified formally that this is the
>> case.
>>
>> Reviewed-by: James Almer <jamrial at gmail.com>
>> Signed-off-by: Ganesh Ajjanagadde <gajjanagadde at gmail.com>
>> ---
>> configure | 1 -
>> libavutil/libm.h | 198 +++++++++++++++++++++++++++++++++++++++++++++++++++++++
>> 2 files changed, 198 insertions(+), 1 deletion(-)
>>
>> diff --git a/configure b/configure
>> index da74616..89c98d1 100755
>> --- a/configure
>> +++ b/configure
>> @@ -2851,7 +2851,6 @@ cropdetect_filter_deps="gpl"
>> delogo_filter_deps="gpl"
>> deshake_filter_select="pixelutils"
>> drawtext_filter_deps="libfreetype"
>> -dynaudnorm_filter_deps="erf"
>> ebur128_filter_deps="gpl"
>> eq_filter_deps="gpl"
>> fftfilt_filter_deps="avcodec"
>> diff --git a/libavutil/libm.h b/libavutil/libm.h
>> index 37b8e86..f849bbb 100644
>> --- a/libavutil/libm.h
>> +++ b/libavutil/libm.h
>> @@ -1,4 +1,5 @@
>> /*
>> + * erf function: Copyright (c) 2006 John Maddock
>> * This file is part of FFmpeg.
>> *
>> * FFmpeg is free software; you can redistribute it and/or
>> @@ -76,6 +77,203 @@ static av_always_inline double copysign(double x, double y)
>> #define cosf(x) ((float)cos(x))
>> #endif
>>
>> +#if !HAVE_ERF
>> +static inline double ff_eval_poly(const double *coeff, int size, double x) {
>> + double sum = coeff[size-1];
>> + int i;
>> + for (i = size-2; i >= 0; --i) {
>> + sum *= x;
>> + sum += coeff[i];
>> + }
>> + return sum;
>> +}
>> +
>> +/**
>> + * erf function
>> + * Algorithm taken from the Boost project, source:
>> + * http://www.boost.org/doc/libs/1_46_1/boost/math/special_functions/erf.hpp
>> + * Use, modification and distribution are subject to the
>> + * Boost Software License, Version 1.0 (see notice below).
>> + * Boost Software License - Version 1.0 - August 17th, 2003
>> +Permission is hereby granted, free of charge, to any person or organization
>> +obtaining a copy of the software and accompanying documentation covered by
>> +this license (the "Software") to use, reproduce, display, distribute,
>> +execute, and transmit the Software, and to prepare derivative works of the
>> +Software, and to permit third-parties to whom the Software is furnished to
>> +do so, all subject to the following:
>> +
>> +The copyright notices in the Software and this entire statement, including
>> +the above license grant, this restriction and the following disclaimer,
>> +must be included in all copies of the Software, in whole or in part, and
>> +all derivative works of the Software, unless such copies or derivative
>> +works are solely in the form of machine-executable object code generated by
>> +a source language processor.
>> +
>> +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
>> +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
>> +FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
>> +SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
>> +FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
>> +ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
>> +DEALINGS IN THE SOFTWARE.
>> + */
>> +static inline double erf(double z)
>> +{
>> + double result;
>> +
>> + /* handle the symmetry: erf(-x) = -erf(x) */
>> + if (z < 0)
>> + return -erf(-z);
>> +
>> + /* branch based on range of z, and pick appropriate approximation */
>> + if (z == 0)
>> + return 0;
>> + else if (z < 1e-10)
>> + return z * 1.125 + z * 0.003379167095512573896158903121545171688;
>> + else if (z < 0.5) {
>> + // Maximum Deviation Found: 1.561e-17
>> + // Expected Error Term: 1.561e-17
>> + // Maximum Relative Change in Control Points: 1.155e-04
>> + // Max Error found at double precision = 2.961182e-17
>> +
>> + static const double y = 1.044948577880859375;
>> + static const double p[] = {
>> + 0.0834305892146531832907,
>> + -0.338165134459360935041,
>> + -0.0509990735146777432841,
>> + -0.00772758345802133288487,
>> + -0.000322780120964605683831,
>> + };
>> + static const double q[] = {
>> + 1,
>> + 0.455004033050794024546,
>> + 0.0875222600142252549554,
>> + 0.00858571925074406212772,
>> + 0.000370900071787748000569,
>> + };
>> + double zz = z * z;
>> + return z * (y + ff_eval_poly(p, FF_ARRAY_ELEMS(p), zz) / ff_eval_poly(q, FF_ARRAY_ELEMS(q), zz));
>> + }
>> + /* here onwards compute erfc */
>> + else if (z < 1.5) {
>> + // Maximum Deviation Found: 3.702e-17
>> + // Expected Error Term: 3.702e-17
>> + // Maximum Relative Change in Control Points: 2.845e-04
>> + // Max Error found at double precision = 4.841816e-17
>> + static const double y = 0.405935764312744140625;
>> + static const double p[] = {
>> + -0.098090592216281240205,
>> + 0.178114665841120341155,
>> + 0.191003695796775433986,
>> + 0.0888900368967884466578,
>> + 0.0195049001251218801359,
>> + 0.00180424538297014223957,
>> + };
>> + static const double q[] = {
>> + 1,
>> + 1.84759070983002217845,
>> + 1.42628004845511324508,
>> + 0.578052804889902404909,
>> + 0.12385097467900864233,
>> + 0.0113385233577001411017,
>> + 0.337511472483094676155e-5,
>> + };
>> + result = y + ff_eval_poly(p, FF_ARRAY_ELEMS(p), z - 0.5) / ff_eval_poly(q, FF_ARRAY_ELEMS(q), z - 0.5);
>> + result *= exp(-z * z) / z;
>> + return 1 - result;
>> + }
>> + else if (z < 2.5) {
>> + // Max Error found at double precision = 6.599585e-18
>> + // Maximum Deviation Found: 3.909e-18
>> + // Expected Error Term: 3.909e-18
>> + // Maximum Relative Change in Control Points: 9.886e-05
>> + static const double y = 0.50672817230224609375;
>> + static const double p[] = {
>> + -0.0243500476207698441272,
>> + 0.0386540375035707201728,
>> + 0.04394818964209516296,
>> + 0.0175679436311802092299,
>> + 0.00323962406290842133584,
>> + 0.000235839115596880717416,
>> + };
>> + static const double q[] = {
>> + 1,
>> + 1.53991494948552447182,
>> + 0.982403709157920235114,
>> + 0.325732924782444448493,
>> + 0.0563921837420478160373,
>> + 0.00410369723978904575884,
>> + };
>> + result = y + ff_eval_poly(p, FF_ARRAY_ELEMS(p), z - 1.5) / ff_eval_poly(q, FF_ARRAY_ELEMS(q), z - 1.5);
>> + result *= exp(-z * z) / z;
>> + return 1 - result;
>> + }
>> + else if (z < 4.5) {
>> + // Maximum Deviation Found: 1.512e-17
>> + // Expected Error Term: 1.512e-17
>> + // Maximum Relative Change in Control Points: 2.222e-04
>> + // Max Error found at double precision = 2.062515e-17
>> + static const double y = 0.5405750274658203125;
>> + static const double p[] = {
>> + 0.00295276716530971662634,
>> + 0.0137384425896355332126,
>> + 0.00840807615555585383007,
>> + 0.00212825620914618649141,
>> + 0.000250269961544794627958,
>> + 0.113212406648847561139e-4,
>> + };
>> + static const double q[] = {
>> + 1,
>> + 1.04217814166938418171,
>> + 0.442597659481563127003,
>> + 0.0958492726301061423444,
>> + 0.0105982906484876531489,
>> + 0.000479411269521714493907,
>> + };
>> + result = y + ff_eval_poly(p, FF_ARRAY_ELEMS(p), z - 3.5) / ff_eval_poly(q, FF_ARRAY_ELEMS(q), z - 3.5);
>> + result *= exp(-z * z) / z;
>> + return 1 - result;
>> + }
>> + /* differ from Boost here, the claim of underflow of erfc(x) past 5.8 is
>> + * slightly incorrect, change to 5.92
>> + * (really somewhere between 5.9125 and 5.925 is when it saturates) */
>> + else if (z < 5.92) {
>> + // Max Error found at double precision = 2.997958e-17
>> + // Maximum Deviation Found: 2.860e-17
>> + // Expected Error Term: 2.859e-17
>> + // Maximum Relative Change in Control Points: 1.357e-05
>> + static const double y = 0.5579090118408203125;
>> + static const double p[] = {
>> + 0.00628057170626964891937,
>> + 0.0175389834052493308818,
>> + -0.212652252872804219852,
>> + -0.687717681153649930619,
>> + -2.5518551727311523996,
>> + -3.22729451764143718517,
>> + -2.8175401114513378771,
>> + };
>> + static const double q[] = {
>> + 1,
>> + 2.79257750980575282228,
>> + 11.0567237927800161565,
>> + 15.930646027911794143,
>> + 22.9367376522880577224,
>> + 13.5064170191802889145,
>> + 5.48409182238641741584,
>> + };
>> + result = y + ff_eval_poly(p, FF_ARRAY_ELEMS(p), 1 / z) / ff_eval_poly(q, FF_ARRAY_ELEMS(q), 1 / z);
>> + result *= exp(-z * z) / z;
>> + return 1 - result;
>> + }
>> + /* handle the nan case, but don't use isnan for max portability */
>> + else if (z != z)
>> + return z;
>> + /* finally return saturated result */
>> + else
>> + return 1;
>> +}
>> +#endif
>> +
>> #if !HAVE_EXPF
>> #undef expf
>> #define expf(x) ((float)exp(x))
>> --
>> 2.6.4
>>
[...]
Pushed with slight modifications.
This is a request to any running MSVC/other platform lacking erf: can
you please test to make sure this works?
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