lavu/libm: add erf hack and make dynaudnorm available everywhere

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.

This function turns out to actually be essentially as 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
occuring at point where libm erf is correctly rounded, this is not.

Illustration of superior rounding of this function:
arg   : 0.83999999999999997
erf   : 0.76514271145499457
boost : 0.76514271145499446
real  : 0.76514271145499446

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.

Reasons it is not easy/worthwhile to create a "correctly rounded"
variant of this function (i.e 0.5ulp):
1. Upstream libm's don't do it anyway, so we can't guarantee this unless
we force this implementation on all platforms. This is not easy, as the
linker would complain unless measures are taken.
2. Nothing in FFmpeg cares or can care about such things, due to the
above and FFmpeg's nature.
3. Creating a correctly rounded function will in practice need some use of long
double/fma. long double, although C89/C90, unfortunately has problems on
ppc. This needs fixing of toolchain flags/configure. In any case this
will be slower for miniscule gain.

Reviewed-by: James Almer <jamrial@gmail.com>
Signed-off-by: Ganesh Ajjanagadde <gajjanagadde@gmail.com>

1 files changed, 201 insertions, 0 deletions

diff --git a/libavutil/libm.h b/libavutil/libm.h
index 37b8e86b5e..146768aac6 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 +77,206 @@ 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)
+{
+#ifndef FF_ARRAY_ELEMS
+#define FF_ARRAY_ELEMS(a) (sizeof(a) / sizeof((a)[0]))
+#endif
+    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))