swresample/resample: improve bessel function accuracy and speed

This improves accuracy for the bessel function at large arguments, and this in turn
should improve the quality of the Kaiser window. It also improves the
performance of the bessel function and hence build_filter by ~ 20%.
Details are given below.

Algorithm: taken from the Boost project, who have done a detailed
investigation of the accuracy of their method, as compared with e.g the
GNU Scientific Library (GSL):
http://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html.
Boost source code (also cited and licensed in the code):
https://searchcode.com/codesearch/view/14918379/.

Accuracy: sample values may be obtained as follows. i0 denotes the old bessel code,
i0_boost the approach here, and i0_real an arbitrary precision result (truncated) from Wolfram Alpha:
type "bessel i0(6.0)" to reproduce. These are evaluation points that occur for
the default kaiser_beta = 9.

Some illustrations:
bessel(8.0)
i0      (8.000000) = 427.564115721804739678191254
i0_boost(8.000000) = 427.564115721804796521610115
i0_real (8.000000) = 427.564115721804785177396791

bessel(6.0)
i0      (6.000000) = 67.234406976477956163762428
i0_boost(6.000000) = 67.234406976477970374617144
i0_real (6.000000) = 67.234406976477975326188025

Reason for accuracy: Main accuracy benefits come at larger bessel arguments, where the
Taylor-Maclaurin method is not that good: 23+ iterations
(at large arguments, since the series is about 0) can cause
significant floating point error accumulation.

Benchmarks: Obtained on x86-64, Haswell, GNU/Linux via a loop calling
build_filter 1000 times:
test: fate-swr-resample-dblp-44100-2626

new:
995894468 decicycles in build_filter(loop 1000),     256 runs,      0 skips
1029719302 decicycles in build_filter(loop 1000),     512 runs,      0 skips
984101131 decicycles in build_filter(loop 1000),    1024 runs,      0 skips

old:
1250020763 decicycles in build_filter(loop 1000),     256 runs,      0 skips
1246353282 decicycles in build_filter(loop 1000),     512 runs,      0 skips
1220017565 decicycles in build_filter(loop 1000),    1024 runs,      0 skips

A further ~ 5% may be squeezed by enabling -ftree-vectorize. However,
this is a separate issue from this patch.

Reviewed-by: Michael Niedermayer <michael@niedermayer.cc>
Signed-off-by: Ganesh Ajjanagadde <gajjanagadde@gmail.com>

libswresample/resample.c

@@ -1,6 +1,7 @@
 /*
  * audio resampling
  * Copyright (c) 2004-2012 Michael Niedermayer <michaelni@gmx.at>
+ * bessel function: Copyright (c) 2006 Xiaogang Zhang
  *
  * This file is part of FFmpeg.
  *
@@ -28,38 +29,108 @@
 #include "libavutil/avassert.h"
 #include "resample.h"
 
+static inline double 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;
+}
+
 /**
  * 0th order modified bessel function of the first kind.
+ * Algorithm taken from the Boost project, source:
+ * https://searchcode.com/codesearch/view/14918379/
+ * 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 double bessel(double x){
-    double lastv=0;
-    double t, v;
-    int i;
-    static const double inv[100]={
- 1.0/( 1* 1), 1.0/( 2* 2), 1.0/( 3* 3), 1.0/( 4* 4), 1.0/( 5* 5), 1.0/( 6* 6), 1.0/( 7* 7), 1.0/( 8* 8), 1.0/( 9* 9), 1.0/(10*10),
- 1.0/(11*11), 1.0/(12*12), 1.0/(13*13), 1.0/(14*14), 1.0/(15*15), 1.0/(16*16), 1.0/(17*17), 1.0/(18*18), 1.0/(19*19), 1.0/(20*20),
- 1.0/(21*21), 1.0/(22*22), 1.0/(23*23), 1.0/(24*24), 1.0/(25*25), 1.0/(26*26), 1.0/(27*27), 1.0/(28*28), 1.0/(29*29), 1.0/(30*30),
- 1.0/(31*31), 1.0/(32*32), 1.0/(33*33), 1.0/(34*34), 1.0/(35*35), 1.0/(36*36), 1.0/(37*37), 1.0/(38*38), 1.0/(39*39), 1.0/(40*40),
- 1.0/(41*41), 1.0/(42*42), 1.0/(43*43), 1.0/(44*44), 1.0/(45*45), 1.0/(46*46), 1.0/(47*47), 1.0/(48*48), 1.0/(49*49), 1.0/(50*50),
- 1.0/(51*51), 1.0/(52*52), 1.0/(53*53), 1.0/(54*54), 1.0/(55*55), 1.0/(56*56), 1.0/(57*57), 1.0/(58*58), 1.0/(59*59), 1.0/(60*60),
- 1.0/(61*61), 1.0/(62*62), 1.0/(63*63), 1.0/(64*64), 1.0/(65*65), 1.0/(66*66), 1.0/(67*67), 1.0/(68*68), 1.0/(69*69), 1.0/(70*70),
- 1.0/(71*71), 1.0/(72*72), 1.0/(73*73), 1.0/(74*74), 1.0/(75*75), 1.0/(76*76), 1.0/(77*77), 1.0/(78*78), 1.0/(79*79), 1.0/(80*80),
- 1.0/(81*81), 1.0/(82*82), 1.0/(83*83), 1.0/(84*84), 1.0/(85*85), 1.0/(86*86), 1.0/(87*87), 1.0/(88*88), 1.0/(89*89), 1.0/(90*90),
- 1.0/(91*91), 1.0/(92*92), 1.0/(93*93), 1.0/(94*94), 1.0/(95*95), 1.0/(96*96), 1.0/(97*97), 1.0/(98*98), 1.0/(99*99), 1.0/(10000)
-    };
 
-    x= x*x/4;
-    t = x;
-    v = 1 + x;
-    for(i=1; v != lastv; i+=2){
-        t *= x*inv[i];
-        v += t;
-        lastv=v;
-        t *= x*inv[i + 1];
-        v += t;
-        av_assert2(i<98);
+static double bessel(double x) {
+// Modified Bessel function of the first kind of order zero
+// minimax rational approximations on intervals, see
+// Blair and Edwards, Chalk River Report AECL-4928, 1974
+    static const double p1[] = {
+        -2.2335582639474375249e+15,
+        -5.5050369673018427753e+14,
+        -3.2940087627407749166e+13,
+        -8.4925101247114157499e+11,
+        -1.1912746104985237192e+10,
+        -1.0313066708737980747e+08,
+        -5.9545626019847898221e+05,
+        -2.4125195876041896775e+03,
+        -7.0935347449210549190e+00,
+        -1.5453977791786851041e-02,
+        -2.5172644670688975051e-05,
+        -3.0517226450451067446e-08,
+        -2.6843448573468483278e-11,
+        -1.5982226675653184646e-14,
+        -5.2487866627945699800e-18,
+    };
+    static const double q1[] = {
+        -2.2335582639474375245e+15,
+         7.8858692566751002988e+12,
+        -1.2207067397808979846e+10,
+         1.0377081058062166144e+07,
+        -4.8527560179962773045e+03,
+         1.0L,
+    };
+    static const double p2[] = {
+        -2.2210262233306573296e-04,
+         1.3067392038106924055e-02,
+        -4.4700805721174453923e-01,
+         5.5674518371240761397e+00,
+        -2.3517945679239481621e+01,
+         3.1611322818701131207e+01,
+        -9.6090021968656180000e+00,
+    };
+    static const double q2[] = {
+        -5.5194330231005480228e-04,
+         3.2547697594819615062e-02,
+        -1.1151759188741312645e+00,
+         1.3982595353892851542e+01,
+        -6.0228002066743340583e+01,
+         8.5539563258012929600e+01,
+        -3.1446690275135491500e+01,
+        1.0L,
+    };
+    double y, r, factor;
+    if (x == 0)
+        return 1.0;
+    x = fabs(x);
+    if (x <= 15) {
+        y = x * x;
+        return eval_poly(p1, FF_ARRAY_ELEMS(p1), y) / eval_poly(q1, FF_ARRAY_ELEMS(q1), y);
+    }
+    else {
+        y = 1 / x - 1.0 / 15;
+        r = eval_poly(p2, FF_ARRAY_ELEMS(p2), y) / eval_poly(q2, FF_ARRAY_ELEMS(q2), y);
+        factor = exp(x) / sqrt(x);
+        return factor * r;
     }
-    return v;
 }
 
 /**