324 lines
8.7 KiB
C
324 lines
8.7 KiB
C
/****************************************************************************
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* libc/math/lib_gamma.c
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*
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* Ported to NuttX from FreeBSD by Alan Carvalho de Assis:
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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*
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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* 3. Neither the name NuttX nor the names of its contributors may be
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* used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
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* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
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* COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
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* BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
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* OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED
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* AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
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* ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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* POSSIBILITY OF SUCH DAMAGE.
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*
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****************************************************************************/
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/* "A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
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* "Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
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* "An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
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*
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* approximation method:
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*
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* (x - 0.5) S(x)
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* Gamma(x) = (x + g - 0.5) * ----------------
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* exp(x + g - 0.5)
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*
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* with
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* a1 a2 a3 aN
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* S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
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* x + 1 x + 2 x + 3 x + N
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*
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* with a0, a1, a2, a3,.. aN constants which depend on g.
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*
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* for x < 0 the following reflection formula is used:
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*
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* Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
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*
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* most ideas and constants are from boost and python
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*/
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/****************************************************************************
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* Included Files
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****************************************************************************/
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#include <nuttx/config.h>
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#include <nuttx/compiler.h>
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#include <sys/types.h>
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#include <math.h>
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#ifdef CONFIG_HAVE_DOUBLE
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/****************************************************************************
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* Pre-processor Definitions
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****************************************************************************/
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#define FORCE_EVAL(x) \
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do \
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{ \
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if (sizeof(x) == sizeof(float)) \
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{ \
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volatile float __x; \
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UNUSED(__x); \
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__x = (x); \
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} \
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else if (sizeof(x) == sizeof(double)) \
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{ \
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volatile double __x; \
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UNUSED(__x); \
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__x = (x); \
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} \
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else \
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{ \
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volatile long double __x; \
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UNUSED(__x); \
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__x = (x); \
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} \
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} \
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while(0)
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#define N 12
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/****************************************************************************
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* Private Data
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****************************************************************************/
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static const double pi = 3.141592653589793238462643383279502884;
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static const double g_gmhalf = 5.524680040776729583740234375;
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static const double g_snum[N + 1] =
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{
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23531376880.410759688572007674451636754734846804940,
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42919803642.649098768957899047001988850926355848959,
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35711959237.355668049440185451547166705960488635843,
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17921034426.037209699919755754458931112671403265390,
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6039542586.3520280050642916443072979210699388420708,
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1439720407.3117216736632230727949123939715485786772,
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248874557.86205415651146038641322942321632125127801,
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31426415.585400194380614231628318205362874684987640,
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2876370.6289353724412254090516208496135991145378768,
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186056.26539522349504029498971604569928220784236328,
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8071.6720023658162106380029022722506138218516325024,
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210.82427775157934587250973392071336271166969580291,
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2.5066282746310002701649081771338373386264310793408,
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};
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static const double g_sden[N + 1] =
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{
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0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535,
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2637558, 357423, 32670, 1925, 66, 1,
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};
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/* n! for small integer n */
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static const double g_fact[] =
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{
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1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0,
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479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0,
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355687428096000.0, 6402373705728000.0, 121645100408832000.0,
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2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0,
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};
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/* S(x) rational function for positive x */
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/****************************************************************************
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* Private Functions
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****************************************************************************/
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/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
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static double sinpi(double x)
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{
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int n;
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/* argument reduction: x = |x| mod 2 */
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/* spurious inexact when x is odd int */
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x = x * 0.5;
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x = 2 * (x - floor(x));
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/* reduce x into [-.25,.25] */
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n = 4 * x;
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n = (n + 1) / 2;
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x -= n * 0.5;
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x *= pi;
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switch (n)
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{
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default: /* case 4 */
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case 0:
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return __sin(x, 0, 0);
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case 1:
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return __cos(x, 0);
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case 2:
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return __sin(-x, 0, 0);
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case 3:
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return -__cos(x, 0);
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}
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}
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static double s(double x)
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{
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double num = 0;
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double den = 0;
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int i;
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/* to avoid overflow handle large x differently */
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if (x < 8)
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{
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for (i = N; i >= 0; i--)
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{
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num = num * x + g_snum[i];
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den = den * x + g_sden[i];
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}
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}
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else
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{
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for (i = 0; i <= N; i++)
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{
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num = num / x + g_snum[i];
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den = den / x + g_sden[i];
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}
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}
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return num/den;
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}
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/****************************************************************************
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* Public Functions
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****************************************************************************/
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double tgamma(double x)
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{
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union
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{
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double f;
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uint64_t i;
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} u;
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u.f = x;
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double absx;
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double y;
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double dy;
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double z;
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double r;
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uint32_t ix = u.i >> 32 & 0x7fffffff;
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int sign = u.i >> 63;
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/* special cases */
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if (ix >= 0x7ff00000)
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{
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/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
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return x + INFINITY;
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}
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if (ix < (0x3ff - 54) << 20)
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{
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/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
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return 1 / x;
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}
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/* integer arguments */
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/* raise inexact when non-integer */
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if (x == floor(x))
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{
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if (sign)
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{
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return 0 / 0.0;
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}
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if (x <= sizeof g_fact / sizeof *g_fact)
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{
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return g_fact[(int)x - 1];
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}
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}
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/* x >= 172: tgamma(x)=inf with overflow */
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/* x =< -184: tgamma(x)=+-0 with underflow */
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if (ix >= 0x40670000)
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{
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/* |x| >= 184 */
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if (sign)
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{
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FORCE_EVAL((float)(0x1p-126 / x));
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if (floor(x) * 0.5 == floor(x * 0.5))
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{
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return 0;
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}
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return -0.0;
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}
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x *= 0x1p1023;
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return x;
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}
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absx = sign ? -x : x;
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/* handle the error of x + g - 0.5 */
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y = absx + g_gmhalf;
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if (absx > g_gmhalf)
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{
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dy = y - absx;
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dy -= g_gmhalf;
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}
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else
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{
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dy = y - g_gmhalf;
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dy -= absx;
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}
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z = absx - 0.5;
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r = s(absx) * exp(-y);
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if (x < 0)
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{
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/* reflection formula for negative x */
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/* sinpi(absx) is not 0, integers are already handled */
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r = -pi / (sinpi(absx) * absx * r);
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dy = -dy;
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z = -z;
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}
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r += dy * (g_gmhalf + 0.5) * r / y;
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z = pow(y, 0.5 * z);
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y = r * z * z;
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return y;
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}
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double gamma(double x)
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{
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return tgamma(x);
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}
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#endif
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