diff --git a/include/cxx/cmath b/include/cxx/cmath index 998406c153..c43b4c59e5 100644 --- a/include/cxx/cmath +++ b/include/cxx/cmath @@ -76,6 +76,8 @@ namespace std using ::sqrtf; using ::tanf; using ::tanhf; + using ::gamma; + using ::lgamma; #endif #ifdef CONFIG_HAVE_DOUBLE diff --git a/include/nuttx/lib/math.h b/include/nuttx/lib/math.h index e9e0bf2596..9623d3d590 100644 --- a/include/nuttx/lib/math.h +++ b/include/nuttx/lib/math.h @@ -230,6 +230,13 @@ long double expl (long double x); #define expm1l(x) (expl(x) - 1.0) #endif +#ifdef CONFIG_HAVE_DOUBLE +double __cos(double x, double y); +double __sin(double x, double y, int iy); +double gamma(double x); +double lgamma(double x); +#endif + float logf (float x); #ifdef CONFIG_HAVE_DOUBLE double log (double x); diff --git a/libc/math.csv b/libc/math.csv index 90b0ce3e0a..ec17f1cbde 100644 --- a/libc/math.csv +++ b/libc/math.csv @@ -37,6 +37,7 @@ "ldexp","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double",int" "ldexpf","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float",int" "ldexpl","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double","int" +"lgamma","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double" "log","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double" "log10","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double" "log10f","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float" @@ -46,6 +47,7 @@ "log2l","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double" "logf","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float" "logl","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double" +"gamma","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double" "modf","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double","double *" "modff","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float","float *" "modfl","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double","long double *" diff --git a/libc/math/Make.defs b/libc/math/Make.defs index 7ead14882a..a1f5697d52 100644 --- a/libc/math/Make.defs +++ b/libc/math/Make.defs @@ -62,6 +62,8 @@ CSRCS += lib_truncl.c CSRCS += lib_libexpi.c lib_libsqrtapprox.c CSRCS += lib_libexpif.c +CSRCS += __cos.c __sin.c lib_gamma.c lib_lgamma.c + # Use the C versions of some functions only if architecture specific # optimized versions are not provided. diff --git a/libc/math/__cos.c b/libc/math/__cos.c new file mode 100644 index 0000000000..4d7b2f410d --- /dev/null +++ b/libc/math/__cos.c @@ -0,0 +1,124 @@ +/**************************************************************************** + * libc/math/__cos.c + * + * Ported to NuttX from FreeBSD by Alan Carvalho de Assis: + * + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in + * the documentation and/or other materials provided with the + * distribution. + * 3. Neither the name NuttX nor the names of its contributors may be + * used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS + * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT + * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS + * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE + * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, + * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS + * OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED + * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN + * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE + * POSSIBILITY OF SUCH DAMAGE. + * + ****************************************************************************/ + +/* __cos( x, y ) + * + * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * + * Algorithm + * 1. Since cos(-x) = cos(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. + * 3. cos(x) is approximated by a polynomial of degree 14 on + * [0,pi/4] + * 4 14 + * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x + * where the remez error is + * + * | 2 4 6 8 10 12 14 | -58 + * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 + * | | + * + * 4 6 8 10 12 14 + * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then + * cos(x) ~ 1 - x*x/2 + r + * since cos(x+y) ~ cos(x) - sin(x)*y + * ~ cos(x) - x*y, + * a correction term is necessary in cos(x) and hence + * cos(x+y) = 1 - (x*x/2 - (r - x*y)) + * For better accuracy, rearrange to + * cos(x+y) ~ w + (tmp + (r-x*y)) + * where w = 1 - x*x/2 and tmp is a tiny correction term + * (1 - x*x/2 == w + tmp exactly in infinite precision). + * The exactness of w + tmp in infinite precision depends on w + * and tmp having the same precision as x. If they have extra + * precision due to compiler bugs, then the extra precision is + * only good provided it is retained in all terms of the final + * expression for cos(). Retention happens in all cases tested + * under FreeBSD, so don't pessimize things by forcibly clipping + * any extra precision in w. + */ + +/**************************************************************************** + * Included Files + ****************************************************************************/ + +#include +#include + +#include +#include + +#ifdef CONFIG_HAVE_DOUBLE + +/**************************************************************************** + * Private Data + ****************************************************************************/ + +static const double g_c1 = 4.16666666666666019037e-02; /* 0x3fa55555, 0x5555554c */ +static const double g_c2 = -1.38888888888741095749e-03; /* 0xbf56C16c, 0x16c15177 */ +static const double g_c3 = 2.48015872894767294178e-05; /* 0x3efa01a0, 0x19cb1590 */ +static const double g_c4 = -2.75573143513906633035e-07; /* 0xbe927e4e, 0x809c52ad */ +static const double g_c5 = 2.08757232129817482790e-09; /* 0x3e21ee9E, 0xbdb4b1c4 */ +static const double g_c6 = -1.13596475577881948265e-11; /* 0xbda8fae9, 0xbe8838d4 */ + +/**************************************************************************** + * Public Functions + ****************************************************************************/ + +double __cos(double x, double y) +{ + double hz; + double z; + double r; + double w; + + z = x * x; + w = z * z; + r = + z * (g_c1 + z * (g_c2 + z * g_c3)) + w * w * (g_c4 + z * (g_c5 + z * g_c6)); + hz = 0.5 * z; + w = 1.0 - hz; + + return w + (((1.0 - w) - hz) + (z * r - x * y)); +} +#endif diff --git a/libc/math/__sin.c b/libc/math/__sin.c new file mode 100644 index 0000000000..b1653ee8b4 --- /dev/null +++ b/libc/math/__sin.c @@ -0,0 +1,121 @@ +/**************************************************************************** + * libc/math/__cos.c + * + * Ported to NuttX from FreeBSD by Alan Carvalho de Assis: + * + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in + * the documentation and/or other materials provided with the + * distribution. + * 3. Neither the name NuttX nor the names of its contributors may be + * used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS + * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT + * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS + * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE + * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, + * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS + * OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED + * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN + * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE + * POSSIBILITY OF SUCH DAMAGE. + * + ****************************************************************************/ + +/* __sin( x, y, iy) + * + * kernel sin function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). + * + * Algorithm + * 1. Since sin(-x) = -sin(x), we need only to consider positive x. + * 2. Callers must return sin(-0) = -0 without calling here since our + * odd polynomial is not evaluated in a way that preserves -0. + * Callers may do the optimization sin(x) ~ x for tiny x. + * 3. sin(x) is approximated by a polynomial of degree 13 on + * [0,pi/4] + * 3 13 + * sin(x) ~ x + S1*x + ... + S6*x + * where + * + * |sin(x) 2 4 6 8 10 12 | -58 + * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 + * | x | + * + * 4. sin(x+y) = sin(x) + sin'(x')*y + * ~ sin(x) + (1-x*x/2)*y + * For better accuracy, let + * 3 2 2 2 2 + * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) + * then 3 2 + * sin(x) = x + (S1*x + (x *(r-y/2)+y)) + */ + +/**************************************************************************** + * Included Files + ****************************************************************************/ + +#include +#include + +#include +#include + +#ifdef CONFIG_HAVE_DOUBLE + +/**************************************************************************** + * Private Data + ****************************************************************************/ + +static const double g_s1 = -1.66666666666666324348e-01; /* 0xbfc55555, 0x55555549 */ +static const double g_s2 = 8.33333333332248946124e-03; /* 0x3f811111, 0x1110f8a6 */ +static const double g_s3 = -1.98412698298579493134e-04; /* 0xbf2a01a0, 0x19c161d5 */ +static const double g_s4 = 2.75573137070700676789e-06; /* 0x3ec71de3, 0x57b1fe7d */ +static const double g_s5 = -2.50507602534068634195e-08; /* 0xbe5ae5e6, 0x8a2b9ceb */ +static const double g_s6 = 1.58969099521155010221e-10; /* 0x3de5d93a, 0x5acfd57c */ + +/**************************************************************************** + * Public Functions + ****************************************************************************/ + +double __sin(double x, double y, int iy) +{ + double z; + double r; + double v; + double w; + + z = x * x; + w = z * z; + r = g_s2 + z * (g_s3 + z * g_s4) + z * w * (g_s5 + z * g_s6); + v = z * x; + + if (iy == 0) + { + return x + v * (g_s1 + z * r); + } + else + { + return x - ((z * (0.5 * y - v * r) - y) - v * g_s1); + } +} +#endif diff --git a/libc/math/lib_gamma.c b/libc/math/lib_gamma.c new file mode 100644 index 0000000000..68f37d9134 --- /dev/null +++ b/libc/math/lib_gamma.c @@ -0,0 +1,323 @@ +/**************************************************************************** + * libc/math/lib_gamma.c + * + * Ported to NuttX from FreeBSD by Alan Carvalho de Assis: + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in + * the documentation and/or other materials provided with the + * distribution. + * 3. Neither the name NuttX nor the names of its contributors may be + * used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS + * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT + * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS + * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE + * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, + * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS + * OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED + * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN + * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE + * POSSIBILITY OF SUCH DAMAGE. + * + ****************************************************************************/ + +/* "A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964) + * "Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001) + * "An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004) + * + * approximation method: + * + * (x - 0.5) S(x) + * Gamma(x) = (x + g - 0.5) * ---------------- + * exp(x + g - 0.5) + * + * with + * a1 a2 a3 aN + * S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ] + * x + 1 x + 2 x + 3 x + N + * + * with a0, a1, a2, a3,.. aN constants which depend on g. + * + * for x < 0 the following reflection formula is used: + * + * Gamma(x)*Gamma(-x) = -pi/(x sin(pi x)) + * + * most ideas and constants are from boost and python + */ + +/**************************************************************************** + * Included Files + ****************************************************************************/ + +#include +#include + +#include +#include + +#ifdef CONFIG_HAVE_DOUBLE + +/**************************************************************************** + * Pre-processor Definitions + ****************************************************************************/ + +#define FORCE_EVAL(x) \ + do \ + { \ + if (sizeof(x) == sizeof(float)) \ + { \ + volatile float __x; \ + UNUSED(__x); \ + __x = (x); \ + } \ + else if (sizeof(x) == sizeof(double)) \ + { \ + volatile double __x; \ + UNUSED(__x); \ + __x = (x); \ + } \ + else \ + { \ + volatile long double __x; \ + UNUSED(__x); \ + __x = (x); \ + } \ + } \ + while(0) + +#define N 12 + +/**************************************************************************** + * Private Data + ****************************************************************************/ + +static const double pi = 3.141592653589793238462643383279502884; + +static const double g_gmhalf = 5.524680040776729583740234375; +static const double g_snum[N + 1] = +{ + 23531376880.410759688572007674451636754734846804940, + 42919803642.649098768957899047001988850926355848959, + 35711959237.355668049440185451547166705960488635843, + 17921034426.037209699919755754458931112671403265390, + 6039542586.3520280050642916443072979210699388420708, + 1439720407.3117216736632230727949123939715485786772, + 248874557.86205415651146038641322942321632125127801, + 31426415.585400194380614231628318205362874684987640, + 2876370.6289353724412254090516208496135991145378768, + 186056.26539522349504029498971604569928220784236328, + 8071.6720023658162106380029022722506138218516325024, + 210.82427775157934587250973392071336271166969580291, + 2.5066282746310002701649081771338373386264310793408, +}; + +static const double g_sden[N + 1] = +{ + 0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535, + 2637558, 357423, 32670, 1925, 66, 1, +}; + +/* n! for small integer n */ + +static const double g_fact[] = +{ + 1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0, + 479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0, + 355687428096000.0, 6402373705728000.0, 121645100408832000.0, + 2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0, +}; + +/* S(x) rational function for positive x */ + +/**************************************************************************** + * Private Functions + ****************************************************************************/ + +/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */ + +static double sinpi(double x) +{ + int n; + + /* argument reduction: x = |x| mod 2 */ + /* spurious inexact when x is odd int */ + + x = x * 0.5; + x = 2 * (x - floor(x)); + + /* reduce x into [-.25,.25] */ + + n = 4 * x; + n = (n + 1) / 2; + x -= n * 0.5; + + x *= pi; + switch (n) + { + default: /* case 4 */ + case 0: + return __sin(x, 0, 0); + + case 1: + return __cos(x, 0); + + case 2: + return __sin(-x, 0, 0); + + case 3: + return -__cos(x, 0); + } +} + +static double s(double x) +{ + double num = 0; + double den = 0; + int i; + + /* to avoid overflow handle large x differently */ + + if (x < 8) + { + for (i = N; i >= 0; i--) + { + num = num * x + g_snum[i]; + den = den * x + g_sden[i]; + } + } + else + { + for (i = 0; i <= N; i++) + { + num = num / x + g_snum[i]; + den = den / x + g_sden[i]; + } + } + + return num/den; +} + +/**************************************************************************** + * Public Functions + ****************************************************************************/ + +double tgamma(double x) +{ + union + { + double f; + uint64_t i; + } u; + u.f = x; + + double absx; + double y; + double dy; + double z; + double r; + uint32_t ix = u.i >> 32 & 0x7fffffff; + int sign = u.i >> 63; + + /* special cases */ + + if (ix >= 0x7ff00000) + { + /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ + + return x + INFINITY; + } + + if (ix < (0x3ff - 54) << 20) + { + /* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */ + return 1 / x; + } + + /* integer arguments */ + /* raise inexact when non-integer */ + + if (x == floor(x)) + { + if (sign) + { + return 0 / 0.0; + } + + if (x <= sizeof g_fact / sizeof *g_fact) + { + return g_fact[(int)x - 1]; + } + } + + /* x >= 172: tgamma(x)=inf with overflow */ + /* x =< -184: tgamma(x)=+-0 with underflow */ + + if (ix >= 0x40670000) + { + /* |x| >= 184 */ + + if (sign) + { + FORCE_EVAL((float)(0x1p-126 / x)); + if (floor(x) * 0.5 == floor(x * 0.5)) + { + return 0; + } + return -0.0; + } + + x *= 0x1p1023; + return x; + } + + absx = sign ? -x : x; + + /* handle the error of x + g - 0.5 */ + + y = absx + g_gmhalf; + if (absx > g_gmhalf) + { + dy = y - absx; + dy -= g_gmhalf; + } + else + { + dy = y - g_gmhalf; + dy -= absx; + } + + z = absx - 0.5; + r = s(absx) * exp(-y); + if (x < 0) + { + /* reflection formula for negative x */ + /* sinpi(absx) is not 0, integers are already handled */ + + r = -pi / (sinpi(absx) * absx * r); + dy = -dy; + z = -z; + } + + r += dy * (g_gmhalf + 0.5) * r / y; + z = pow(y, 0.5 * z); + y = r * z * z; + + return y; +} + +double gamma(double x) +{ + return tgamma(x); +} +#endif diff --git a/libc/math/lib_lgamma.c b/libc/math/lib_lgamma.c new file mode 100644 index 0000000000..598e68936e --- /dev/null +++ b/libc/math/lib_lgamma.c @@ -0,0 +1,456 @@ +/**************************************************************************** + * libc/math/lib_gamma.c + * + * Ported to NuttX from FreeBSD by Alan Carvalho de Assis: + * + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in + * the documentation and/or other materials provided with the + * distribution. + * 3. Neither the name NuttX nor the names of its contributors may be + * used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS + * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT + * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS + * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE + * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, + * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS + * OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED + * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN + * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE + * POSSIBILITY OF SUCH DAMAGE. + * + ****************************************************************************/ + +/* lgamma_r(x, signgamp) + * + * Reentrant version of the logarithm of the Gamma function + * with user provide pointer for the sign of Gamma(x). + * + * Method: + * 1. Argument Reduction for 0 < x <= 8 + * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may + * reduce x to a number in [1.5,2.5] by + * lgamma(1+s) = log(s) + lgamma(s) + * for example, + * lgamma(7.3) = log(6.3) + lgamma(6.3) + * = log(6.3*5.3) + lgamma(5.3) + * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) + * 2. Polynomial approximation of lgamma around its + * minimun ymin=1.461632144968362245 to maintain monotonicity. + * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use + * Let z = x-ymin; + * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) + * where + * poly(z) is a 14 degree polynomial. + * 2. Rational approximation in the primary interval [2,3] + * We use the following approximation: + * s = x-2.0; + * lgamma(x) = 0.5*s + s*P(s)/Q(s) + * with accuracy + * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 + * Our algorithms are based on the following observation + * + * zeta(2)-1 2 zeta(3)-1 3 + * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... + * 2 3 + * + * where Euler = 0.5771... is the Euler constant, which is very + * close to 0.5. + * + * 3. For x>=8, we have + * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... + * (better formula: + * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) + * Let z = 1/x, then we approximation + * f(z) = lgamma(x) - (x-0.5)(log(x)-1) + * by + * 3 5 11 + * w = w0 + w1*z + w2*z + w3*z + ... + w6*z + * where + * |w - f(z)| < 2**-58.74 + * + * 4. For negative x, since (G is gamma function) + * -x*G(-x)*G(x) = pi/sin(pi*x), + * we have + * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) + * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 + * Hence, for x<0, signgam = sign(sin(pi*x)) and + * lgamma(x) = log(|Gamma(x)|) + * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); + * Note: one should avoid compute pi*(-x) directly in the + * computation of sin(pi*(-x)). + * + * 5. Special Cases + * lgamma(2+s) ~ s*(1-Euler) for tiny s + * lgamma(1) = lgamma(2) = 0 + * lgamma(x) ~ -log(|x|) for tiny x + * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero + * lgamma(inf) = inf + * lgamma(-inf) = inf (bug for bug compatible with C99!?) + */ + +/**************************************************************************** + * Included Files + ****************************************************************************/ + +#include +#include + +#include +#include + +#ifdef CONFIG_HAVE_DOUBLE + +/**************************************************************************** + * Private Data + ****************************************************************************/ + +static int g_signgam = 0; + +static const double g_pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */ +static const double g_a0 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */ +static const double g_a1 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */ +static const double g_a2 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */ +static const double g_a3 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */ +static const double g_a4 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */ +static const double g_a5 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */ +static const double g_a6 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */ +static const double g_a7 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */ +static const double g_a8 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */ +static const double g_a9 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */ +static const double g_a10 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */ +static const double g_a11 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */ +static const double g_tc = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */ +static const double g_tf = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */ + +/* tt = -(tail of tf) */ + +static const double g_tt = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */ +static const double g_t0 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */ +static const double g_t1 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */ +static const double g_t2 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */ +static const double g_t3 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */ +static const double g_t4 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */ +static const double g_t5 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */ +static const double g_t6 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */ +static const double g_t7 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */ +static const double g_t8 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */ +static const double g_t9 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */ +static const double g_t10 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */ +static const double g_t11 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */ +static const double g_t12 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */ +static const double g_t13 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */ +static const double g_t14 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */ +static const double g_u0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ +static const double g_u1 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */ +static const double g_u2 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */ +static const double g_u3 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */ +static const double g_u4 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */ +static const double g_u5 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */ +static const double g_v1 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */ +static const double g_v2 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */ +static const double g_v3 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */ +static const double g_v4 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */ +static const double g_v5 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */ +static const double g_s0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ +static const double g_s1 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */ +static const double g_s2 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */ +static const double g_s3 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */ +static const double g_s4 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */ +static const double g_s5 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */ +static const double g_s6 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */ +static const double g_r1 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */ +static const double g_r2 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */ +static const double g_r3 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */ +static const double g_r4 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */ +static const double g_r5 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */ +static const double g_r6 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */ +static const double g_w0 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */ +static const double g_w1 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */ +static const double g_w2 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */ +static const double g_w3 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */ +static const double g_w4 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */ +static const double g_w5 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */ +static const double g_w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ + +/**************************************************************************** + * Private Functions + ****************************************************************************/ + +/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */ + +static double sin_pi(double x) +{ + int n; + + /* spurious inexact if odd int */ + + x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */ + + n = (int)(x * 4.0); + n = (n + 1) / 2; + x -= n * 0.5f; + x *= g_pi; + + switch (n) + { + default: /* case 4: */ + case 0: + return __sin(x, 0.0, 0); + + case 1: + return __cos(x, 0.0); + + case 2: + return __sin(-x, 0.0, 0); + + case 3: + return -__cos(x, 0.0); + } +} + +/**************************************************************************** + * Public Functions + ****************************************************************************/ + +double lgamma_r(double x, int *signgamp) +{ + union + { + double f; + uint64_t i; + } u; + u.f = x; + + double t; + double y; + double z; + double nadj; + double p; + double p1; + double p2; + double p3; + double q; + double r; + double w; + uint32_t ix; + int sign; + int i; + + /* purge off +-inf, NaN, +-0, tiny and negative arguments */ + + *signgamp = 1; + sign = u.i >> 63; + + ix = u.i >> 32 & 0x7fffffff; + if (ix >= 0x7ff00000) + { + return x * x; + } + + /* |x|<2**-70, return -log(|x|) */ + + if (ix < (0x3ff - 70) << 20) + { + if (sign) + { + x = -x; + *signgamp = -1; + } + return -log(x); + } + + if (sign) + { + x = -x; + t = sin_pi(x); + + if (t == 0.0) + { + /* -integer */ + return 1.0 / (x - x); + } + + if (t > 0.0) + { + *signgamp = -1; + } + else + { + t = -t; + } + + nadj = log(g_pi / (t * x)); + } + + /* purge off 1 and 2 */ + + if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t) u.i == 0) + { + r = 0; + } + else /* for x < 2.0 */ + { + if (ix < 0x40000000) + { + if (ix <= 0x3feccccc) + { + /* lgamma(x) = lgamma(x+1)-log(x) */ + + r = -log(x); + + if (ix >= 0x3FE76944) + { + y = 1.0 - x; + i = 0; + } + else + { + if (ix >= 0x3FCDA661) + { + y = x - (g_tc - 1.0); + i = 1; + } + else + { + y = x; + i = 2; + } + } + } + else + { + r = 0.0; + + if (ix >= 0x3FFBB4C3) + { + /* [1.7316,2] */ + y = 2.0 - x; + i = 0; + } + else + { + if (ix >= 0x3FF3B4C4) + { + /* [1.23,1.73] */ + y = x - g_tc; + i = 1; + } + else + { + y = x - 1.0; + i = 2; + } + } + } + + switch (i) + { + case 0: + z = y*y; + p1 = g_a0+z*(g_a2+z*(g_a4+z*(g_a6+z*(g_a8+z*g_a10)))); + p2 = z*(g_a1+z*(g_a3+z*(g_a5+z*(g_a7+z*(g_a9+z*g_a11))))); + p = y*p1+p2; + r += (p-0.5*y); + break; + + case 1: + z = y*y; + w = z*y; + p1 = g_t0+w*(g_t3+w*(g_t6+w*(g_t9+w*g_t12))); /* parallel comp */ + p2 = g_t1+w*(g_t4+w*(g_t7+w*(g_t10+w*g_t13))); + p3 = g_t2+w*(g_t5+w*(g_t8+w*(g_t11+w*g_t14))); + p = z*p1-(g_tt-w*(p2+y*p3)); + r += g_tf + p; + break; + + case 2: + p1 = y*(g_u0+y*(g_u1+y*(g_u2+y*(g_u3+y*(g_u4+y*g_u5))))); + p2 = 1.0+y*(g_v1+y*(g_v2+y*(g_v3+y*(g_v4+y*g_v5)))); + r += -0.5*y + p1/p2; + } + } + else + { + if (ix < 0x40200000) + { + /* x < 8.0 */ + + i = (int)x; + y = x - (double)i; + p = y*(g_s0+y*(g_s1+y*(g_s2+y*(g_s3+y*(g_s4+y*(g_s5+y*g_s6)))))); + q = 1.0+y*(g_r1+y*(g_r2+y*(g_r3+y*(g_r4+y*(g_r5+y*g_r6))))); + r = 0.5*y+p/q; + z = 1.0; + + /* lgamma(1+s) = log(s) + lgamma(s) */ + + switch (i) + { + case 7: + z *= y + 6.0; /* FALLTHRU */ + case 6: + z *= y + 5.0; /* FALLTHRU */ + case 5: + z *= y + 4.0; /* FALLTHRU */ + case 4: + z *= y + 3.0; /* FALLTHRU */ + case 3: + z *= y + 2.0; /* FALLTHRU */ + r += log(z); + break; + } + } + else + { + if (ix < 0x43900000) + { + /* 8.0 <= x < 2**58 */ + + t = log(x); + z = 1.0 / x; + y = z * z; + w = g_w0+z*(g_w1+y*(g_w2+y*(g_w3+y*(g_w4+y*(g_w5+y*g_w6))))); + r = (x-0.5)*(t-1.0)+w; + } + else + { + /* 2**58 <= x <= inf */ + + r = x * (log(x) - 1.0); + } + } + } + } + + if (sign) + { + r = nadj - r; + } + + return r; +} + +double lgamma(double x) +{ + return lgamma_r(x, &g_signgam); +} +#endif