/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_tgammal.c */ /* * Copyright (c) 2008 Stephen L. Moshier * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ /* * Gamma function * * * SYNOPSIS: * * long double x, y, tgammal(); * * y = tgammal( x ); * * * DESCRIPTION: * * Returns gamma function of the argument. The result is * correctly signed. * * Arguments |x| <= 13 are reduced by recurrence and the function * approximated by a rational function of degree 7/8 in the * interval (2,3). Large arguments are handled by Stirling's * formula. Large negative arguments are made positive using * a reflection formula. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -40,+40 10000 3.6e-19 7.9e-20 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 * * Accuracy for large arguments is dominated by error in powl(). * */ #include "libm.h" #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 long double tgammal(long double x) { return tgamma(x); } #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 /* tgamma(x+2) = tgamma(x+2) P(x)/Q(x) 0 <= x <= 1 Relative error n=7, d=8 Peak error = 1.83e-20 Relative error spread = 8.4e-23 */ static const long double P[8] = { 4.212760487471622013093E-5L, 4.542931960608009155600E-4L, 4.092666828394035500949E-3L, 2.385363243461108252554E-2L, 1.113062816019361559013E-1L, 3.629515436640239168939E-1L, 8.378004301573126728826E-1L, 1.000000000000000000009E0L, }; static const long double Q[9] = { -1.397148517476170440917E-5L, 2.346584059160635244282E-4L, -1.237799246653152231188E-3L, -7.955933682494738320586E-4L, 2.773706565840072979165E-2L, -4.633887671244534213831E-2L, -2.243510905670329164562E-1L, 4.150160950588455434583E-1L, 9.999999999999999999908E-1L, }; /* static const long double P[] = { -3.01525602666895735709e0L, -3.25157411956062339893e1L, -2.92929976820724030353e2L, -1.70730828800510297666e3L, -7.96667499622741999770e3L, -2.59780216007146401957e4L, -5.99650230220855581642e4L, -7.15743521530849602425e4L }; static const long double Q[] = { 1.00000000000000000000e0L, -1.67955233807178858919e1L, 8.85946791747759881659e1L, 5.69440799097468430177e1L, -1.98526250512761318471e3L, 3.31667508019495079814e3L, 1.60577839621734713377e4L, -2.97045081369399940529e4L, -7.15743521530849602412e4L }; */ #define MAXGAML 1755.455L /*static const long double LOGPI = 1.14472988584940017414L;*/ /* Stirling's formula for the gamma function tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) z(x) = x 13 <= x <= 1024 Relative error n=8, d=0 Peak error = 9.44e-21 Relative error spread = 8.8e-4 */ static const long double STIR[9] = { 7.147391378143610789273E-4L, -2.363848809501759061727E-5L, -5.950237554056330156018E-4L, 6.989332260623193171870E-5L, 7.840334842744753003862E-4L, -2.294719747873185405699E-4L, -2.681327161876304418288E-3L, 3.472222222230075327854E-3L, 8.333333333333331800504E-2L, }; #define MAXSTIR 1024.0L static const long double SQTPI = 2.50662827463100050242E0L; /* 1/tgamma(x) = z P(z) * z(x) = 1/x * 0 < x < 0.03125 * Peak relative error 4.2e-23 */ static const long double S[9] = { -1.193945051381510095614E-3L, 7.220599478036909672331E-3L, -9.622023360406271645744E-3L, -4.219773360705915470089E-2L, 1.665386113720805206758E-1L, -4.200263503403344054473E-2L, -6.558780715202540684668E-1L, 5.772156649015328608253E-1L, 1.000000000000000000000E0L, }; /* 1/tgamma(-x) = z P(z) * z(x) = 1/x * 0 < x < 0.03125 * Peak relative error 5.16e-23 * Relative error spread = 2.5e-24 */ static const long double SN[9] = { 1.133374167243894382010E-3L, 7.220837261893170325704E-3L, 9.621911155035976733706E-3L, -4.219773343731191721664E-2L, -1.665386113944413519335E-1L, -4.200263503402112910504E-2L, 6.558780715202536547116E-1L, 5.772156649015328608727E-1L, -1.000000000000000000000E0L, }; static const long double PIL = 3.1415926535897932384626L; /* Gamma function computed by Stirling's formula. */ static long double stirf(long double x) { long double y, w, v; w = 1.0/x; /* For large x, use rational coefficients from the analytical expansion. */ if (x > 1024.0) w = (((((6.97281375836585777429E-5L * w + 7.84039221720066627474E-4L) * w - 2.29472093621399176955E-4L) * w - 2.68132716049382716049E-3L) * w + 3.47222222222222222222E-3L) * w + 8.33333333333333333333E-2L) * w + 1.0; else w = 1.0 + w * __polevll(w, STIR, 8); y = expl(x); if (x > MAXSTIR) { /* Avoid overflow in pow() */ v = powl(x, 0.5L * x - 0.25L); y = v * (v / y); } else { y = powl(x, x - 0.5L) / y; } y = SQTPI * y * w; return y; } long double tgammal(long double x) { long double p, q, z; if (!isfinite(x)) return x + INFINITY; q = fabsl(x); if (q > 13.0) { if (x < 0.0) { p = floorl(q); z = q - p; if (z == 0) return 0 / z; if (q > MAXGAML) { z = 0; } else { if (z > 0.5) { p += 1.0; z = q - p; } z = q * sinl(PIL * z); z = fabsl(z) * stirf(q); z = PIL/z; } if (0.5 * p == floorl(q * 0.5)) z = -z; } else if (x > MAXGAML) { z = x * 0x1p16383L; } else { z = stirf(x); } return z; } z = 1.0; while (x >= 3.0) { x -= 1.0; z *= x; } while (x < -0.03125L) { z /= x; x += 1.0; } if (x <= 0.03125L) goto small; while (x < 2.0) { z /= x; x += 1.0; } if (x == 2.0) return z; x -= 2.0; p = __polevll(x, P, 7); q = __polevll(x, Q, 8); z = z * p / q; return z; small: /* z==1 if x was originally +-0 */ if (x == 0 && z != 1) return x / x; if (x < 0.0) { x = -x; q = z / (x * __polevll(x, SN, 8)); } else q = z / (x * __polevll(x, S, 8)); return q; } #endif