1 files changed, 287 insertions, 0 deletions
diff --git a/src/math/tgammal.c b/src/math/tgammal.c
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+/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_tgammal.c */
+/*
+ * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
+ *
+ * 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();
+ * extern int signgam;
+ *
+ * y = tgammal( x );
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns gamma function of the argument.  The result is
+ * correctly signed, and the sign (+1 or -1) is also
+ * returned in a global (extern) variable named signgam.
+ * This variable is also filled in by the logarithmic gamma
+ * function lgamma().
+ *
+ * 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 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 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 long double P[] = {
+-3.01525602666895735709e0L,
+-3.25157411956062339893e1L,
+-2.92929976820724030353e2L,
+-1.70730828800510297666e3L,
+-7.96667499622741999770e3L,
+-2.59780216007146401957e4L,
+-5.99650230220855581642e4L,
+-7.15743521530849602425e4L
+};
+static 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 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 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 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 = 1.0L/x;
+	/* For large x, use rational coefficients from the analytical expansion.  */
+	if (x > 1024.0L)
+		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
+		 + 1.0L;
+	else
+		w = 1.0L + 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;
+	int i;
+
+	signgam = 1;
+	if (isnan(x))
+		return NAN;
+	if (x == INFINITY)
+		return INFINITY;
+	if (x == -INFINITY)
+		return x - x;
+	q = fabsl(x);
+	if (q > 13.0L) {
+		if (q > MAXGAML)
+			goto goverf;
+		if (x < 0.0L) {
+			p = floorl(q);
+			if (p == q)
+				return (x - x) / (x - x);
+			i = p;
+			if ((i & 1) == 0)
+				signgam = -1;
+			z = q - p;
+			if (z > 0.5L) {
+				p += 1.0L;
+				z = q - p;
+			}
+			z = q * sinl(PIL * z);
+			z = fabsl(z) * stirf(q);
+			if (z <= PIL/LDBL_MAX) {
+goverf:
+				return signgam * INFINITY;
+			}
+			z = PIL/z;
+		} else {
+			z = stirf(x);
+		}
+		return signgam * z;
+	}
+
+	z = 1.0L;
+	while (x >= 3.0L) {
+		x -= 1.0L;
+		z *= x;
+	}
+	while (x < -0.03125L) {
+		z /= x;
+		x += 1.0L;
+	}
+	if (x <= 0.03125L)
+		goto small;
+	while (x < 2.0L) {
+		z /= x;
+		x += 1.0L;
+	}
+	if (x == 2.0L)
+		return z;
+
+	x -= 2.0L;
+	p = __polevll(x, P, 7);
+	q = __polevll(x, Q, 8);
+	z = z * p / q;
+	if(z < 0)
+		signgam = -1;
+	return z;
+
+small:
+	if (x == 0.0L)
+		return (x - x) / (x - x);
+	if (x < 0.0L) {
+		x = -x;
+		q = z / (x * __polevll(x, SN, 8));
+		signgam = -1;
+	} else
+		q = z / (x * __polevll(x, S, 8));
+	return q;
+}
+#endif