Add a "fork" of musl as //fusl.

fusl/src/math/lgammal.c

Index: fusl/src/math/lgammal.c
diff --git a/fusl/src/math/lgammal.c b/fusl/src/math/lgammal.c
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+/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.c */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/*
+ * 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.
+ */
+/* lgammal(x)
+ * 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)
+ *   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)
+ *      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
+ *
+ *   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(inf) = inf
+ *              lgamma(-integer) = +-inf
+ *
+ */
+
+#define _GNU_SOURCE
+#include "libm.h"
+#include "libc.h"
+
+#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
+double __lgamma_r(double x, int *sg);
+
+long double __lgammal_r(long double x, int *sg)
+{
+	return __lgamma_r(x, sg);
+}
+#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
+static const long double
+pi = 3.14159265358979323846264L,
+
+/* lgam(1+x) = 0.5 x + x a(x)/b(x)
+    -0.268402099609375 <= x <= 0
+    peak relative error 6.6e-22 */
+a0 = -6.343246574721079391729402781192128239938E2L,
+a1 =  1.856560238672465796768677717168371401378E3L,
+a2 =  2.404733102163746263689288466865843408429E3L,
+a3 =  8.804188795790383497379532868917517596322E2L,
+a4 =  1.135361354097447729740103745999661157426E2L,
+a5 =  3.766956539107615557608581581190400021285E0L,
+
+b0 =  8.214973713960928795704317259806842490498E3L,
+b1 =  1.026343508841367384879065363925870888012E4L,
+b2 =  4.553337477045763320522762343132210919277E3L,
+b3 =  8.506975785032585797446253359230031874803E2L,
+b4 =  6.042447899703295436820744186992189445813E1L,
+/* b5 =  1.000000000000000000000000000000000000000E0 */
+
+
+tc =  1.4616321449683623412626595423257213284682E0L,
+tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */
+/* tt = (tail of tf), i.e. tf + tt has extended precision. */
+tt = 3.3649914684731379602768989080467587736363E-18L,
+/* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
+-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
+
+/* lgam (x + tc) = tf + tt + x g(x)/h(x)
+    -0.230003726999612341262659542325721328468 <= x
+       <= 0.2699962730003876587373404576742786715318
+     peak relative error 2.1e-21 */
+g0 = 3.645529916721223331888305293534095553827E-18L,
+g1 = 5.126654642791082497002594216163574795690E3L,
+g2 = 8.828603575854624811911631336122070070327E3L,
+g3 = 5.464186426932117031234820886525701595203E3L,
+g4 = 1.455427403530884193180776558102868592293E3L,
+g5 = 1.541735456969245924860307497029155838446E2L,
+g6 = 4.335498275274822298341872707453445815118E0L,
+
+h0 = 1.059584930106085509696730443974495979641E4L,
+h1 = 2.147921653490043010629481226937850618860E4L,
+h2 = 1.643014770044524804175197151958100656728E4L,
+h3 = 5.869021995186925517228323497501767586078E3L,
+h4 = 9.764244777714344488787381271643502742293E2L,
+h5 = 6.442485441570592541741092969581997002349E1L,
+/* h6 = 1.000000000000000000000000000000000000000E0 */
+
+
+/* lgam (x+1) = -0.5 x + x u(x)/v(x)
+    -0.100006103515625 <= x <= 0.231639862060546875
+    peak relative error 1.3e-21 */
+u0 = -8.886217500092090678492242071879342025627E1L,
+u1 =  6.840109978129177639438792958320783599310E2L,
+u2 =  2.042626104514127267855588786511809932433E3L,
+u3 =  1.911723903442667422201651063009856064275E3L,
+u4 =  7.447065275665887457628865263491667767695E2L,
+u5 =  1.132256494121790736268471016493103952637E2L,
+u6 =  4.484398885516614191003094714505960972894E0L,
+
+v0 =  1.150830924194461522996462401210374632929E3L,
+v1 =  3.399692260848747447377972081399737098610E3L,
+v2 =  3.786631705644460255229513563657226008015E3L,
+v3 =  1.966450123004478374557778781564114347876E3L,
+v4 =  4.741359068914069299837355438370682773122E2L,
+v5 =  4.508989649747184050907206782117647852364E1L,
+/* v6 =  1.000000000000000000000000000000000000000E0 */
+
+
+/* lgam (x+2) = .5 x + x s(x)/r(x)
+     0 <= x <= 1
+     peak relative error 7.2e-22 */
+s0 =  1.454726263410661942989109455292824853344E6L,
+s1 = -3.901428390086348447890408306153378922752E6L,
+s2 = -6.573568698209374121847873064292963089438E6L,
+s3 = -3.319055881485044417245964508099095984643E6L,
+s4 = -7.094891568758439227560184618114707107977E5L,
+s5 = -6.263426646464505837422314539808112478303E4L,
+s6 = -1.684926520999477529949915657519454051529E3L,
+
+r0 = -1.883978160734303518163008696712983134698E7L,
+r1 = -2.815206082812062064902202753264922306830E7L,
+r2 = -1.600245495251915899081846093343626358398E7L,
+r3 = -4.310526301881305003489257052083370058799E6L,
+r4 = -5.563807682263923279438235987186184968542E5L,
+r5 = -3.027734654434169996032905158145259713083E4L,
+r6 = -4.501995652861105629217250715790764371267E2L,
+/* r6 =  1.000000000000000000000000000000000000000E0 */
+
+
+/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
+    x >= 8
+    Peak relative error 1.51e-21
+w0 = LS2PI - 0.5 */
+w0 =  4.189385332046727417803e-1L,
+w1 =  8.333333333333331447505E-2L,
+w2 = -2.777777777750349603440E-3L,
+w3 =  7.936507795855070755671E-4L,
+w4 = -5.952345851765688514613E-4L,
+w5 =  8.412723297322498080632E-4L,
+w6 = -1.880801938119376907179E-3L,
+w7 =  4.885026142432270781165E-3L;
+
+/* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
+static long double sin_pi(long double x)
+{
+	int n;
+
+	/* spurious inexact if odd int */
+	x *= 0.5;
+	x = 2.0*(x - floorl(x));  /* x mod 2.0 */
+
+	n = (int)(x*4.0);
+	n = (n+1)/2;
+	x -= n*0.5f;
+	x *= pi;
+
+	switch (n) {
+	default: /* case 4: */
+	case 0: return __sinl(x, 0.0, 0);
+	case 1: return __cosl(x, 0.0);
+	case 2: return __sinl(-x, 0.0, 0);
+	case 3: return -__cosl(x, 0.0);
+	}
+}
+
+long double __lgammal_r(long double x, int *sg) {
+	long double t, y, z, nadj, p, p1, p2, q, r, w;
+	union ldshape u = {x};
+	uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
+	int sign = u.i.se >> 15;
+	int i;
+
+	*sg = 1;
+
+	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
+	if (ix >= 0x7fff0000)
+		return x * x;
+	if (ix < 0x3fc08000) {  /* |x|<2**-63, return -log(|x|) */
+		if (sign) {
+			*sg = -1;
+			x = -x;
+		}
+		return -logl(x);
+	}
+	if (sign) {
+		x = -x;
+		t = sin_pi(x);
+		if (t == 0.0)
+			return 1.0 / (x-x); /* -integer */
+		if (t > 0.0)
+			*sg = -1;
+		else
+			t = -t;
+		nadj = logl(pi / (t * x));
+	}
+
+	/* purge off 1 and 2 (so the sign is ok with downward rounding) */
+	if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) {
+		r = 0;
+	} else if (ix < 0x40008000) {  /* x < 2.0 */
+		if (ix <= 0x3ffee666) {  /* 8.99993896484375e-1 */
+			/* lgamma(x) = lgamma(x+1) - log(x) */
+			r = -logl(x);
+			if (ix >= 0x3ffebb4a) {  /* 7.31597900390625e-1 */
+				y = x - 1.0;
+				i = 0;
+			} else if (ix >= 0x3ffced33) {  /* 2.31639862060546875e-1 */
+				y = x - (tc - 1.0);
+				i = 1;
+			} else { /* x < 0.23 */
+				y = x;
+				i = 2;
+			}
+		} else {
+			r = 0.0;
+			if (ix >= 0x3fffdda6) {  /* 1.73162841796875 */
+				/* [1.7316,2] */
+				y = x - 2.0;
+				i = 0;
+			} else if (ix >= 0x3fff9da6) {  /* 1.23162841796875 */
+				/* [1.23,1.73] */
+				y = x - tc;
+				i = 1;
+			} else {
+				/* [0.9, 1.23] */
+				y = x - 1.0;
+				i = 2;
+			}
+		}
+		switch (i) {
+		case 0:
+			p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
+			p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
+			r += 0.5 * y + y * p1/p2;
+			break;
+		case 1:
+			p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
+			p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
+			p = tt + y * p1/p2;
+			r += (tf + p);
+			break;
+		case 2:
+			p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
+			p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
+			r += (-0.5 * y + p1 / p2);
+		}
+	} else if (ix < 0x40028000) {  /* 8.0 */
+		/* x < 8.0 */
+		i = (int)x;
+		y = x - (double)i;
+		p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
+		q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
+		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 += logl(z);
+			break;
+		}
+	} else if (ix < 0x40418000) {  /* 2^66 */
+		/* 8.0 <= x < 2**66 */
+		t = logl(x);
+		z = 1.0 / x;
+		y = z * z;
+		w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
+		r = (x - 0.5) * (t - 1.0) + w;
+	} else /* 2**66 <= x <= inf */
+		r = x * (logl(x) - 1.0);
+	if (sign)
+		r = nadj - r;
+	return r;
+}
+#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
+// TODO: broken implementation to make things compile
+double __lgamma_r(double x, int *sg);
+
+long double __lgammal_r(long double x, int *sg)
+{
+	return __lgamma_r(x, sg);
+}
+#endif
+
+extern int __signgam;
+
+long double lgammal(long double x)
+{
+	return __lgammal_r(x, &__signgam);
+}
+
+weak_alias(__lgammal_r, lgammal_r);