first commit of the new libm!

thanks to the hard work of Szabolcs Nagy (nsz), identifying the best
(from correctness and license standpoint) implementations from freebsd
and openbsd and cleaning them up! musl should now fully support c99
float and long double math functions, and has near-complete complex
math support. tgmath should also work (fully on gcc-compatible
compilers, and mostly on any c99 compiler).

based largely on commit 0376d44a890fea261506f1fc63833e7a686dca19 from
nsz's libm git repo, with some additions (dummy versions of a few
missing long double complex functions, etc.) by me.

various cleanups still need to be made, including re-adding (if
they're correct) some asm functions that were dropped.

1 files changed, 393 insertions, 0 deletions

diff --git a/src/math/lgammal.c b/src/math/lgammal.c
new file mode 100644
index 00000000..603477c9
--- /dev/null
+++ b/src/math/lgammal.c
@@ -0,0 +1,393 @@
+/* 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
+ *
+ */
+
+#include "libm.h"
+
+long double lgammal(long double x)
+{
+	return lgammal_r(x, &signgam);
+}
+
+#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
+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
+half = 0.5L,
+one = 1.0L,
+pi = 3.14159265358979323846264L,
+two63 = 9.223372036854775808e18L,
+
+/* 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;
+
+static const long double zero = 0.0L;
+
+static long double sin_pi(long double x)
+{
+	long double y, z;
+	int n, ix;
+	uint32_t se, i0, i1;
+
+	GET_LDOUBLE_WORDS(se, i0, i1, x);
+	ix = se & 0x7fff;
+	ix = (ix << 16) | (i0 >> 16);
+	if (ix < 0x3ffd8000)  /* 0.25 */
+		return sinl(pi * x);
+	y = -x;  /* x is assume negative */
+
+	/*
+	 * argument reduction, make sure inexact flag not raised if input
+	 * is an integer
+	 */
+	z = floorl(y);
+	if (z != y) {  /* inexact anyway */
+		y *= 0.5;
+		y = 2.0*(y - floorl(y));/* y = |x| mod 2.0 */
+		n = (int) (y*4.0);
+	} else {
+		if (ix >= 0x403f8000) {  /* 2^64 */
+			y = zero;  /* y must be even */
+			n = 0;
+		} else {
+			if (ix < 0x403e8000)  /* 2^63 */
+				z = y + two63;  /* exact */
+			GET_LDOUBLE_WORDS(se, i0, i1, z);
+			n = i1 & 1;
+			y = n;
+			n <<= 2;
+		}
+	}
+
+	switch (n) {
+	case 0:
+		y = sinl(pi * y);
+		break;
+	case 1:
+	case 2:
+		y = cosl(pi * (half - y));
+		break;
+	case 3:
+	case 4:
+		y = sinl(pi * (one - y));
+		break;
+	case 5:
+	case 6:
+		y = -cosl(pi * (y - 1.5));
+		break;
+	default:
+		y = sinl(pi * (y - 2.0));
+		break;
+	}
+	return -y;
+}
+
+long double lgammal_r(long double x, int *sg) {
+	long double t, y, z, nadj, p, p1, p2, q, r, w;
+	int i, ix;
+	uint32_t se, i0, i1;
+
+	*sg = 1;
+	GET_LDOUBLE_WORDS(se, i0, i1, x);
+	ix = se & 0x7fff;
+
+	if ((ix | i0 | i1) == 0) {
+		if (se & 0x8000)
+			*sg = -1;
+		return one / fabsl(x);
+	}
+
+	ix = (ix << 16) | (i0 >> 16);
+
+	/* purge off +-inf, NaN, +-0, and negative arguments */
+	if (ix >= 0x7fff0000)
+		return x * x;
+
+	if (ix < 0x3fc08000) {  /* |x|<2**-63, return -log(|x|) */
+		if (se & 0x8000) {
+			*sg = -1;
+			return -logl(-x);
+		}
+		return -logl(x);
+	}
+	if (se & 0x8000) {
+		t = sin_pi (x);
+		if (t == zero)
+			return one / fabsl(t); /* -integer */
+		nadj = logl(pi / fabsl(t * x));
+		if (t < zero)
+			*sg = -1;
+		x = -x;
+	}
+
+	/* purge off 1 and 2 */
+	if ((((ix - 0x3fff8000) | i0 | i1) == 0) ||
+	    (((ix - 0x40008000) | i0 | i1) == 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 - one;
+				i = 0;
+			} else if (ix >= 0x3ffced33) {  /* 2.31639862060546875e-1 */
+				y = x - (tc - one);
+				i = 1;
+			} else { /* x < 0.23 */
+				y = x;
+				i = 2;
+			}
+		} else {
+			r = zero;
+			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 - one;
+				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 += half * 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 += (-half * y + p1 / p2);
+		}
+	} else if (ix < 0x40028000) {  /* 8.0 */
+		/* x < 8.0 */
+		i = (int)x;
+		t = zero;
+		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 = half * y + p / q;
+		z = one;/* 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 = one / x;
+		y = z * z;
+		w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
+		r = (x - half) * (t - one) + w;
+	} else /* 2**66 <= x <= inf */
+		r = x * (logl (x) - one);
+	if (se & 0x8000)
+		r = nadj - r;
+	return r;
+}
+#endif