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, 315 insertions, 0 deletions

diff --git a/src/math/lgamma_r.c b/src/math/lgamma_r.c
new file mode 100644
index 00000000..6baa0e52
--- /dev/null
+++ b/src/math/lgamma_r.c
@@ -0,0 +1,315 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ *
+ */
+/* 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!?)
+ *
+ */
+
+#include "libm.h"
+
+static const double
+two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
+half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
+one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
+pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
+a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
+a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
+a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
+a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
+a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
+a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
+a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
+a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
+a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
+a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
+a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
+a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
+tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
+tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
+/* tt = -(tail of tf) */
+tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
+t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
+t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
+t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
+t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
+t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
+t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
+t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
+t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
+t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
+t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
+t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
+t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
+t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
+t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
+t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
+u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
+u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
+u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
+u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
+u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
+u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
+v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
+v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
+v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
+v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
+v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
+s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
+s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
+s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
+s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
+s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
+s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
+s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
+r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
+r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
+r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
+r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
+r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
+r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
+w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
+w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
+w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
+w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
+w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
+w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
+w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
+
+static const double zero = 0.00000000000000000000e+00;
+
+static double sin_pi(double x)
+{
+	double y,z;
+	int n,ix;
+
+	GET_HIGH_WORD(ix, x);
+	ix &= 0x7fffffff;
+
+	if (ix < 0x3fd00000)
+		return __sin(pi*x, zero, 0);
+
+	y = -x;  /* negative x is assumed */
+
+	/*
+	 * argument reduction, make sure inexact flag not raised if input
+	 * is an integer
+	 */
+	z = floor(y);
+	if (z != y) {    /* inexact anyway */
+		y *= 0.5;
+		y  = 2.0*(y - floor(y));   /* y = |x| mod 2.0 */
+		n  = (int)(y*4.0);
+	} else {
+		if (ix >= 0x43400000) {
+			y = zero;    /* y must be even */
+			n = 0;
+		} else {
+			if (ix < 0x43300000)
+				z = y + two52;  /* exact */
+			GET_LOW_WORD(n, z);
+			n &= 1;
+			y = n;
+			n <<= 2;
+		}
+	}
+	switch (n) {
+	case 0:  y =  __sin(pi*y, zero, 0); break;
+	case 1:
+	case 2:  y =  __cos(pi*(0.5-y), zero); break;
+	case 3:
+	case 4:  y =  __sin(pi*(one-y), zero, 0); break;
+	case 5:
+	case 6:  y = -__cos(pi*(y-1.5), zero); break;
+	default: y =  __sin(pi*(y-2.0), zero, 0); break;
+	}
+	return -y;
+}
+
+
+double lgamma_r(double x, int *signgamp)
+{
+	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
+	int32_t hx;
+	int i,lx,ix;
+
+	EXTRACT_WORDS(hx, lx, x);
+
+	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
+	*signgamp = 1;
+	ix = hx & 0x7fffffff;
+	if (ix >= 0x7ff00000)
+		return x*x;
+	if ((ix|lx) == 0)
+		return one/zero;
+	if (ix < 0x3b900000) {  /* |x|<2**-70, return -log(|x|) */
+		if(hx < 0) {
+			*signgamp = -1;
+			return -log(-x);
+		}
+		return -log(x);
+	}
+	if (hx < 0) {
+		if (ix >= 0x43300000)  /* |x|>=2**52, must be -integer */
+			return one/zero;
+		t = sin_pi(x);
+		if (t == zero) /* -integer */
+			return one/zero;
+		nadj = log(pi/fabs(t*x));
+		if (t < zero)
+			*signgamp = -1;
+		x = -x;
+	}
+
+	/* purge off 1 and 2 */
+	if (((ix - 0x3ff00000)|lx) == 0 || ((ix - 0x40000000)|lx) == 0)
+		r = 0;
+	/* for x < 2.0 */
+	else if (ix < 0x40000000) {
+		if (ix <= 0x3feccccc) {   /* lgamma(x) = lgamma(x+1)-log(x) */
+			r = -log(x);
+			if (ix >= 0x3FE76944) {
+				y = one - x;
+				i = 0;
+			} else if (ix >= 0x3FCDA661) {
+				y = x - (tc-one);
+				i = 1;
+			} else {
+				y = x;
+				i = 2;
+			}
+		} else {
+			r = zero;
+			if (ix >= 0x3FFBB4C3) {  /* [1.7316,2] */
+				y = 2.0 - x;
+				i = 0;
+			} else if(ix >= 0x3FF3B4C4) {  /* [1.23,1.73] */
+				y = x - tc;
+				i = 1;
+			} else {
+				y = x - one;
+				i = 2;
+			}
+		}
+		switch (i) {
+		case 0:
+			z = y*y;
+			p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
+			p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
+			p = y*p1+p2;
+			r += (p-0.5*y);
+			break;
+		case 1:
+			z = y*y;
+			w = z*y;
+			p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
+			p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
+			p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
+			p = z*p1-(tt-w*(p2+y*p3));
+			r += tf + p;
+			break;
+		case 2:
+			p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
+			p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
+			r += -0.5*y + p1/p2;
+		}
+	} else if (ix < 0x40200000) {  /* 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 = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
+		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 += log(z);
+			break;
+		}
+	} else if (ix < 0x43900000) {  /* 8.0 <= x < 2**58 */
+		t = log(x);
+		z = one/x;
+		y = z*z;
+		w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
+		r = (x-half)*(t-one)+w;
+	} else                         /* 2**58 <= x <= inf */
+		r =  x*(log(x)-one);
+	if (hx < 0)
+		r = nadj - r;
+	return r;
+}