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.

src/math/__tan.c

@@ -0,0 +1,122 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
+/*
+ * ====================================================
+ * Copyright 2004 Sun Microsystems, Inc.  All Rights Reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/* __tan( x, y, k )
+ * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
+ * Input x is assumed to be bounded by ~pi/4 in magnitude.
+ * Input y is the tail of x.
+ * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
+ *
+ * Algorithm
+ *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
+ *      2. Callers must return tan(-0) = -0 without calling here since our
+ *         odd polynomial is not evaluated in a way that preserves -0.
+ *         Callers may do the optimization tan(x) ~ x for tiny x.
+ *      3. tan(x) is approximated by a odd polynomial of degree 27 on
+ *         [0,0.67434]
+ *                               3             27
+ *              tan(x) ~ x + T1*x + ... + T13*x
+ *         where
+ *
+ *              |tan(x)         2     4            26   |     -59.2
+ *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
+ *              |  x                                    |
+ *
+ *         Note: tan(x+y) = tan(x) + tan'(x)*y
+ *                        ~ tan(x) + (1+x*x)*y
+ *         Therefore, for better accuracy in computing tan(x+y), let
+ *                   3      2      2       2       2
+ *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
+ *         then
+ *                                  3    2
+ *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
+ *
+ *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
+ *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
+ *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
+ */
+
+#include "libm.h"
+
+static const double T[] = {
+             3.33333333333334091986e-01, /* 3FD55555, 55555563 */
+             1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
+             5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
+             2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
+             8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
+             3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
+             1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
+             5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
+             2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
+             7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
+             7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
+            -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
+             2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
+/* one */    1.00000000000000000000e+00, /* 3FF00000, 00000000 */
+/* pio4 */   7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
+/* pio4lo */ 3.06161699786838301793e-17  /* 3C81A626, 33145C07 */
+};
+#define one     T[13]
+#define pio4    T[14]
+#define pio4lo  T[15]
+
+double __tan(double x, double y, int iy)
+{
+	double z, r, v, w, s, sign;
+	int32_t ix, hx;
+
+	GET_HIGH_WORD(hx,x);
+	ix = hx & 0x7fffffff;    /* high word of |x| */
+	if (ix >= 0x3FE59428) {  /* |x| >= 0.6744 */
+		if (hx < 0) {
+			x = -x;
+			y = -y;
+		}
+		z = pio4 - x;
+		w = pio4lo - y;
+		x = z + w;
+		y = 0.0;
+	}
+	z = x * x;
+	w = z * z;
+	/*
+	 * Break x^5*(T[1]+x^2*T[2]+...) into
+	 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
+	 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
+	 */
+	r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
+	v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
+	s = z * x;
+	r = y + z * (s * (r + v) + y);
+	r += T[0] * s;
+	w = x + r;
+	if (ix >= 0x3FE59428) {
+		v = iy;
+		sign = 1 - ((hx >> 30) & 2);
+		return sign * (v - 2.0 * (x - (w * w / (w + v) - r)));
+	}
+	if (iy == 1)
+		return w;
+	else {
+		/*
+		 * if allow error up to 2 ulp, simply return
+		 * -1.0 / (x+r) here
+		 */
+		/* compute -1.0 / (x+r) accurately */
+		double a, t;
+		z = w;
+		SET_LOW_WORD(z,0);
+		v = r - (z - x);        /* z+v = r+x */
+		t = a = -1.0 / w;       /* a = -1.0/w */
+		SET_LOW_WORD(t,0);
+		s = 1.0 + t * z;
+		return t + a * (s + t * v);
+	}
+}