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, 157 insertions, 0 deletions

diff --git a/src/math/exp.c b/src/math/exp.c
new file mode 100644
index 00000000..c1c9a63c
--- /dev/null
+++ b/src/math/exp.c
@@ -0,0 +1,157 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
+/*
+ * ====================================================
+ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/* exp(x)
+ * Returns the exponential of x.
+ *
+ * Method
+ *   1. Argument reduction:
+ *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+ *      Given x, find r and integer k such that
+ *
+ *               x = k*ln2 + r,  |r| <= 0.5*ln2.
+ *
+ *      Here r will be represented as r = hi-lo for better
+ *      accuracy.
+ *
+ *   2. Approximation of exp(r) by a special rational function on
+ *      the interval [0,0.34658]:
+ *      Write
+ *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
+ *      We use a special Remes algorithm on [0,0.34658] to generate
+ *      a polynomial of degree 5 to approximate R. The maximum error
+ *      of this polynomial approximation is bounded by 2**-59. In
+ *      other words,
+ *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+ *      (where z=r*r, and the values of P1 to P5 are listed below)
+ *      and
+ *          |                  5          |     -59
+ *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
+ *          |                             |
+ *      The computation of exp(r) thus becomes
+ *                             2*r
+ *              exp(r) = 1 + -------
+ *                            R - r
+ *                                 r*R1(r)
+ *                     = 1 + r + ----------- (for better accuracy)
+ *                                2 - R1(r)
+ *      where
+ *                               2       4             10
+ *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
+ *
+ *   3. Scale back to obtain exp(x):
+ *      From step 1, we have
+ *         exp(x) = 2^k * exp(r)
+ *
+ * Special cases:
+ *      exp(INF) is INF, exp(NaN) is NaN;
+ *      exp(-INF) is 0, and
+ *      for finite argument, only exp(0)=1 is exact.
+ *
+ * Accuracy:
+ *      according to an error analysis, the error is always less than
+ *      1 ulp (unit in the last place).
+ *
+ * Misc. info.
+ *      For IEEE double
+ *          if x >  7.09782712893383973096e+02 then exp(x) overflow
+ *          if x < -7.45133219101941108420e+02 then exp(x) underflow
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+
+#include "libm.h"
+
+static const double
+one     = 1.0,
+halF[2] = {0.5,-0.5,},
+huge    = 1.0e+300,
+o_threshold =  7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
+u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
+ln2HI[2]   = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
+              -6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */
+ln2LO[2]   = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
+              -1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */
+invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
+P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
+P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
+P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
+P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
+P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
+
+static volatile double
+twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */
+
+double exp(double x)
+{
+	double y,hi=0.0,lo=0.0,c,t,twopk;
+	int32_t k=0,xsb;
+	uint32_t hx;
+
+	GET_HIGH_WORD(hx, x);
+	xsb = (hx>>31)&1;  /* sign bit of x */
+	hx &= 0x7fffffff;  /* high word of |x| */
+
+	/* filter out non-finite argument */
+	if (hx >= 0x40862E42) {  /* if |x| >= 709.78... */
+		if (hx >= 0x7ff00000) {
+			uint32_t lx;
+	
+			GET_LOW_WORD(lx,x);
+			if (((hx&0xfffff)|lx) != 0)  /* NaN */
+				 return x+x;
+			return xsb==0 ? x : 0.0;  /* exp(+-inf)={inf,0} */
+		}
+		if (x > o_threshold)
+			return huge*huge; /* overflow */
+		if (x < u_threshold)
+			return twom1000*twom1000; /* underflow */
+	}
+
+	/* argument reduction */
+	if (hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
+		if (hx < 0x3FF0A2B2) {  /* and |x| < 1.5 ln2 */
+			hi = x-ln2HI[xsb];
+			lo = ln2LO[xsb];
+			k = 1 - xsb - xsb;
+		} else {
+			k  = (int)(invln2*x+halF[xsb]);
+			t  = k;
+			hi = x - t*ln2HI[0];  /* t*ln2HI is exact here */
+			lo = t*ln2LO[0];
+		}
+		STRICT_ASSIGN(double, x, hi - lo);
+	} else if(hx < 0x3e300000)  {  /* |x| < 2**-28 */
+		/* raise inexact */
+		if (huge+x > one)
+			return one+x;
+	} else
+		k = 0;
+
+	/* x is now in primary range */
+	t  = x*x;
+	if (k >= -1021)
+		INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);
+	else
+		INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);
+	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
+	if (k == 0)
+		return one - ((x*c)/(c-2.0) - x);
+	y = one-((lo-(x*c)/(2.0-c))-hi);
+	if (k < -1021)
+		return y*twopk*twom1000;
+	if (k == 1024)
+		return y*2.0*0x1p1023;
+	return y*twopk;
+}