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, 306 insertions, 0 deletions

diff --git a/src/math/erf.c b/src/math/erf.c
new file mode 100644
index 00000000..18ee01cf
--- /dev/null
+++ b/src/math/erf.c
@@ -0,0 +1,306 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/s_erf.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.
+ * ====================================================
+ */
+/* double erf(double x)
+ * double erfc(double x)
+ *                           x
+ *                    2      |\
+ *     erf(x)  =  ---------  | exp(-t*t)dt
+ *                 sqrt(pi) \|
+ *                           0
+ *
+ *     erfc(x) =  1-erf(x)
+ *  Note that
+ *              erf(-x) = -erf(x)
+ *              erfc(-x) = 2 - erfc(x)
+ *
+ * Method:
+ *      1. For |x| in [0, 0.84375]
+ *          erf(x)  = x + x*R(x^2)
+ *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
+ *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
+ *         where R = P/Q where P is an odd poly of degree 8 and
+ *         Q is an odd poly of degree 10.
+ *                                               -57.90
+ *                      | R - (erf(x)-x)/x | <= 2
+ *
+ *
+ *         Remark. The formula is derived by noting
+ *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
+ *         and that
+ *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
+ *         is close to one. The interval is chosen because the fix
+ *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
+ *         near 0.6174), and by some experiment, 0.84375 is chosen to
+ *         guarantee the error is less than one ulp for erf.
+ *
+ *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
+ *         c = 0.84506291151 rounded to single (24 bits)
+ *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
+ *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
+ *                        1+(c+P1(s)/Q1(s))    if x < 0
+ *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
+ *         Remark: here we use the taylor series expansion at x=1.
+ *              erf(1+s) = erf(1) + s*Poly(s)
+ *                       = 0.845.. + P1(s)/Q1(s)
+ *         That is, we use rational approximation to approximate
+ *                      erf(1+s) - (c = (single)0.84506291151)
+ *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
+ *         where
+ *              P1(s) = degree 6 poly in s
+ *              Q1(s) = degree 6 poly in s
+ *
+ *      3. For x in [1.25,1/0.35(~2.857143)],
+ *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
+ *              erf(x)  = 1 - erfc(x)
+ *         where
+ *              R1(z) = degree 7 poly in z, (z=1/x^2)
+ *              S1(z) = degree 8 poly in z
+ *
+ *      4. For x in [1/0.35,28]
+ *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
+ *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
+ *                      = 2.0 - tiny            (if x <= -6)
+ *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
+ *              erf(x)  = sign(x)*(1.0 - tiny)
+ *         where
+ *              R2(z) = degree 6 poly in z, (z=1/x^2)
+ *              S2(z) = degree 7 poly in z
+ *
+ *      Note1:
+ *         To compute exp(-x*x-0.5625+R/S), let s be a single
+ *         precision number and s := x; then
+ *              -x*x = -s*s + (s-x)*(s+x)
+ *              exp(-x*x-0.5626+R/S) =
+ *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
+ *      Note2:
+ *         Here 4 and 5 make use of the asymptotic series
+ *                        exp(-x*x)
+ *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
+ *                        x*sqrt(pi)
+ *         We use rational approximation to approximate
+ *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
+ *         Here is the error bound for R1/S1 and R2/S2
+ *              |R1/S1 - f(x)|  < 2**(-62.57)
+ *              |R2/S2 - f(x)|  < 2**(-61.52)
+ *
+ *      5. For inf > x >= 28
+ *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
+ *              erfc(x) = tiny*tiny (raise underflow) if x > 0
+ *                      = 2 - tiny if x<0
+ *
+ *      7. Special case:
+ *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
+ *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
+ *              erfc/erf(NaN) is NaN
+ */
+
+#include "libm.h"
+
+static const double
+tiny = 1e-300,
+half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
+one  = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
+two  = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
+/* c = (float)0.84506291151 */
+erx  = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
+/*
+ * Coefficients for approximation to  erf on [0,0.84375]
+ */
+efx  =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
+efx8 =  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
+pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
+pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
+pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
+pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
+pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
+qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
+qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
+qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
+qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
+qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
+/*
+ * Coefficients for approximation to  erf  in [0.84375,1.25]
+ */
+pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
+pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
+pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
+pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
+pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
+pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
+pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
+qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
+qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
+qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
+qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
+qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
+qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
+/*
+ * Coefficients for approximation to  erfc in [1.25,1/0.35]
+ */
+ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
+ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
+ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
+ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
+ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
+ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
+ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
+ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
+sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
+sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
+sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
+sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
+sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
+sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
+sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
+sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
+/*
+ * Coefficients for approximation to  erfc in [1/.35,28]
+ */
+rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
+rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
+rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
+rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
+rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
+rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
+rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
+sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
+sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
+sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
+sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
+sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
+sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
+sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
+
+double erf(double x)
+{
+	int32_t hx,ix,i;
+	double R,S,P,Q,s,y,z,r;
+
+	GET_HIGH_WORD(hx, x);
+	ix = hx & 0x7fffffff;
+	if (ix >= 0x7ff00000) {
+		/* erf(nan)=nan, erf(+-inf)=+-1 */
+		i = ((uint32_t)hx>>31)<<1;
+		return (double)(1-i) + one/x;
+	}
+	if (ix < 0x3feb0000) {  /* |x|<0.84375 */
+		if (ix < 0x3e300000) {  /* |x|<2**-28 */
+			if (ix < 0x00800000)
+				/* avoid underflow */
+				return 0.125*(8.0*x + efx8*x);
+			return x + efx*x;
+		}
+		z = x*x;
+		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
+		s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+		y = r/s;
+		return x + x*y;
+	}
+	if (ix < 0x3ff40000) {  /* 0.84375 <= |x| < 1.25 */
+		s = fabs(x)-one;
+		P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
+		Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+		if (hx >= 0)
+			return erx + P/Q;
+		return -erx - P/Q;
+	}
+	if (ix >= 0x40180000) {  /* inf > |x| >= 6 */
+		if (hx >= 0)
+			return one-tiny;
+		return tiny-one;
+	}
+	x = fabs(x);
+	s = one/(x*x);
+	if (ix < 0x4006DB6E) {  /* |x| < 1/0.35 */
+		R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
+		     ra5+s*(ra6+s*ra7))))));
+		S = one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
+		     sa5+s*(sa6+s*(sa7+s*sa8)))))));
+	} else {                /* |x| >= 1/0.35 */
+		R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
+		     rb5+s*rb6)))));
+		S = one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
+		     sb5+s*(sb6+s*sb7))))));
+	}
+	z = x;
+	SET_LOW_WORD(z,0);
+	r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
+	if (hx >= 0)
+		return one-r/x;
+	return r/x-one;
+}
+
+double erfc(double x)
+{
+	int32_t hx,ix;
+	double R,S,P,Q,s,y,z,r;
+
+	GET_HIGH_WORD(hx, x);
+	ix = hx & 0x7fffffff;
+	if (ix >= 0x7ff00000) {
+		/* erfc(nan)=nan, erfc(+-inf)=0,2 */
+		return (double)(((uint32_t)hx>>31)<<1) + one/x;
+	}
+	if (ix < 0x3feb0000) {  /* |x| < 0.84375 */
+		if (ix < 0x3c700000)  /* |x| < 2**-56 */
+			return one - x;
+		z = x*x;
+		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
+		s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+		y = r/s;
+		if (hx < 0x3fd00000) {  /* x < 1/4 */
+			return one - (x+x*y);
+		} else {
+			r = x*y;
+			r += x-half;
+			return half - r ;
+		}
+	}
+	if (ix < 0x3ff40000) {  /* 0.84375 <= |x| < 1.25 */
+		s = fabs(x)-one;
+		P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
+		Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+		if (hx >= 0) {
+			z = one-erx;
+			return z - P/Q;
+		} else {
+			z = erx+P/Q;
+			return one+z;
+		}
+	}
+	if (ix < 0x403c0000) {  /* |x| < 28 */
+		x = fabs(x);
+		s = one/(x*x);
+		if (ix < 0x4006DB6D) {  /* |x| < 1/.35 ~ 2.857143*/
+			R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
+			     ra5+s*(ra6+s*ra7))))));
+			S = one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
+			     sa5+s*(sa6+s*(sa7+s*sa8)))))));
+		} else {                /* |x| >= 1/.35 ~ 2.857143 */
+			if (hx < 0 && ix >= 0x40180000)  /* x < -6 */
+				return two-tiny;
+			R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
+			     rb5+s*rb6)))));
+			S = one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
+			     sb5+s*(sb6+s*sb7))))));
+		}
+		z = x;
+		SET_LOW_WORD(z, 0);
+		r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
+		if (hx > 0)
+			return r/x;
+		return two-r/x;
+	}
+	if (hx > 0)
+		return tiny*tiny;
+	return two-tiny;
+}