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, 390 insertions, 0 deletions

diff --git a/src/math/erfl.c b/src/math/erfl.c
new file mode 100644
index 00000000..c38d7450
--- /dev/null
+++ b/src/math/erfl.c
@@ -0,0 +1,390 @@
+/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_erfl.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.
+ */
+/* 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]
+ *         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
+ *         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)
+ *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
+ *
+ *      3. For x in [1.25,1/0.35(~2.857143)],
+ *      erfc(x) = (1/x)*exp(-x*x-0.5625+R1(z)/S1(z))
+ *              z=1/x^2
+ *      erf(x)  = 1 - erfc(x)
+ *
+ *      4. For x in [1/0.35,107]
+ *      erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
+ *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2(z)/S2(z))
+ *                             if -6.666<x<0
+ *                      = 2.0 - tiny            (if x <= -6.666)
+ *              z=1/x^2
+ *      erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6.666, else
+ *      erf(x)  = sign(x)*(1.0 - tiny)
+ *      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)
+ *
+ *      5. For inf > x >= 107
+ *      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"
+
+#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
+long double erfl(long double x)
+{
+	return erfl(x);
+}
+#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
+static const long double
+tiny = 1e-4931L,
+half = 0.5L,
+one = 1.0L,
+two = 2.0L,
+/* c = (float)0.84506291151 */
+erx = 0.845062911510467529296875L,
+
+/*
+ * Coefficients for approximation to  erf on [0,0.84375]
+ */
+/* 2/sqrt(pi) - 1 */
+efx = 1.2837916709551257389615890312154517168810E-1L,
+/* 8 * (2/sqrt(pi) - 1) */
+efx8 = 1.0270333367641005911692712249723613735048E0L,
+pp[6] = {
+	1.122751350964552113068262337278335028553E6L,
+	-2.808533301997696164408397079650699163276E6L,
+	-3.314325479115357458197119660818768924100E5L,
+	-6.848684465326256109712135497895525446398E4L,
+	-2.657817695110739185591505062971929859314E3L,
+	-1.655310302737837556654146291646499062882E2L,
+},
+qq[6] = {
+	8.745588372054466262548908189000448124232E6L,
+	3.746038264792471129367533128637019611485E6L,
+	7.066358783162407559861156173539693900031E5L,
+	7.448928604824620999413120955705448117056E4L,
+	4.511583986730994111992253980546131408924E3L,
+	1.368902937933296323345610240009071254014E2L,
+	/* 1.000000000000000000000000000000000000000E0 */
+},
+
+/*
+ * Coefficients for approximation to  erf  in [0.84375,1.25]
+ */
+/* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
+   -0.15625 <= x <= +.25
+   Peak relative error 8.5e-22  */
+pa[8] = {
+	-1.076952146179812072156734957705102256059E0L,
+	 1.884814957770385593365179835059971587220E2L,
+	-5.339153975012804282890066622962070115606E1L,
+	 4.435910679869176625928504532109635632618E1L,
+	 1.683219516032328828278557309642929135179E1L,
+	-2.360236618396952560064259585299045804293E0L,
+	 1.852230047861891953244413872297940938041E0L,
+	 9.394994446747752308256773044667843200719E-2L,
+},
+qa[7] =  {
+	4.559263722294508998149925774781887811255E2L,
+	3.289248982200800575749795055149780689738E2L,
+	2.846070965875643009598627918383314457912E2L,
+	1.398715859064535039433275722017479994465E2L,
+	6.060190733759793706299079050985358190726E1L,
+	2.078695677795422351040502569964299664233E1L,
+	4.641271134150895940966798357442234498546E0L,
+	/* 1.000000000000000000000000000000000000000E0 */
+},
+
+/*
+ * Coefficients for approximation to  erfc in [1.25,1/0.35]
+ */
+/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
+   1/2.85711669921875 < 1/x < 1/1.25
+   Peak relative error 3.1e-21  */
+ra[] = {
+	1.363566591833846324191000679620738857234E-1L,
+	1.018203167219873573808450274314658434507E1L,
+	1.862359362334248675526472871224778045594E2L,
+	1.411622588180721285284945138667933330348E3L,
+	5.088538459741511988784440103218342840478E3L,
+	8.928251553922176506858267311750789273656E3L,
+	7.264436000148052545243018622742770549982E3L,
+	2.387492459664548651671894725748959751119E3L,
+	2.220916652813908085449221282808458466556E2L,
+},
+sa[] = {
+	-1.382234625202480685182526402169222331847E1L,
+	-3.315638835627950255832519203687435946482E2L,
+	-2.949124863912936259747237164260785326692E3L,
+	-1.246622099070875940506391433635999693661E4L,
+	-2.673079795851665428695842853070996219632E4L,
+	-2.880269786660559337358397106518918220991E4L,
+	-1.450600228493968044773354186390390823713E4L,
+	-2.874539731125893533960680525192064277816E3L,
+	-1.402241261419067750237395034116942296027E2L,
+	/* 1.000000000000000000000000000000000000000E0 */
+},
+
+/*
+ * Coefficients for approximation to  erfc in [1/.35,107]
+ */
+/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
+   1/6.6666259765625 < 1/x < 1/2.85711669921875
+   Peak relative error 4.2e-22  */
+rb[] = {
+	-4.869587348270494309550558460786501252369E-5L,
+	-4.030199390527997378549161722412466959403E-3L,
+	-9.434425866377037610206443566288917589122E-2L,
+	-9.319032754357658601200655161585539404155E-1L,
+	-4.273788174307459947350256581445442062291E0L,
+	-8.842289940696150508373541814064198259278E0L,
+	-7.069215249419887403187988144752613025255E0L,
+	-1.401228723639514787920274427443330704764E0L,
+},
+sb[] = {
+	4.936254964107175160157544545879293019085E-3L,
+	1.583457624037795744377163924895349412015E-1L,
+	1.850647991850328356622940552450636420484E0L,
+	9.927611557279019463768050710008450625415E0L,
+	2.531667257649436709617165336779212114570E1L,
+	2.869752886406743386458304052862814690045E1L,
+	1.182059497870819562441683560749192539345E1L,
+	/* 1.000000000000000000000000000000000000000E0 */
+},
+/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
+   1/107 <= 1/x <= 1/6.6666259765625
+   Peak relative error 1.1e-21  */
+rc[] = {
+	-8.299617545269701963973537248996670806850E-5L,
+	-6.243845685115818513578933902532056244108E-3L,
+	-1.141667210620380223113693474478394397230E-1L,
+	-7.521343797212024245375240432734425789409E-1L,
+	-1.765321928311155824664963633786967602934E0L,
+	-1.029403473103215800456761180695263439188E0L,
+},
+sc[] = {
+	8.413244363014929493035952542677768808601E-3L,
+	2.065114333816877479753334599639158060979E-1L,
+	1.639064941530797583766364412782135680148E0L,
+	4.936788463787115555582319302981666347450E0L,
+	5.005177727208955487404729933261347679090E0L,
+	/* 1.000000000000000000000000000000000000000E0 */
+};
+
+long double erfl(long double x)
+{
+	long double R, S, P, Q, s, y, z, r;
+	int32_t ix, i;
+	uint32_t se, i0, i1;
+
+	GET_LDOUBLE_WORDS (se, i0, i1, x);
+	ix = se & 0x7fff;
+
+	if (ix >= 0x7fff) {  /* erf(nan)=nan */
+		i = ((se & 0xffff) >> 15) << 1;
+		return (long double)(1 - i) + one / x;  /* erf(+-inf)=+-1 */
+	}
+
+	ix = (ix << 16) | (i0 >> 16);
+	if (ix < 0x3ffed800) {  /* |x| < 0.84375 */
+		if (ix < 0x3fde8000) {  /* |x| < 2**-33 */
+			if (ix < 0x00080000)
+				return 0.125 * (8.0 * x + efx8 * x);  /* avoid underflow */
+			return x + efx * x;
+		}
+		z = x * x;
+		r = pp[0] + z * (pp[1] +
+		     z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
+		s = qq[0] + z * (qq[1] +
+		     z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
+		y = r / s;
+		return x + x * y;
+	}
+	if (ix < 0x3fffa000) {  /* 0.84375 <= |x| < 1.25 */
+		s = fabsl (x) - one;
+		P = pa[0] + s * (pa[1] + s * (pa[2] +
+		     s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
+		Q = qa[0] + s * (qa[1] + s * (qa[2] +
+		     s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
+		if ((se & 0x8000) == 0)
+			return erx + P / Q;
+		return -erx - P / Q;
+	}
+	if (ix >= 0x4001d555) {  /* inf > |x| >= 6.6666259765625 */
+		if ((se & 0x8000) == 0)
+			return one - tiny;
+		return tiny - one;
+	}
+	x = fabsl (x);
+	s = one / (x * x);
+	if (ix < 0x4000b6db) {  /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
+		R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
+		     s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
+		S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
+		     s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
+	} else { /* 2.857 <= |x| < 6.667 */
+		R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
+		     s * (rb[5] + s * (rb[6] + s * rb[7]))))));
+		S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
+		     s * (sb[5] + s * (sb[6] + s))))));
+	}
+	z = x;
+	GET_LDOUBLE_WORDS(i, i0, i1, z);
+	i1 = 0;
+	SET_LDOUBLE_WORDS(z, i, i0, i1);
+	r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S);
+	if ((se & 0x8000) == 0)
+		return one - r / x;
+	return r / x - one;
+}
+
+long double erfcl(long double x)
+{
+	int32_t hx, ix;
+	long double R, S, P, Q, s, y, z, r;
+	uint32_t se, i0, i1;
+
+	GET_LDOUBLE_WORDS (se, i0, i1, x);
+	ix = se & 0x7fff;
+	if (ix >= 0x7fff) {  /* erfc(nan) = nan, erfc(+-inf) = 0,2 */
+		return (long double)(((se & 0xffff) >> 15) << 1) + one / x;
+	}
+
+	ix = (ix << 16) | (i0 >> 16);
+	if (ix < 0x3ffed800) {  /* |x| < 0.84375 */
+		if (ix < 0x3fbe0000)  /* |x| < 2**-65 */
+			return one - x;
+		z = x * x;
+		r = pp[0] + z * (pp[1] +
+		     z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
+		s = qq[0] + z * (qq[1] +
+		     z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
+		y = r / s;
+		if (ix < 0x3ffd8000) /* x < 1/4 */
+			return one - (x + x * y);
+		r = x * y;
+		r += x - half;
+		return half - r;
+	}
+	if (ix < 0x3fffa000) {  /* 0.84375 <= |x| < 1.25 */
+		s = fabsl (x) - one;
+		P = pa[0] + s * (pa[1] + s * (pa[2] +
+		     s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
+		Q = qa[0] + s * (qa[1] + s * (qa[2] +
+		     s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
+		if ((se & 0x8000) == 0) {
+			z = one - erx;
+			return z - P / Q;
+		}
+		z = erx + P / Q;
+		return one + z;
+	}
+	if (ix < 0x4005d600) {  /* |x| < 107 */
+		x = fabsl (x);
+		s = one / (x * x);
+		if (ix < 0x4000b6db) {  /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
+			R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
+			     s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
+			S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
+			     s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
+		} else if (ix < 0x4001d555) {  /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */
+			R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
+			     s * (rb[5] + s * (rb[6] + s * rb[7]))))));
+			S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
+			     s * (sb[5] + s * (sb[6] + s))))));
+		} else { /* 107 > |x| >= 6.666 */
+			if (se & 0x8000)
+				return two - tiny;/* x < -6.666 */
+			R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
+			     s * (rc[4] + s * rc[5]))));
+			S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
+			     s * (sc[4] + s))));
+		}
+		z = x;
+		GET_LDOUBLE_WORDS (hx, i0, i1, z);
+		i1 = 0;
+		i0 &= 0xffffff00;
+		SET_LDOUBLE_WORDS (z, hx, i0, i1);
+		r = expl (-z * z - 0.5625) *
+		expl ((z - x) * (z + x) + R / S);
+		if ((se & 0x8000) == 0)
+			return r / x;
+		return two - r / x;
+	}
+
+	if ((se & 0x8000) == 0)
+		return tiny * tiny;
+	return two - tiny;
+}
+#endif