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, 385 insertions, 0 deletions

diff --git a/src/math/j1.c b/src/math/j1.c
new file mode 100644
index 00000000..29ccff0c
--- /dev/null
+++ b/src/math/j1.c
@@ -0,0 +1,385 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/e_j1.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.
+ * ====================================================
+ */
+/* j1(x), y1(x)
+ * Bessel function of the first and second kinds of order zero.
+ * Method -- j1(x):
+ *      1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
+ *      2. Reduce x to |x| since j1(x)=-j1(-x),  and
+ *         for x in (0,2)
+ *              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
+ *         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
+ *         for x in (2,inf)
+ *              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
+ *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+ *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+ *         as follow:
+ *              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+ *                      =  1/sqrt(2) * (sin(x) - cos(x))
+ *              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+ *                      = -1/sqrt(2) * (sin(x) + cos(x))
+ *         (To avoid cancellation, use
+ *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+ *          to compute the worse one.)
+ *
+ *      3 Special cases
+ *              j1(nan)= nan
+ *              j1(0) = 0
+ *              j1(inf) = 0
+ *
+ * Method -- y1(x):
+ *      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
+ *      2. For x<2.
+ *         Since
+ *              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
+ *         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
+ *         We use the following function to approximate y1,
+ *              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
+ *         where for x in [0,2] (abs err less than 2**-65.89)
+ *              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
+ *              V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
+ *         Note: For tiny x, 1/x dominate y1 and hence
+ *              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
+ *      3. For x>=2.
+ *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+ *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+ *         by method mentioned above.
+ */
+
+#include "libm.h"
+
+static double pone(double), qone(double);
+
+static const double
+huge      = 1e300,
+one       = 1.0,
+invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
+tpi       = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
+/* R0/S0 on [0,2] */
+r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
+r01 =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
+r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
+r03 =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
+s01 =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
+s02 =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
+s03 =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
+s04 =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
+s05 =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
+
+static const double zero = 0.0;
+
+double j1(double x)
+{
+	double z,s,c,ss,cc,r,u,v,y;
+	int32_t hx,ix;
+
+	GET_HIGH_WORD(hx, x);
+	ix = hx & 0x7fffffff;
+	if (ix >= 0x7ff00000)
+		return one/x;
+	y = fabs(x);
+	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
+		s = sin(y);
+		c = cos(y);
+		ss = -s-c;
+		cc = s-c;
+		if (ix < 0x7fe00000) {  /* make sure y+y not overflow */
+			z = cos(y+y);
+			if (s*c > zero)
+				cc = z/ss;
+			else
+				ss = z/cc;
+		}
+		/*
+		 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
+		 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
+		 */
+		if (ix > 0x48000000)
+			z = (invsqrtpi*cc)/sqrt(y);
+		else {
+			u = pone(y);
+			v = qone(y);
+			z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
+		}
+		if (hx < 0)
+			return -z;
+		else
+			return  z;
+	}
+	if (ix < 0x3e400000) {  /* |x| < 2**-27 */
+		/* raise inexact if x!=0 */
+		if (huge+x > one)
+			return 0.5*x;
+	}
+	z = x*x;
+	r = z*(r00+z*(r01+z*(r02+z*r03)));
+	s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
+	r *= x;
+	return x*0.5 + r/s;
+}
+
+static const double U0[5] = {
+ -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
+  5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
+ -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
+  2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
+ -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
+};
+static const double V0[5] = {
+  1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
+  2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
+  1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
+  6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
+  1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
+};
+
+
+double y1(double x)
+{
+	double z,s,c,ss,cc,u,v;
+	int32_t hx,ix,lx;
+
+	EXTRACT_WORDS(hx, lx, x);
+	ix = 0x7fffffff & hx;
+	/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
+	if (ix >= 0x7ff00000)
+		return one/(x+x*x);
+	if ((ix|lx) == 0)
+		return -one/zero;
+	if (hx < 0)
+		return zero/zero;
+	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
+		s = sin(x);
+		c = cos(x);
+		ss = -s-c;
+		cc = s-c;
+		if (ix < 0x7fe00000) {  /* make sure x+x not overflow */
+			z = cos(x+x);
+			if (s*c > zero)
+				cc = z/ss;
+			else
+				ss = z/cc;
+		}
+		/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
+		 * where x0 = x-3pi/4
+		 *      Better formula:
+		 *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+		 *                      =  1/sqrt(2) * (sin(x) - cos(x))
+		 *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+		 *                      = -1/sqrt(2) * (cos(x) + sin(x))
+		 * To avoid cancellation, use
+		 *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+		 * to compute the worse one.
+		 */
+		if (ix > 0x48000000)
+			z = (invsqrtpi*ss)/sqrt(x);
+		else {
+			u = pone(x);
+			v = qone(x);
+			z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
+		}
+		return z;
+	}
+	if (ix <= 0x3c900000)  /* x < 2**-54 */
+		return -tpi/x;
+	z = x*x;
+	u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
+	v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
+	return x*(u/v) + tpi*(j1(x)*log(x)-one/x);
+}
+
+/* For x >= 8, the asymptotic expansions of pone is
+ *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
+ * We approximate pone by
+ *      pone(x) = 1 + (R/S)
+ * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
+ *        S = 1 + ps0*s^2 + ... + ps4*s^10
+ * and
+ *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
+ */
+
+static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
+  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
+  1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
+  1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
+  4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
+  3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
+  7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
+};
+static const double ps8[5] = {
+  1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
+  3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
+  3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
+  9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
+  3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
+};
+
+static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
+  1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
+  1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
+  6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
+  1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
+  5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
+  5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
+};
+static const double ps5[5] = {
+  5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
+  9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
+  5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
+  7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
+  1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
+};
+
+static const double pr3[6] = {
+  3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
+  1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
+  3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
+  3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
+  9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
+  4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
+};
+static const double ps3[5] = {
+  3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
+  3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
+  1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
+  8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
+  1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
+};
+
+static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
+  1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
+  1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
+  2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
+  1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
+  1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
+  5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
+};
+static const double ps2[5] = {
+  2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
+  1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
+  2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
+  1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
+  8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
+};
+
+static double pone(double x)
+{
+	const double *p,*q;
+	double z,r,s;
+	int32_t ix;
+
+	GET_HIGH_WORD(ix, x);
+	ix &= 0x7fffffff;
+	if      (ix >= 0x40200000){p = pr8; q = ps8;}
+	else if (ix >= 0x40122E8B){p = pr5; q = ps5;}
+	else if (ix >= 0x4006DB6D){p = pr3; q = ps3;}
+	else if (ix >= 0x40000000){p = pr2; q = ps2;}
+	z = one/(x*x);
+	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
+	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
+	return one+ r/s;
+}
+
+/* For x >= 8, the asymptotic expansions of qone is
+ *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
+ * We approximate pone by
+ *      qone(x) = s*(0.375 + (R/S))
+ * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
+ *        S = 1 + qs1*s^2 + ... + qs6*s^12
+ * and
+ *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
+ */
+
+static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
+  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
+ -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
+ -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
+ -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
+ -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
+ -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
+};
+static const double qs8[6] = {
+  1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
+  7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
+  1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
+  7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
+  6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
+ -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
+};
+
+static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
+ -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
+ -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
+ -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
+ -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
+ -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
+ -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
+};
+static const double qs5[6] = {
+  8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
+  1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
+  1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
+  4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
+  2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
+ -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
+};
+
+static const double qr3[6] = {
+ -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
+ -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
+ -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
+ -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
+ -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
+ -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
+};
+static const double qs3[6] = {
+  4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
+  6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
+  3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
+  5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
+  1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
+ -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
+};
+
+static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
+ -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
+ -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
+ -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
+ -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
+ -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
+ -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
+};
+static const double qs2[6] = {
+  2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
+  2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
+  7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
+  7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
+  1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
+ -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
+};
+
+static double qone(double x)
+{
+	const double *p,*q;
+	double  s,r,z;
+	int32_t ix;
+
+	GET_HIGH_WORD(ix, x);
+	ix &= 0x7fffffff;
+	if      (ix >= 0x40200000){p = qr8; q = qs8;}
+	else if (ix >= 0x40122E8B){p = qr5; q = qs5;}
+	else if (ix >= 0x4006DB6D){p = qr3; q = qs3;}
+	else if (ix >= 0x40000000){p = qr2; q = qs2;}
+	z = one/(x*x);
+	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
+	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
+	return (.375 + r/s)/x;
+}