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, 389 insertions, 0 deletions

diff --git a/src/math/j0.c b/src/math/j0.c
new file mode 100644
index 00000000..b5490641
--- /dev/null
+++ b/src/math/j0.c
@@ -0,0 +1,389 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.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.
+ * ====================================================
+ */
+/* j0(x), y0(x)
+ * Bessel function of the first and second kinds of order zero.
+ * Method -- j0(x):
+ *      1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
+ *      2. Reduce x to |x| since j0(x)=j0(-x),  and
+ *         for x in (0,2)
+ *              j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
+ *         (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
+ *         for x in (2,inf)
+ *              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
+ *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
+ *         as follow:
+ *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
+ *                      = 1/sqrt(2) * (cos(x) + sin(x))
+ *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/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
+ *              j0(nan)= nan
+ *              j0(0) = 1
+ *              j0(inf) = 0
+ *
+ * Method -- y0(x):
+ *      1. For x<2.
+ *         Since
+ *              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
+ *         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
+ *         We use the following function to approximate y0,
+ *              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
+ *         where
+ *              U(z) = u00 + u01*z + ... + u06*z^6
+ *              V(z) = 1  + v01*z + ... + v04*z^4
+ *         with absolute approximation error bounded by 2**-72.
+ *         Note: For tiny x, U/V = u0 and j0(x)~1, hence
+ *              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
+ *      2. For x>=2.
+ *              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
+ *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
+ *         by the method mentioned above.
+ *      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
+ */
+
+#include "libm.h"
+
+static double pzero(double), qzero(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, 2.00] */
+R02 =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
+R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
+R04 =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
+R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
+S01 =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
+S02 =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
+S03 =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
+S04 =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
+
+static const double zero = 0.0;
+
+double j0(double x)
+{
+	double z, s,c,ss,cc,r,u,v;
+	int32_t hx,ix;
+
+	GET_HIGH_WORD(hx, x);
+	ix = hx & 0x7fffffff;
+	if (ix >= 0x7ff00000)
+		return one/(x*x);
+	x = fabs(x);
+	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 does not overflow */
+			z = -cos(x+x);
+			if ((s*c) < zero)
+				cc = z/ss;
+			else
+				ss = z/cc;
+		}
+		/*
+		 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
+		 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
+		 */
+		if (ix > 0x48000000)
+			z = (invsqrtpi*cc)/sqrt(x);
+		else {
+			u = pzero(x);
+			v = qzero(x);
+			z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
+		}
+		return z;
+	}
+	if (ix < 0x3f200000) {  /* |x| < 2**-13 */
+		/* raise inexact if x != 0 */
+		if (huge+x > one) {
+			if (ix < 0x3e400000)  /* |x| < 2**-27 */
+				return one;
+			return one - 0.25*x*x;
+		}
+	}
+	z = x*x;
+	r = z*(R02+z*(R03+z*(R04+z*R05)));
+	s = one+z*(S01+z*(S02+z*(S03+z*S04)));
+	if (ix < 0x3FF00000) {   /* |x| < 1.00 */
+		return one + z*(-0.25+(r/s));
+	} else {
+		u = 0.5*x;
+		return (one+u)*(one-u) + z*(r/s);
+	}
+}
+
+static const double
+u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
+u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
+u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
+u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
+u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
+u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
+u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
+v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
+v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
+v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
+v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
+
+double y0(double x)
+{
+	double z,s,c,ss,cc,u,v;
+	int32_t hx,ix,lx;
+
+	EXTRACT_WORDS(hx, lx, x);
+	ix = 0x7fffffff & hx;
+	/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(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 */
+		/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
+		 * where x0 = x-pi/4
+		 *      Better formula:
+		 *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
+		 *                      =  1/sqrt(2) * (sin(x) + cos(x))
+		 *              sin(x0) = 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.
+		 */
+		s = sin(x);
+		c = cos(x);
+		ss = s-c;
+		cc = s+c;
+		/*
+		 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
+		 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
+		 */
+		if (ix < 0x7fe00000) {  /* make sure x+x does not overflow */
+			z = -cos(x+x);
+			if (s*c < zero)
+				cc = z/ss;
+			else
+				ss = z/cc;
+		}
+		if (ix > 0x48000000)
+			z = (invsqrtpi*ss)/sqrt(x);
+		else {
+			u = pzero(x);
+			v = qzero(x);
+			z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
+		}
+		return z;
+	}
+	if (ix <= 0x3e400000) {  /* x < 2**-27 */
+		return u00 + tpi*log(x);
+	}
+	z = x*x;
+	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
+	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
+	return u/v + tpi*(j0(x)*log(x));
+}
+
+/* The asymptotic expansions of pzero is
+ *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
+ * For x >= 2, We approximate pzero by
+ *      pzero(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
+ *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
+ */
+static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
+  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
+ -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
+ -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
+ -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
+ -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
+ -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
+};
+static const double pS8[5] = {
+  1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
+  3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
+  4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
+  1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
+  4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
+};
+
+static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
+ -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
+ -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
+ -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
+ -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
+ -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
+ -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
+};
+static const double pS5[5] = {
+  6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
+  1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
+  5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
+  9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
+  2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
+};
+
+static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
+ -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
+ -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
+ -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
+ -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
+ -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
+ -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
+};
+static const double pS3[5] = {
+  3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
+  3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
+  1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
+  1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
+  1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
+};
+
+static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
+ -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
+ -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
+ -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
+ -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
+ -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
+ -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
+};
+static const double pS2[5] = {
+  2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
+  1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
+  2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
+  1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
+  1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
+};
+
+static double pzero(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 qzero is
+ *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
+ * We approximate pzero by
+ *      qzero(x) = s*(-1.25 + (R/S))
+ * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
+ *        S = 1 + qS0*s^2 + ... + qS5*s^12
+ * and
+ *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
+ */
+static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
+  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
+  7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
+  1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
+  5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
+  8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
+  3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
+};
+static const double qS8[6] = {
+  1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
+  8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
+  1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
+  8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
+  8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
+ -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
+};
+
+static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
+  1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
+  7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
+  5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
+  1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
+  1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
+  1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
+};
+static const double qS5[6] = {
+  8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
+  2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
+  1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
+  5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
+  3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
+ -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
+};
+
+static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
+  4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
+  7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
+  3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
+  4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
+  1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
+  1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
+};
+static const double qS3[6] = {
+  4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
+  7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
+  3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
+  6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
+  2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
+ -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
+};
+
+static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
+  1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
+  7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
+  1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
+  1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
+  3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
+  1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
+};
+static const double qS2[6] = {
+  3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
+  2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
+  8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
+  8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
+  2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
+ -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
+};
+
+static double qzero(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 (-.125 + r/s)/x;
+}