src/math/j1f.c

/* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
/*
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */

#include "libm.h"

static float ponef(float), qonef(float);

static const float
huge      = 1e30,
one       = 1.0,
invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
tpi       = 6.3661974669e-01, /* 0x3f22f983 */
/* R0/S0 on [0,2] */
r00 = -6.2500000000e-02, /* 0xbd800000 */
r01 =  1.4070566976e-03, /* 0x3ab86cfd */
r02 = -1.5995563444e-05, /* 0xb7862e36 */
r03 =  4.9672799207e-08, /* 0x335557d2 */
s01 =  1.9153760746e-02, /* 0x3c9ce859 */
s02 =  1.8594678841e-04, /* 0x3942fab6 */
s03 =  1.1771846857e-06, /* 0x359dffc2 */
s04 =  5.0463624390e-09, /* 0x31ad6446 */
s05 =  1.2354227016e-11; /* 0x2d59567e */

static const float zero = 0.0;

float j1f(float x)
{
	float z,s,c,ss,cc,r,u,v,y;
	int32_t hx,ix;

	GET_FLOAT_WORD(hx, x);
	ix = hx & 0x7fffffff;
	if (ix >= 0x7f800000)
		return one/x;
	y = fabsf(x);
	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
		s = sinf(y);
		c = cosf(y);
		ss = -s-c;
		cc = s-c;
		if (ix < 0x7f000000) {  /* make sure y+y not overflow */
			z = cosf(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 > 0x80000000)
			z = (invsqrtpi*cc)/sqrtf(y);
		else {
			u = ponef(y);
			v = qonef(y);
			z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
		}
		if (hx < 0)
			return -z;
		return  z;
	}
	if (ix < 0x32000000) {  /* |x| < 2**-27 */
		/* raise inexact if x!=0 */
		if (huge+x > one)
			return 0.5f*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 0.5f*x + r/s;
}

static const float U0[5] = {
 -1.9605709612e-01, /* 0xbe48c331 */
  5.0443872809e-02, /* 0x3d4e9e3c */
 -1.9125689287e-03, /* 0xbafaaf2a */
  2.3525259166e-05, /* 0x37c5581c */
 -9.1909917899e-08, /* 0xb3c56003 */
};
static const float V0[5] = {
  1.9916731864e-02, /* 0x3ca3286a */
  2.0255257550e-04, /* 0x3954644b */
  1.3560879779e-06, /* 0x35b602d4 */
  6.2274145840e-09, /* 0x31d5f8eb */
  1.6655924903e-11, /* 0x2d9281cf */
};

float y1f(float x)
{
	float z,s,c,ss,cc,u,v;
	int32_t hx,ix;

	GET_FLOAT_WORD(hx, x);
	ix = 0x7fffffff & hx;
	/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
	if (ix >= 0x7f800000)
		return one/(x+x*x);
	if (ix == 0)
		return -one/zero;
	if (hx < 0)
		return zero/zero;
	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
		s = sinf(x);
		c = cosf(x);
		ss = -s-c;
		cc = s-c;
		if (ix < 0x7f000000) {  /* make sure x+x not overflow */
			z = cosf(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)/sqrtf(x);
		else {
			u = ponef(x);
			v = qonef(x);
			z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
		}
		return z;
	}
	if (ix <= 0x24800000)  /* 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*(j1f(x)*logf(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 float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  0.0000000000e+00, /* 0x00000000 */
  1.1718750000e-01, /* 0x3df00000 */
  1.3239480972e+01, /* 0x4153d4ea */
  4.1205184937e+02, /* 0x43ce06a3 */
  3.8747453613e+03, /* 0x45722bed */
  7.9144794922e+03, /* 0x45f753d6 */
};
static const float ps8[5] = {
  1.1420736694e+02, /* 0x42e46a2c */
  3.6509309082e+03, /* 0x45642ee5 */
  3.6956207031e+04, /* 0x47105c35 */
  9.7602796875e+04, /* 0x47bea166 */
  3.0804271484e+04, /* 0x46f0a88b */
};

static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  1.3199052094e-11, /* 0x2d68333f */
  1.1718749255e-01, /* 0x3defffff */
  6.8027510643e+00, /* 0x40d9b023 */
  1.0830818176e+02, /* 0x42d89dca */
  5.1763616943e+02, /* 0x440168b7 */
  5.2871520996e+02, /* 0x44042dc6 */
};
static const float ps5[5] = {
  5.9280597687e+01, /* 0x426d1f55 */
  9.9140142822e+02, /* 0x4477d9b1 */
  5.3532670898e+03, /* 0x45a74a23 */
  7.8446904297e+03, /* 0x45f52586 */
  1.5040468750e+03, /* 0x44bc0180 */
};

static const float pr3[6] = {
  3.0250391081e-09, /* 0x314fe10d */
  1.1718686670e-01, /* 0x3defffab */
  3.9329774380e+00, /* 0x407bb5e7 */
  3.5119403839e+01, /* 0x420c7a45 */
  9.1055007935e+01, /* 0x42b61c2a */
  4.8559066772e+01, /* 0x42423c7c */
};
static const float ps3[5] = {
  3.4791309357e+01, /* 0x420b2a4d */
  3.3676245117e+02, /* 0x43a86198 */
  1.0468714600e+03, /* 0x4482dbe3 */
  8.9081134033e+02, /* 0x445eb3ed */
  1.0378793335e+02, /* 0x42cf936c */
};

static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  1.0771083225e-07, /* 0x33e74ea8 */
  1.1717621982e-01, /* 0x3deffa16 */
  2.3685150146e+00, /* 0x401795c0 */
  1.2242610931e+01, /* 0x4143e1bc */
  1.7693971634e+01, /* 0x418d8d41 */
  5.0735230446e+00, /* 0x40a25a4d */
};
static const float ps2[5] = {
  2.1436485291e+01, /* 0x41ab7dec */
  1.2529022980e+02, /* 0x42fa9499 */
  2.3227647400e+02, /* 0x436846c7 */
  1.1767937469e+02, /* 0x42eb5bd7 */
  8.3646392822e+00, /* 0x4105d590 */
};

static float ponef(float x)
{
	const float *p,*q;
	float z,r,s;
	int32_t ix;

	GET_FLOAT_WORD(ix, x);
	ix &= 0x7fffffff;
	if      (ix >= 0x41000000){p = pr8; q = ps8;}
	else if (ix >= 0x40f71c58){p = pr5; q = ps5;}
	else if (ix >= 0x4036db68){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 float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  0.0000000000e+00, /* 0x00000000 */
 -1.0253906250e-01, /* 0xbdd20000 */
 -1.6271753311e+01, /* 0xc1822c8d */
 -7.5960174561e+02, /* 0xc43de683 */
 -1.1849806641e+04, /* 0xc639273a */
 -4.8438511719e+04, /* 0xc73d3683 */
};
static const float qs8[6] = {
  1.6139537048e+02, /* 0x43216537 */
  7.8253862305e+03, /* 0x45f48b17 */
  1.3387534375e+05, /* 0x4802bcd6 */
  7.1965775000e+05, /* 0x492fb29c */
  6.6660125000e+05, /* 0x4922be94 */
 -2.9449025000e+05, /* 0xc88fcb48 */
};

static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 -2.0897993405e-11, /* 0xadb7d219 */
 -1.0253904760e-01, /* 0xbdd1fffe */
 -8.0564479828e+00, /* 0xc100e736 */
 -1.8366960144e+02, /* 0xc337ab6b */
 -1.3731937256e+03, /* 0xc4aba633 */
 -2.6124443359e+03, /* 0xc523471c */
};
static const float qs5[6] = {
  8.1276550293e+01, /* 0x42a28d98 */
  1.9917987061e+03, /* 0x44f8f98f */
  1.7468484375e+04, /* 0x468878f8 */
  4.9851425781e+04, /* 0x4742bb6d */
  2.7948074219e+04, /* 0x46da5826 */
 -4.7191835938e+03, /* 0xc5937978 */
};

static const float qr3[6] = {
 -5.0783124372e-09, /* 0xb1ae7d4f */
 -1.0253783315e-01, /* 0xbdd1ff5b */
 -4.6101160049e+00, /* 0xc0938612 */
 -5.7847221375e+01, /* 0xc267638e */
 -2.2824453735e+02, /* 0xc3643e9a */
 -2.1921012878e+02, /* 0xc35b35cb */
};
static const float qs3[6] = {
  4.7665153503e+01, /* 0x423ea91e */
  6.7386511230e+02, /* 0x4428775e */
  3.3801528320e+03, /* 0x45534272 */
  5.5477290039e+03, /* 0x45ad5dd5 */
  1.9031191406e+03, /* 0x44ede3d0 */
 -1.3520118713e+02, /* 0xc3073381 */
};

static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 -1.7838172539e-07, /* 0xb43f8932 */
 -1.0251704603e-01, /* 0xbdd1f475 */
 -2.7522056103e+00, /* 0xc0302423 */
 -1.9663616180e+01, /* 0xc19d4f16 */
 -4.2325313568e+01, /* 0xc2294d1f */
 -2.1371921539e+01, /* 0xc1aaf9b2 */
};
static const float qs2[6] = {
  2.9533363342e+01, /* 0x41ec4454 */
  2.5298155212e+02, /* 0x437cfb47 */
  7.5750280762e+02, /* 0x443d602e */
  7.3939318848e+02, /* 0x4438d92a */
  1.5594900513e+02, /* 0x431bf2f2 */
 -4.9594988823e+00, /* 0xc09eb437 */
};

static float qonef(float x)
{
	const float *p,*q;
	float s,r,z;
	int32_t ix;

	GET_FLOAT_WORD(ix, x);
	ix &= 0x7fffffff;
	if      (ix >= 0x40200000){p = qr8; q = qs8;}
	else if (ix >= 0x40f71c58){p = qr5; q = qs5;}
	else if (ix >= 0x4036db68){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 (.375f + r/s)/x;
}