path: root/src/math/j1.c

/* 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,
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 */

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 1.0/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 > 0.0)
				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 > 1.0)
			return 0.5*x;
	}
	z = x*x;
	r = z*(r00+z*(r01+z*(r02+z*r03)));
	s = 1.0+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 1.0/(x+x*x);
	if ((ix|lx) == 0)
		return -1.0/0.0;
	if (hx < 0)
		return 0.0/0.0;
	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 > 0.0)
				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 = 1.0+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
	return x*(u/v) + tpi*(j1(x)*log(x)-1.0/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 = 1.0/(x*x);
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
	s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
	return 1.0+ 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 = 1.0/(x*x);
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
	s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
	return (.375 + r/s)/x;
}