[uclibc-ng.git] / libm / e_j0.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.
 * ====================================================
 */

/* __ieee754_j0(x), __ieee754_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 "math.h"
#include "math_private.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 attribute_hidden __ieee754_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 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 */
	    if(huge+x>one) {	/* raise inexact if x != 0 */
	        if(ix<0x3e400000) return one;	/* |x|<2**-27 */
	        else 	      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));
	}
}

/*
 * wrapper j0(double x)
 */
#ifndef _IEEE_LIBM
double j0(double x)
{
	double z = __ieee754_j0(x);
	if (_LIB_VERSION == _IEEE_ || isnan(x))
		return z;
	if (fabs(x) > X_TLOSS)
		return __kernel_standard(x, x, 34); /* j0(|x|>X_TLOSS) */
	return z;
}
#else
strong_alias(__ieee754_j0, j0)
#endif

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 attribute_hidden __ieee754_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 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*__ieee754_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*(__ieee754_j0(x)*__ieee754_log(x)));
}

/*
 * wrapper y0(double x)
 */
#ifndef _IEEE_LIBM
double y0(double x)
{
	double z = __ieee754_y0(x);
	if (_LIB_VERSION == _IEEE_ || isnan(x))
		return z;
	if (x <= 0.0) {
		if (x == 0.0) /* d= -one/(x-x); */
			return __kernel_standard(x, x, 8);
		/* d = zero/(x-x); */
		return __kernel_standard(x, x, 9);
	}
	if (x > X_TLOSS)
		return __kernel_standard(x, x, 35); /* y0(x>X_TLOSS) */
	return z;
}
#else
strong_alias(__ieee754_y0, y0)
#endif


/* 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 = 0,*q = 0;
	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=0,*q=0;
	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;
}
Commit	Line	Data
7ce331c0 EA	1	/*
	2	* ====================================================
	3	* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
	4	*
	5	* Developed at SunPro, a Sun Microsystems, Inc. business.
	6	* Permission to use, copy, modify, and distribute this
c4e44e97	7	* software is freely granted, provided that this notice
7ce331c0 EA	8	* is preserved.
	9	* ====================================================
	10	*/
	11
7ce331c0 EA	12	/* __ieee754_j0(x), __ieee754_y0(x)
	13	* Bessel function of the first and second kinds of order zero.
	14	* Method -- j0(x):
	15	* 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
	16	* 2. Reduce x to \|x\| since j0(x)=j0(-x), and
	17	* for x in (0,2)
	18	* j0(x) = 1-z/4+ z^2R0/S0, where z = xx;
	19	* (precision: \|j0-1+z/4-z^2R0/S0 \|<2**-63.67 )
	20	* for x in (2,inf)
	21	* j0(x) = sqrt(2/(pix))(p0(x)cos(x0)-q0(x)sin(x0))
	22	* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
	23	* as follow:
	24	* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
	25	* = 1/sqrt(2) * (cos(x) + sin(x))
	26	* sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
	27	* = 1/sqrt(2) * (sin(x) - cos(x))
	28	* (To avoid cancellation, use
	29	* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
	30	* to compute the worse one.)
c4e44e97	31	*
7ce331c0 EA	32	* 3 Special cases
	33	* j0(nan)= nan
	34	* j0(0) = 1
	35	* j0(inf) = 0
c4e44e97	36	*
7ce331c0 EA	37	* Method -- y0(x):
7ce331c0 EA	38	* 1. For x<2.
c4e44e97	39	* Since
7ce331c0 EA	40	* y0(x) = 2/pi(j0(x)(ln(x/2)+Euler) + x^2/4 - ...)
	41	* therefore y0(x)-2/pij0(x)ln(x) is an even function.
	42	* We use the following function to approximate y0,
	43	* y0(x) = U(z)/V(z) + (2/pi)(j0(x)ln(x)), z= x^2
c4e44e97	44	* where
7ce331c0 EA	45	* U(z) = u00 + u01z + ... + u06z^6
	46	* V(z) = 1 + v01z + ... + v04z^4
	47	* with absolute approximation error bounded by 2**-72.
	48	* Note: For tiny x, U/V = u0 and j0(x)~1, hence
	49	* y0(tiny) = u0 + (2/pi)ln(tiny), (choose tiny<2*-27)
	50	* 2. For x>=2.
	51	* y0(x) = sqrt(2/(pix))(p0(x)cos(x0)+q0(x)sin(x0))
	52	* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
	53	* by the method mentioned above.
	54	* 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
	55	*/
	56
	57	#include "math.h"
	58	#include "math_private.h"
	59
7ce331c0	60	static double pzero(double), qzero(double);
7ce331c0	61
c4e44e97	62	static const double
7ce331c0 EA	63	huge = 1e300,
	64	one = 1.0,
	65	invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
	66	tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
	67	/* R0/S0 on [0, 2.00] */
	68	R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
	69	R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
	70	R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
	71	R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
	72	S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
	73	S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
	74	S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
	75	S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
	76
7ce331c0	77	static const double zero = 0.0;
7ce331c0	78
38b7304e	79	double attribute_hidden __ieee754_j0(double x)
7ce331c0 EA	80	{
	81	double z, s,c,ss,cc,r,u,v;
	82	int32_t hx,ix;
	83
	84	GET_HIGH_WORD(hx,x);
	85	ix = hx&0x7fffffff;
	86	if(ix>=0x7ff00000) return one/(x*x);
	87	x = fabs(x);
	88	if(ix >= 0x40000000) { /* \|x\| >= 2.0 */
	89	s = sin(x);
	90	c = cos(x);
	91	ss = s-c;
	92	cc = s+c;
	93	if(ix<0x7fe00000) { /* make sure x+x not overflow */
	94	z = -cos(x+x);
	95	if ((s*c)<zero) cc = z/ss;
	96	else ss = z/cc;
	97	}
	98	/*
	99	* j0(x) = 1/sqrt(pi) * (P(0,x)cc - Q(0,x)ss) / sqrt(x)
	100	* y0(x) = 1/sqrt(pi) * (P(0,x)ss + Q(0,x)cc) / sqrt(x)
	101	*/
	102	if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
	103	else {
	104	u = pzero(x); v = qzero(x);
	105	z = invsqrtpi(ucc-v*ss)/sqrt(x);
	106	}
	107	return z;
	108	}
	109	if(ix<0x3f200000) { /* \|x\| < 2*-13 /
	110	if(huge+x>one) { /* raise inexact if x != 0 */
	111	if(ix<0x3e400000) return one; /* \|x\|<2*-27 /
	112	else return one - 0.25xx;
	113	}
	114	}
	115	z = x*x;
	116	r = z(R02+z(R03+z(R04+zR05)));
	117	s = one+z(S01+z(S02+z(S03+zS04)));
	118	if(ix < 0x3FF00000) { /* \|x\| < 1.00 */
	119	return one + z*(-0.25+(r/s));
	120	} else {
	121	u = 0.5*x;
	122	return((one+u)(one-u)+z(r/s));
	123	}
	124	}
	125
30bd4a6c DV	126	/*
	127	* wrapper j0(double x)
	128	*/
	129	#ifndef _IEEE_LIBM
	130	double j0(double x)
	131	{
	132	double z = __ieee754_j0(x);
	133	if (_LIB_VERSION == _IEEE_ \|\| isnan(x))
	134	return z;
	135	if (fabs(x) > X_TLOSS)
	136	return __kernel_standard(x, x, 34); /* j0(\|x\|>X_TLOSS) */
	137	return z;
	138	}
	139	#else
	140	strong_alias(__ieee754_j0, j0)
	141	#endif
	142
7ce331c0	143	static const double
7ce331c0 EA	144	u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
	145	u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
	146	u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
	147	u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
	148	u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
	149	u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
	150	u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
	151	v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
	152	v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
	153	v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
	154	v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
	155
38b7304e	156	double attribute_hidden __ieee754_y0(double x)
7ce331c0 EA	157	{
	158	double z, s,c,ss,cc,u,v;
	159	int32_t hx,ix,lx;
	160
	161	EXTRACT_WORDS(hx,lx,x);
	162	ix = 0x7fffffff&hx;
	163	/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
c4e44e97	164	if(ix>=0x7ff00000) return one/(x+x*x);
7ce331c0 EA	165	if((ix\|lx)==0) return -one/zero;
	166	if(hx<0) return zero/zero;
	167	if(ix >= 0x40000000) { /* \|x\| >= 2.0 */
	168	/* y0(x) = sqrt(2/(pix))(p0(x)sin(x0)+q0(x)cos(x0))
	169	* where x0 = x-pi/4
	170	* Better formula:
	171	* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
	172	* = 1/sqrt(2) * (sin(x) + cos(x))
	173	* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
	174	* = 1/sqrt(2) * (sin(x) - cos(x))
	175	* To avoid cancellation, use
	176	* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
	177	* to compute the worse one.
	178	*/
	179	s = sin(x);
	180	c = cos(x);
	181	ss = s-c;
	182	cc = s+c;
	183	/*
	184	* j0(x) = 1/sqrt(pi) * (P(0,x)cc - Q(0,x)ss) / sqrt(x)
	185	* y0(x) = 1/sqrt(pi) * (P(0,x)ss + Q(0,x)cc) / sqrt(x)
	186	*/
	187	if(ix<0x7fe00000) { /* make sure x+x not overflow */
	188	z = -cos(x+x);
	189	if ((s*c)<zero) cc = z/ss;
	190	else ss = z/cc;
	191	}
	192	if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
	193	else {
	194	u = pzero(x); v = qzero(x);
	195	z = invsqrtpi(uss+v*cc)/sqrt(x);
	196	}
	197	return z;
	198	}
	199	if(ix<=0x3e400000) { /* x < 2*-27 /
	200	return(u00 + tpi*__ieee754_log(x));
	201	}
	202	z = x*x;
	203	u = u00+z(u01+z(u02+z(u03+z(u04+z(u05+zu06)))));
	204	v = one+z(v01+z(v02+z(v03+zv04)));
	205	return(u/v + tpi(__ieee754_j0(x)__ieee754_log(x)));
	206	}
	207
30bd4a6c DV	208	/*
	209	* wrapper y0(double x)
	210	*/
	211	#ifndef _IEEE_LIBM
	212	double y0(double x)
	213	{
	214	double z = __ieee754_y0(x);
	215	if (_LIB_VERSION == _IEEE_ \|\| isnan(x))
	216	return z;
	217	if (x <= 0.0) {
	218	if (x == 0.0) /* d= -one/(x-x); */
	219	return __kernel_standard(x, x, 8);
	220	/* d = zero/(x-x); */
	221	return __kernel_standard(x, x, 9);
	222	}
	223	if (x > X_TLOSS)
	224	return __kernel_standard(x, x, 35); /* y0(x>X_TLOSS) */
	225	return z;
	226	}
	227	#else
	228	strong_alias(__ieee754_y0, y0)
	229	#endif
	230
	231
7ce331c0 EA	232	/* The asymptotic expansions of pzero is
	233	* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
	234	* For x >= 2, We approximate pzero by
	235	* pzero(x) = 1 + (R/S)
	236	* where R = pR0 + pR1s^2 + pR2s^4 + ... + pR5*s^10
	237	* S = 1 + pS0s^2 + ... + pS4s^10
	238	* and
	239	* \| pzero(x)-1-R/S \| <= 2 ** ( -60.26)
	240	*/
7ce331c0	241	static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
7ce331c0 EA	242	0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
	243	-7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
	244	-8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
	245	-2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
	246	-2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
	247	-5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
	248	};
7ce331c0	249	static const double pS8[5] = {
7ce331c0 EA	250	1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
	251	3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
	252	4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
	253	1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
	254	4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
	255	};
	256
7ce331c0	257	static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
7ce331c0 EA	258	-1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
	259	-7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
	260	-4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
	261	-6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
	262	-3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
	263	-3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
	264	};
7ce331c0	265	static const double pS5[5] = {
7ce331c0 EA	266	6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
	267	1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
	268	5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
	269	9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
	270	2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
	271	};
	272
7ce331c0	273	static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
7ce331c0 EA	274	-2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
	275	-7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
	276	-2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
	277	-2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
	278	-5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
	279	-3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
	280	};
7ce331c0	281	static const double pS3[5] = {
7ce331c0 EA	282	3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
	283	3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
	284	1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
	285	1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
	286	1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
	287	};
	288
7ce331c0	289	static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
7ce331c0 EA	290	-8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
	291	-7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
	292	-1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
	293	-7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
	294	-1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
	295	-3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
	296	};
7ce331c0	297	static const double pS2[5] = {
7ce331c0 EA	298	2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
	299	1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
	300	2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
	301	1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
	302	1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
	303	};
	304
30bd4a6c	305	static double pzero(double x)
7ce331c0	306	{
82ba14bc	307	const double p = 0,q = 0;
7ce331c0 EA	308	double z,r,s;
	309	int32_t ix;
	310	GET_HIGH_WORD(ix,x);
	311	ix &= 0x7fffffff;
	312	if(ix>=0x40200000) {p = pR8; q= pS8;}
	313	else if(ix>=0x40122E8B){p = pR5; q= pS5;}
	314	else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
	315	else if(ix>=0x40000000){p = pR2; q= pS2;}
	316	z = one/(x*x);
	317	r = p[0]+z(p[1]+z(p[2]+z(p[3]+z(p[4]+z*p[5]))));
	318	s = one+z(q[0]+z(q[1]+z(q[2]+z(q[3]+z*q[4]))));
	319	return one+ r/s;
	320	}
c4e44e97	321
7ce331c0 EA	322
	323	/* For x >= 8, the asymptotic expansions of qzero is
	324	* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
	325	* We approximate pzero by
	326	* qzero(x) = s*(-1.25 + (R/S))
	327	* where R = qR0 + qR1s^2 + qR2s^4 + ... + qR5*s^10
	328	* S = 1 + qS0s^2 + ... + qS5s^12
	329	* and
	330	* \| qzero(x)/s +1.25-R/S \| <= 2 ** ( -61.22)
	331	*/
7ce331c0	332	static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
7ce331c0 EA	333	0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
	334	7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
	335	1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
	336	5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
	337	8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
	338	3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
	339	};
7ce331c0	340	static const double qS8[6] = {
7ce331c0 EA	341	1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
	342	8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
	343	1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
	344	8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
	345	8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
	346	-3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
	347	};
	348
7ce331c0	349	static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
7ce331c0 EA	350	1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
	351	7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
	352	5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
	353	1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
	354	1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
	355	1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
	356	};
7ce331c0	357	static const double qS5[6] = {
7ce331c0 EA	358	8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
	359	2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
	360	1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
	361	5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
	362	3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
	363	-5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
	364	};
	365
7ce331c0	366	static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
7ce331c0 EA	367	4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
	368	7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
	369	3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
	370	4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
	371	1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
	372	1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
	373	};
7ce331c0	374	static const double qS3[6] = {
7ce331c0 EA	375	4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
	376	7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
	377	3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
	378	6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
	379	2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
	380	-1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
	381	};
	382
7ce331c0	383	static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
7ce331c0 EA	384	1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
	385	7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
	386	1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
	387	1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
	388	3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
	389	1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
	390	};
7ce331c0	391	static const double qS2[6] = {
7ce331c0 EA	392	3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
	393	2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
	394	8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
	395	8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
	396	2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
	397	-5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
	398	};
	399
30bd4a6c	400	static double qzero(double x)
7ce331c0	401	{
82ba14bc	402	const double p=0,q=0;
7ce331c0 EA	403	double s,r,z;
	404	int32_t ix;
	405	GET_HIGH_WORD(ix,x);
	406	ix &= 0x7fffffff;
	407	if(ix>=0x40200000) {p = qR8; q= qS8;}
	408	else if(ix>=0x40122E8B){p = qR5; q= qS5;}
	409	else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
	410	else if(ix>=0x40000000){p = qR2; q= qS2;}
	411	z = one/(x*x);
	412	r = p[0]+z(p[1]+z(p[2]+z(p[3]+z(p[4]+z*p[5]))));
	413	s = one+z(q[0]+z(q[1]+z(q[2]+z(q[3]+z(q[4]+zq[5])))));
	414	return (-.125 + r/s)/x;
	415	}