glibc/sysdeps/ieee754/dbl-64/e_j0.c

/* @(#)e_j0.c 5.1 93/09/24 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
   for performance improvement on pipelined processors.
 */

/* __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-barriers.h>
#include <math_private.h>
#include <libm-alias-finite.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] */
  R[] = { 0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
	  -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
	  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
	  -4.61832688532103189199e-09 }, /* 0xBE33D5E7, 0x73D63FCE */
  S[] = { 0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
	  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
	  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
	  1.16614003333790000205e-09 }; /* 0x3E1408BC, 0xF4745D8F */

static const double zero = 0.0;

double
__ieee754_j0 (double x)
{
  double z, s, c, ss, cc, r, u, v, r1, r2, s1, s2, z2, z4;
  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 */
    {
      __sincos (x, &s, &c);
      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 */
    {
      math_force_eval (huge + x);       /* raise inexact if x != 0 */
      if (ix < 0x3e400000)
	return one;                     /* |x|<2**-27 */
      else
	return one - 0.25 * x * x;
    }
  z = x * x;
  r1 = z * R[2]; z2 = z * z;
  r2 = R[3] + z * R[4]; z4 = z2 * z2;
  r = r1 + z2 * r2 + z4 * R[5];
  s1 = one + z * S[1];
  s2 = S[2] + z * S[3];
  s = s1 + z2 * s2 + z4 * S[4];
  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));
    }
}
libm_alias_finite (__ieee754_j0, __j0)

static const double
U[] = { -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
	 1.76666452509181115538e-01,  /* 0x3FC69D01, 0x9DE9E3FC */
	-1.38185671945596898896e-02,  /* 0xBF8C4CE8, 0xB16CFA97 */
	 3.47453432093683650238e-04,  /* 0x3F36C54D, 0x20B29B6B */
	-3.81407053724364161125e-06,  /* 0xBECFFEA7, 0x73D25CAD */
	 1.95590137035022920206e-08,  /* 0x3E550057, 0x3B4EABD4 */
	-3.98205194132103398453e-11 }, /* 0xBDC5E43D, 0x693FB3C8 */
V[] = { 1.27304834834123699328e-02,   /* 0x3F8A1270, 0x91C9C71A */
	 7.60068627350353253702e-05,   /* 0x3F13ECBB, 0xF578C6C1 */
	 2.59150851840457805467e-07,   /* 0x3E91642D, 0x7FF202FD */
	 4.41110311332675467403e-10 }; /* 0x3DFE5018, 0x3BD6D9EF */

double
__ieee754_y0 (double x)
{
  double z, s, c, ss, cc, u, v, z2, z4, z6, u1, u2, u3, v1, v2;
  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, y0(0) is -inf.  */
  if (ix >= 0x7ff00000)
    return one / (x + x * x);
  if ((ix | lx) == 0)
    return -1 / zero; /* -inf and divide by zero exception.  */
  if (hx < 0)
    return zero / (zero * x);
  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.
		 */
      __sincos (x, &s, &c);
      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 (U[0] + tpi * __ieee754_log (x));
    }
  z = x * x;
  u1 = U[0] + z * U[1]; z2 = z * z;
  u2 = U[2] + z * U[3]; z4 = z2 * z2;
  u3 = U[4] + z * U[5]; z6 = z4 * z2;
  u = u1 + z2 * u2 + z4 * u3 + z6 * U[6];
  v1 = one + z * V[0];
  v2 = V[1] + z * V[2];
  v = v1 + z2 * v2 + z4 * V[3];
  return (u / v + tpi * (__ieee754_j0 (x) * __ieee754_log (x)));
}
libm_alias_finite (__ieee754_y0, __y0)

/* 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, z2, z4, r1, r2, r3, s1, s2, s3;
  int32_t ix;
  GET_HIGH_WORD (ix, x);
  ix &= 0x7fffffff;
  /* ix >= 0x40000000 for all calls to this function.  */
  if (ix >= 0x41b00000)
    {
      return one;
    }
  else if (ix >= 0x40200000)
    {
      p = pR8; q = pS8;
    }
  else if (ix >= 0x40122E8B)
    {
      p = pR5; q = pS5;
    }
  else if (ix >= 0x4006DB6D)
    {
      p = pR3; q = pS3;
    }
  else
    {
      p = pR2; q = pS2;
    }
  z = one / (x * x);
  r1 = p[0] + z * p[1]; z2 = z * z;
  r2 = p[2] + z * p[3]; z4 = z2 * z2;
  r3 = p[4] + z * p[5];
  r = r1 + z2 * r2 + z4 * r3;
  s1 = one + z * q[0];
  s2 = q[1] + z * q[2];
  s3 = q[3] + z * q[4];
  s = s1 + z2 * s2 + z4 * s3;
  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, z2, z4, z6, r1, r2, r3, s1, s2, s3;
  int32_t ix;
  GET_HIGH_WORD (ix, x);
  ix &= 0x7fffffff;
  /* ix >= 0x40000000 for all calls to this function.  */
  if (ix >= 0x41b00000)
    {
      return -.125 / x;
    }
  else if (ix >= 0x40200000)
    {
      p = qR8; q = qS8;
    }
  else if (ix >= 0x40122E8B)
    {
      p = qR5; q = qS5;
    }
  else if (ix >= 0x4006DB6D)
    {
      p = qR3; q = qS3;
    }
  else
    {
      p = qR2; q = qS2;
    }
  z = one / (x * x);
  r1 = p[0] + z * p[1]; z2 = z * z;
  r2 = p[2] + z * p[3]; z4 = z2 * z2;
  r3 = p[4] + z * p[5]; z6 = z4 * z2;
  r = r1 + z2 * r2 + z4 * r3;
  s1 = one + z * q[0];
  s2 = q[1] + z * q[2];
  s3 = q[3] + z * q[4];
  s = s1 + z2 * s2 + z4 * s3 + z6 * q[5];
  return (-.125 + r / s) / x;
}