glibc/sysdeps/ieee754/ldbl-96/e_j0l.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.
 * ====================================================
 */

/* Long double expansions contributed by
   Stephen L. Moshier <moshier@na-net.ornl.gov>  */

/* __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;
 *	   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
 *
 *	   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"

#ifdef __STDC__
static long double pzero (long double), qzero (long double);
#else
static long double pzero (), qzero ();
#endif

#ifdef __STDC__
static const long double
#else
static long double
#endif
  huge = 1e4930L,
  one = 1.0L,
  invsqrtpi = 5.6418958354775628694807945156077258584405e-1L,
  tpi = 6.3661977236758134307553505349005744813784e-1L,
  j0z1 = 2.40482555769577276862163187932650662155139L,
  j0z2 = 5.520078110286310649596604112813027425221865L,

  /* J0(x) = 1 - x^2 / 4 + x^4 R0(x^2) / S0(x^2)
     0 <= x <= 2
     peak relative error 1.41e-22 */
  R[5] = {
  4.287176872744686992880841716723478740566E7L,
  -6.652058897474241627570911531740907185772E5L,
  7.011848381719789863458364584613651091175E3L,
  -3.168040850193372408702135490809516253693E1L,
  6.030778552661102450545394348845599300939E-2L,
},

 S[4] = {
   2.743793198556599677955266341699130654342E9L,
   3.364330079384816249840086842058954076201E7L,
   1.924119649412510777584684927494642526573E5L,
   6.239282256012734914211715620088714856494E2L,
   /*   1.000000000000000000000000000000000000000E0L,*/
};

#ifdef __STDC__
static const long double zero = 0.0;
#else
static long double zero = 0.0;
#endif

#ifdef __STDC__
long double
__ieee754_j0l (long double x)
#else
long double
__ieee754_j0l (x)
     long double x;
#endif
{
  long double z, s, c, ss, cc, r, u, v;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x7fff)
    return one / (x * x);
  x = fabsl (x);
  if (ix >= 0x4000)		/* |x| >= 2.0 */
    {
      __sincosl (x, &s, &c);
      ss = s - c;
      cc = s + c;
      if (ix < 0x7ffe)
	{			/* make sure x+x not overflow */
	  z = -__cosl (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 > 0x4080)	/* 2^129 */
	z = (invsqrtpi * cc) / __ieee754_sqrtl (x);
      else
	{
	  u = pzero (x);
	  v = qzero (x);
	  z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrtl (x);
	}
      return z;
    }
  if (ix < 0x3fef) /* |x| < 2**-16 */
    {
      if (huge + x > one)
	{			/* raise inexact if x != 0 */
	  if (ix < 0x3fde) /* |x| < 2^-33 */
	    return one;
	  else
	    return one - 0.25 * x * x;
	}
    }
  z = x * x;
  r = z * (R[0] + z * (R[1] + z * (R[2] + z * (R[3] + z * R[4]))));
  s = S[0] + z * (S[1] + z * (S[2] + z * (S[3] + z)));
  if (ix < 0x3fff)
    {				/* |x| < 1.00 */
      return (one - 0.25 * z + z * (r / s));
    }
  else
    {
      u = 0.5 * x;
      return ((one + u) * (one - u) + z * (r / s));
    }
}


/* y0(x) = 2/pi ln(x) J0(x) + U(x^2)/V(x^2)
   0 < x <= 2
   peak relative error 1.7e-21 */
#ifdef __STDC__
static const long double
#else
static long double
#endif
U[6] = {
  -1.054912306975785573710813351985351350861E10L,
  2.520192609749295139432773849576523636127E10L,
  -1.856426071075602001239955451329519093395E9L,
  4.079209129698891442683267466276785956784E7L,
  -3.440684087134286610316661166492641011539E5L,
  1.005524356159130626192144663414848383774E3L,
};
#ifdef __STDC__
static const long double
#else
static long double
#endif
V[5] = {
  1.429337283720789610137291929228082613676E11L,
  2.492593075325119157558811370165695013002E9L,
  2.186077620785925464237324417623665138376E7L,
  1.238407896366385175196515057064384929222E5L,
  4.693924035211032457494368947123233101664E2L,
  /*  1.000000000000000000000000000000000000000E0L */
};

#ifdef __STDC__
long double
__ieee754_y0l (long double x)
#else
long double
__ieee754_y0l (x)
     long double x;
#endif
{
  long double z, s, c, ss, cc, u, v;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
  if (se & 0x8000)
    return zero / zero;
  if (ix >= 0x7fff)
    return one / (x + x * x);
  if ((i0 | i1) == 0)
    return -one / zero;
  if (ix >= 0x4000)
    {				/* |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.
       */
      __sincosl (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 < 0x7ffe)
	{			/* make sure x+x not overflow */
	  z = -__cosl (x + x);
	  if ((s * c) < zero)
	    cc = z / ss;
	  else
	    ss = z / cc;
	}
      if (ix > 0x4080)	/* 1e39 */
	z = (invsqrtpi * ss) / __ieee754_sqrtl (x);
      else
	{
	  u = pzero (x);
	  v = qzero (x);
	  z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrtl (x);
	}
      return z;
    }
  if (ix <= 0x3fde) /* x < 2^-33 */
    {
      z = -7.380429510868722527629822444004602747322E-2L
	+ tpi * __ieee754_logl (x);
      return z;
    }
  z = x * x;
  u = U[0] + z * (U[1] + z * (U[2] + z * (U[3] + z * (U[4] + z * U[5]))));
  v = V[0] + z * (V[1] + z * (V[2] + z * (V[3] + z * (V[4] + z))));
  return (u / v + tpi * (__ieee754_j0l (x) * __ieee754_logl (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 + s^2 R(s^2) / S(s^2)
 */
#ifdef __STDC__
static const long double pR8[7] = {
#else
static long double pR8[7] = {
#endif
  /* 8 <= x <= inf
     Peak relative error 4.62 */
  -4.094398895124198016684337960227780260127E-9L,
  -8.929643669432412640061946338524096893089E-7L,
  -6.281267456906136703868258380673108109256E-5L,
  -1.736902783620362966354814353559382399665E-3L,
  -1.831506216290984960532230842266070146847E-2L,
  -5.827178869301452892963280214772398135283E-2L,
  -2.087563267939546435460286895807046616992E-2L,
};
#ifdef __STDC__
static const long double pS8[6] = {
#else
static long double pS8[6] = {
#endif
  5.823145095287749230197031108839653988393E-8L,
  1.279281986035060320477759999428992730280E-5L,
  9.132668954726626677174825517150228961304E-4L,
  2.606019379433060585351880541545146252534E-2L,
  2.956262215119520464228467583516287175244E-1L,
  1.149498145388256448535563278632697465675E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static const long double pR5[7] = {
#else
static long double pR5[7] = {
#endif
  /* 4.54541015625 <= x <= 8
     Peak relative error 6.51E-22 */
  -2.041226787870240954326915847282179737987E-7L,
  -2.255373879859413325570636768224534428156E-5L,
  -7.957485746440825353553537274569102059990E-4L,
  -1.093205102486816696940149222095559439425E-2L,
  -5.657957849316537477657603125260701114646E-2L,
  -8.641175552716402616180994954177818461588E-2L,
  -1.354654710097134007437166939230619726157E-2L,
};
#ifdef __STDC__
static const long double pS5[6] = {
#else
static long double pS5[6] = {
#endif
  2.903078099681108697057258628212823545290E-6L,
  3.253948449946735405975737677123673867321E-4L,
  1.181269751723085006534147920481582279979E-2L,
  1.719212057790143888884745200257619469363E-1L,
  1.006306498779212467670654535430694221924E0L,
  2.069568808688074324555596301126375951502E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static const long double pR3[7] = {
#else
static long double pR3[7] = {
#endif
  /* 2.85711669921875 <= x <= 4.54541015625
     peak relative error 5.25e-21 */
  -5.755732156848468345557663552240816066802E-6L,
  -3.703675625855715998827966962258113034767E-4L,
  -7.390893350679637611641350096842846433236E-3L,
  -5.571922144490038765024591058478043873253E-2L,
  -1.531290690378157869291151002472627396088E-1L,
  -1.193350853469302941921647487062620011042E-1L,
  -8.567802507331578894302991505331963782905E-3L,
};
#ifdef __STDC__
static const long double pS3[6] = {
#else
static long double pS3[6] = {
#endif
  8.185931139070086158103309281525036712419E-5L,
  5.398016943778891093520574483111255476787E-3L,
  1.130589193590489566669164765853409621081E-1L,
  9.358652328786413274673192987670237145071E-1L,
  3.091711512598349056276917907005098085273E0L,
  3.594602474737921977972586821673124231111E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static const long double pR2[7] = {
#else
static long double pR2[7] = {
#endif
  /* 2 <= x <= 2.85711669921875
     peak relative error 2.64e-21 */
  -1.219525235804532014243621104365384992623E-4L,
  -4.838597135805578919601088680065298763049E-3L,
  -5.732223181683569266223306197751407418301E-2L,
  -2.472947430526425064982909699406646503758E-1L,
  -3.753373645974077960207588073975976327695E-1L,
  -1.556241316844728872406672349347137975495E-1L,
  -5.355423239526452209595316733635519506958E-3L,
};
#ifdef __STDC__
static const long double pS2[6] = {
#else
static long double pS2[6] = {
#endif
  1.734442793664291412489066256138894953823E-3L,
  7.158111826468626405416300895617986926008E-2L,
  9.153839713992138340197264669867993552641E-1L,
  4.539209519433011393525841956702487797582E0L,
  8.868932430625331650266067101752626253644E0L,
  6.067161890196324146320763844772857713502E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static long double
pzero (long double x)
#else
static long double
pzero (x)
     long double x;
#endif
{
#ifdef __STDC__
  const long double *p, *q;
#else
  long double *p, *q;
#endif
  long double z, r, s;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x4002)
    {
      p = pR8;
      q = pS8;
    }				/* x >= 8 */
  else
    {
      i1 = (ix << 16) | (i0 >> 16);
      if (i1 >= 0x40019174)	/* x >= 4.54541015625 */
	{
	  p = pR5;
	  q = pS5;
	}
      else if (i1 >= 0x4000b6db)	/* x >= 2.85711669921875 */
	{
	  p = pR3;
	  q = pS3;
	}
      else if (ix >= 0x4000)	/* x better be >= 2 */
	{
	  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] + z * p[6])))));
  s =
    q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * (q[5] + z)))));
  return (one + z * r / s);
}


/* For x >= 8, the asymptotic expansions of qzero is
 *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 * We approximate qzero by
 * 	qzero(x) = s*(-.125 + R(s^2) / S(s^2))
 */
#ifdef __STDC__
static const long double qR8[7] = {
#else
static long double qR8[7] = {
#endif
  /* 8 <= x <= inf
     peak relative error 2.23e-21 */
  3.001267180483191397885272640777189348008E-10L,
  8.693186311430836495238494289942413810121E-8L,
  8.496875536711266039522937037850596580686E-6L,
  3.482702869915288984296602449543513958409E-4L,
  6.036378380706107692863811938221290851352E-3L,
  3.881970028476167836382607922840452192636E-2L,
  6.132191514516237371140841765561219149638E-2L,
};
#ifdef __STDC__
static const long double qS8[7] = {
#else
static long double qS8[7] = {
#endif
  4.097730123753051126914971174076227600212E-9L,
  1.199615869122646109596153392152131139306E-6L,
  1.196337580514532207793107149088168946451E-4L,
  5.099074440112045094341500497767181211104E-3L,
  9.577420799632372483249761659674764460583E-2L,
  7.385243015344292267061953461563695918646E-1L,
  1.917266424391428937962682301561699055943E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static const long double qR5[7] = {
#else
static long double qR5[7] = {
#endif
  /* 4.54541015625 <= x <= 8
     peak relative error 1.03e-21 */
  3.406256556438974327309660241748106352137E-8L,
  4.855492710552705436943630087976121021980E-6L,
  2.301011739663737780613356017352912281980E-4L,
  4.500470249273129953870234803596619899226E-3L,
  3.651376459725695502726921248173637054828E-2L,
  1.071578819056574524416060138514508609805E-1L,
  7.458950172851611673015774675225656063757E-2L,
};
#ifdef __STDC__
static const long double qS5[7] = {
#else
static long double qS5[7] = {
#endif
  4.650675622764245276538207123618745150785E-7L,
  6.773573292521412265840260065635377164455E-5L,
  3.340711249876192721980146877577806687714E-3L,
  7.036218046856839214741678375536970613501E-2L,
  6.569599559163872573895171876511377891143E-1L,
  2.557525022583599204591036677199171155186E0L,
  3.457237396120935674982927714210361269133E0L,
  /* 1.000000000000000000000000000000000000000E0L,*/
};

#ifdef __STDC__
static const long double qR3[7] = {
#else
static long double qR3[7] = {
#endif
  /* 2.85711669921875 <= x <= 4.54541015625
     peak relative error 5.24e-21 */
  1.749459596550816915639829017724249805242E-6L,
  1.446252487543383683621692672078376929437E-4L,
  3.842084087362410664036704812125005761859E-3L,
  4.066369994699462547896426554180954233581E-2L,
  1.721093619117980251295234795188992722447E-1L,
  2.538595333972857367655146949093055405072E-1L,
  8.560591367256769038905328596020118877936E-2L,
};
#ifdef __STDC__
static const long double qS3[7] = {
#else
static long double qS3[7] = {
#endif
  2.388596091707517488372313710647510488042E-5L,
  2.048679968058758616370095132104333998147E-3L,
  5.824663198201417760864458765259945181513E-2L,
  6.953906394693328750931617748038994763958E-1L,
  3.638186936390881159685868764832961092476E0L,
  7.900169524705757837298990558459547842607E0L,
  5.992718532451026507552820701127504582907E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static const long double qR2[7] = {
#else
static long double qR2[7] = {
#endif
  /* 2 <= x <= 2.85711669921875
     peak relative error 1.58e-21  */
  6.306524405520048545426928892276696949540E-5L,
  3.209606155709930950935893996591576624054E-3L,
  5.027828775702022732912321378866797059604E-2L,
  3.012705561838718956481911477587757845163E-1L,
  6.960544893905752937420734884995688523815E-1L,
  5.431871999743531634887107835372232030655E-1L,
  9.447736151202905471899259026430157211949E-2L,
};
#ifdef __STDC__
static const long double qS2[7] = {
#else
static long double qS2[7] = {
#endif
  8.610579901936193494609755345106129102676E-4L,
  4.649054352710496997203474853066665869047E-2L,
  8.104282924459837407218042945106320388339E-1L,
  5.807730930825886427048038146088828206852E0L,
  1.795310145936848873627710102199881642939E1L,
  2.281313316875375733663657188888110605044E1L,
  1.011242067883822301487154844458322200143E1L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

#ifdef __STDC__
static long double
qzero (long double x)
#else
static long double
qzero (x)
     long double x;
#endif
{
#ifdef __STDC__
  const long double *p, *q;
#else
  long double *p, *q;
#endif
  long double s, r, z;
  int32_t ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x4002)		/* x >= 8 */
    {
      p = qR8;
      q = qS8;
    }
  else
    {
      i1 = (ix << 16) | (i0 >> 16);
      if (i1 >= 0x40019174)	/* x >= 4.54541015625 */
	{
	  p = qR5;
	  q = qS5;
	}
      else if (i1 >= 0x4000b6db)	/* x >= 2.85711669921875 */
	{
	  p = qR3;
	  q = qS3;
	}
      else if (ix >= 0x4000)	/* x better be >= 2 */
	{
	  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] + z * p[6])))));
  s =
    q[0] + z * (q[1] +
		z * (q[2] +
		     z * (q[3] + z * (q[4] + z * (q[5] + z * (q[6] + z))))));
  return (-.125 + z * r / s) / x;
}