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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 are
  Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
  and are incorporated herein by permission of the author.  The author 
  reserves the right to distribute this material elsewhere under different
  copying permissions.  These modifications are distributed here under 
  the following terms:

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */

/* __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,

  /* 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;
}

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