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e_j1l.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_j1(x), __ieee754_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;
 *       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
 *       Note: For tiny x, 1/x dominate y1 and hence
 *          y1(tiny) = -2/pi/tiny
 *    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 "math.h"
#include "math_private.h"

#ifdef __STDC__
static long double pone (long double), qone (long double);
#else
static long double pone (), qone ();
#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,

  /* J1(x) = .5 x + x x^2 R(x^2) / S(x^2)
     0 <= x <= 2
     Peak relative error 4.5e-21 */
R[5] = {
    -9.647406112428107954753770469290757756814E7L,
    2.686288565865230690166454005558203955564E6L,
    -3.689682683905671185891885948692283776081E4L,
    2.195031194229176602851429567792676658146E2L,
    -5.124499848728030297902028238597308971319E-1L,
},

  S[4] =
{
  1.543584977988497274437410333029029035089E9L,
  2.133542369567701244002565983150952549520E7L,
  1.394077011298227346483732156167414670520E5L,
  5.252401789085732428842871556112108446506E2L,
  /*  1.000000000000000000000000000000000000000E0L, */
};

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


#ifdef __STDC__
long double
__ieee754_j1l (long double x)
#else
long double
__ieee754_j1l (x)
     long double x;
#endif
{
  long double z, c, r, s, ss, cc, u, v, y;
  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;
  y = fabsl (x);
  if (ix >= 0x4000)
    {                   /* |x| >= 2.0 */
      __sincosl (y, &s, &c);
      ss = -s - c;
      cc = s - c;
      if (ix < 0x7ffe)
      {                 /* make sure y+y not overflow */
        z = __cosl (y + y);
        if ((s * c) > zero)
          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 > 0x4080)
      z = (invsqrtpi * cc) / __ieee754_sqrtl (y);
      else
      {
        u = pone (y);
        v = qone (y);
        z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrtl (y);
      }
      if (se & 0x8000)
      return -z;
      else
      return z;
    }
  if (ix < 0x3fde) /* |x| < 2^-33 */
    {
      if (huge + x > one)
      return 0.5 * x;         /* inexact if x!=0 necessary */
    }
  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)));
  r *= x;
  return (x * 0.5 + r / s);
}


/* Y1(x) = 2/pi * (log(x) * j1(x) - 1/x) + x R(x^2)
   0 <= x <= 2
   Peak relative error 2.3e-23 */
#ifdef __STDC__
static const long double U0[6] = {
#else
static long double U0[6] = {
#endif
  -5.908077186259914699178903164682444848615E10L,
  1.546219327181478013495975514375773435962E10L,
  -6.438303331169223128870035584107053228235E8L,
  9.708540045657182600665968063824819371216E6L,
  -6.138043997084355564619377183564196265471E4L,
  1.418503228220927321096904291501161800215E2L,
};
#ifdef __STDC__
static const long double V0[5] = {
#else
static long double V0[5] = {
#endif
  3.013447341682896694781964795373783679861E11L,
  4.669546565705981649470005402243136124523E9L,
  3.595056091631351184676890179233695857260E7L,
  1.761554028569108722903944659933744317994E5L,
  5.668480419646516568875555062047234534863E2L,
  /*  1.000000000000000000000000000000000000000E0L, */
};


#ifdef __STDC__
long double
__ieee754_y1l (long double x)
#else
long double
__ieee754_y1l (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;
  /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(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 */
      __sincosl (x, &s, &c);
      ss = -s - c;
      cc = s - c;
      if (ix < 0x7fe00000)
      {                 /* make sure x+x not overflow */
        z = __cosl (x + x);
        if ((s * c) > zero)
          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 > 0x4080)
      z = (invsqrtpi * ss) / __ieee754_sqrtl (x);
      else
      {
        u = pone (x);
        v = qone (x);
        z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrtl (x);
      }
      return z;
    }
  if (ix <= 0x3fbe)
    {                   /* x < 2**-65 */
      return (-tpi / x);
    }
  z = x * x;
 u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * (U0[4] + z * U0[5]))));
 v = V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * (V0[4] + z))));
  return (x * (u / v) +
        tpi * (__ieee754_j1l (x) * __ieee754_logl (x) - one / 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)
 */

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   8 <= x <= inf  (0 <= z <= 0.125)
   Peak relative error 5.2e-22  */

#ifdef __STDC__
static const long double pr8[7] = {
#else
static long double pr8[7] = {
#endif
  8.402048819032978959298664869941375143163E-9L,
  1.813743245316438056192649247507255996036E-6L,
  1.260704554112906152344932388588243836276E-4L,
  3.439294839869103014614229832700986965110E-3L,
  3.576910849712074184504430254290179501209E-2L,
  1.131111483254318243139953003461511308672E-1L,
  4.480715825681029711521286449131671880953E-2L,
};
#ifdef __STDC__
static const long double ps8[6] = {
#else
static long double ps8[6] = {
#endif
  7.169748325574809484893888315707824924354E-8L,
  1.556549720596672576431813934184403614817E-5L,
  1.094540125521337139209062035774174565882E-3L,
  3.060978962596642798560894375281428805840E-2L,
  3.374146536087205506032643098619414507024E-1L,
  1.253830208588979001991901126393231302559E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   4.54541015625 <= x <= 8
   Peak relative error 7.7e-22  */
#ifdef __STDC__
static const long double pr5[7] = {
#else
static long double pr5[7] = {
#endif
  4.318486887948814529950980396300969247900E-7L,
  4.715341880798817230333360497524173929315E-5L,
  1.642719430496086618401091544113220340094E-3L,
  2.228688005300803935928733750456396149104E-2L,
  1.142773760804150921573259605730018327162E-1L,
  1.755576530055079253910829652698703791957E-1L,
  3.218803858282095929559165965353784980613E-2L,
};
#ifdef __STDC__
static const long double ps5[6] = {
#else
static long double ps5[6] = {
#endif
  3.685108812227721334719884358034713967557E-6L,
  4.069102509511177498808856515005792027639E-4L,
  1.449728676496155025507893322405597039816E-2L,
  2.058869213229520086582695850441194363103E-1L,
  1.164890985918737148968424972072751066553E0L,
  2.274776933457009446573027260373361586841E0L,
  /*  1.000000000000000000000000000000000000000E0L,*/
};

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   2.85711669921875 <= x <= 4.54541015625
   Peak relative error 6.5e-21  */
#ifdef __STDC__
static const long double pr3[7] = {
#else
static long double pr3[7] = {
#endif
  1.265251153957366716825382654273326407972E-5L,
  8.031057269201324914127680782288352574567E-4L,
  1.581648121115028333661412169396282881035E-2L,
  1.179534658087796321928362981518645033967E-1L,
  3.227936912780465219246440724502790727866E-1L,
  2.559223765418386621748404398017602935764E-1L,
  2.277136933287817911091370397134882441046E-2L,
};
#ifdef __STDC__
static const long double ps3[6] = {
#else
static long double ps3[6] = {
#endif
  1.079681071833391818661952793568345057548E-4L,
  6.986017817100477138417481463810841529026E-3L,
  1.429403701146942509913198539100230540503E-1L,
  1.148392024337075609460312658938700765074E0L,
  3.643663015091248720208251490291968840882E0L,
  3.990702269032018282145100741746633960737E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* J1(x) cosX + Y1(x) sinX  =  sqrt( 2/(pi x)) P1(x)
   P1(x) = 1 + z^2 R(z^2), z=1/x
   2 <= x <= 2.85711669921875
   Peak relative error 3.5e-21  */
#ifdef __STDC__
static const long double pr2[7] = {
#else
static long double pr2[7] = {
#endif
  2.795623248568412225239401141338714516445E-4L,
  1.092578168441856711925254839815430061135E-2L,
  1.278024620468953761154963591853679640560E-1L,
  5.469680473691500673112904286228351988583E-1L,
  8.313769490922351300461498619045639016059E-1L,
  3.544176317308370086415403567097130611468E-1L,
  1.604142674802373041247957048801599740644E-2L,
};
#ifdef __STDC__
static const long double ps2[6] = {
#else
static long double ps2[6] = {
#endif
  2.385605161555183386205027000675875235980E-3L,
  9.616778294482695283928617708206967248579E-2L,
  1.195215570959693572089824415393951258510E0L,
  5.718412857897054829999458736064922974662E0L,
  1.065626298505499086386584642761602177568E1L,
  6.809140730053382188468983548092322151791E0L,
 /* 1.000000000000000000000000000000000000000E0L, */
};


#ifdef __STDC__
static long double
pone (long double x)
#else
static long double
pone (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) /* x >= 8 */
    {
      p = pr8;
      q = ps8;
    }
  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 qone is
 *    3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 * We approximate pone by
 *    qone(x) = s*(0.375 + (R/S))
 */

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   8 <= x <= inf
   Peak relative error 8.3e-22 */

#ifdef __STDC__
static const long double qr8[7] = {
#else
static long double qr8[7] = {
#endif
  -5.691925079044209246015366919809404457380E-10L,
  -1.632587664706999307871963065396218379137E-7L,
  -1.577424682764651970003637263552027114600E-5L,
  -6.377627959241053914770158336842725291713E-4L,
  -1.087408516779972735197277149494929568768E-2L,
  -6.854943629378084419631926076882330494217E-2L,
  -1.055448290469180032312893377152490183203E-1L,
};
#ifdef __STDC__
static const long double qs8[7] = {
#else
static long double qs8[7] = {
#endif
  5.550982172325019811119223916998393907513E-9L,
  1.607188366646736068460131091130644192244E-6L,
  1.580792530091386496626494138334505893599E-4L,
  6.617859900815747303032860443855006056595E-3L,
  1.212840547336984859952597488863037659161E-1L,
  9.017885953937234900458186716154005541075E-1L,
  2.201114489712243262000939120146436167178E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   4.54541015625 <= x <= 8
   Peak relative error 4.1e-22 */
#ifdef __STDC__
static const long double qr5[7] = {
#else
static long double qr5[7] = {
#endif
  -6.719134139179190546324213696633564965983E-8L,
  -9.467871458774950479909851595678622044140E-6L,
  -4.429341875348286176950914275723051452838E-4L,
  -8.539898021757342531563866270278505014487E-3L,
  -6.818691805848737010422337101409276287170E-2L,
  -1.964432669771684034858848142418228214855E-1L,
  -1.333896496989238600119596538299938520726E-1L,
};
#ifdef __STDC__
static const long double qs5[7] = {
#else
static long double qs5[7] = {
#endif
  6.552755584474634766937589285426911075101E-7L,
  9.410814032118155978663509073200494000589E-5L,
  4.561677087286518359461609153655021253238E-3L,
  9.397742096177905170800336715661091535805E-2L,
  8.518538116671013902180962914473967738771E-1L,
  3.177729183645800174212539541058292579009E0L,
  4.006745668510308096259753538973038902990E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   2.85711669921875 <= x <= 4.54541015625
   Peak relative error 2.2e-21 */
#ifdef __STDC__
static const long double qr3[7] = {
#else
static long double qr3[7] = {
#endif
  -3.618746299358445926506719188614570588404E-6L,
  -2.951146018465419674063882650970344502798E-4L,
  -7.728518171262562194043409753656506795258E-3L,
  -8.058010968753999435006488158237984014883E-2L,
  -3.356232856677966691703904770937143483472E-1L,
  -4.858192581793118040782557808823460276452E-1L,
  -1.592399251246473643510898335746432479373E-1L,
};
#ifdef __STDC__
static const long double qs3[7] = {
#else
static long double qs3[7] = {
#endif
  3.529139957987837084554591421329876744262E-5L,
  2.973602667215766676998703687065066180115E-3L,
  8.273534546240864308494062287908662592100E-2L,
  9.613359842126507198241321110649974032726E-1L,
  4.853923697093974370118387947065402707519E0L,
  1.002671608961669247462020977417828796933E1L,
  7.028927383922483728931327850683151410267E0L,
  /* 1.000000000000000000000000000000000000000E0L, */
};

/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
   Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
   2 <= x <= 2.85711669921875
   Peak relative error 6.9e-22 */
#ifdef __STDC__
static const long double qr2[7] = {
#else
static long double qr2[7] = {
#endif
  -1.372751603025230017220666013816502528318E-4L,
  -6.879190253347766576229143006767218972834E-3L,
  -1.061253572090925414598304855316280077828E-1L,
  -6.262164224345471241219408329354943337214E-1L,
  -1.423149636514768476376254324731437473915E0L,
  -1.087955310491078933531734062917489870754E0L,
  -1.826821119773182847861406108689273719137E-1L,
};
#ifdef __STDC__
static const long double qs2[7] = {
#else
static long double qs2[7] = {
#endif
  1.338768933634451601814048220627185324007E-3L,
  7.071099998918497559736318523932241901810E-2L,
  1.200511429784048632105295629933382142221E0L,
  8.327301713640367079030141077172031825276E0L,
  2.468479301872299311658145549931764426840E1L,
  2.961179686096262083509383820557051621644E1L,
  1.201402313144305153005639494661767354977E1L,
 /* 1.000000000000000000000000000000000000000E0L, */
};


#ifdef __STDC__
static long double
qone (long double x)
#else
static long double
qone (x)
     long double x;
#endif
{
#ifdef __STDC__
  const long double *p, *q;
#else
  long double *p, *q;
#endif
  static 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 (.375 + z * r / s) / x;
}

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