text: /* origin: FreeBSD /usr/src/lib/msun/src/e_j1.c */
  /*
   * ====================================================
   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   *
   * Developed at SunSoft, a Sun Microsystems, Inc. business.
   * Permission to use, copy, modify, and distribute this
   * software is freely granted, provided that this notice
   * is preserved.
   * ====================================================
   */
  /* j1(x), 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;
   *         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
   *         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
   *         where for x in [0,2] (abs err less than 2**-65.89)
   *              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
   *              V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
   *         Note: For tiny x, 1/x dominate y1 and hence
   *              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
   *      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 "libm.h"
  
  static double pone(double), qone(double);
  
  static const double invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
      tpi = 6.36619772367581382433e-01;                       /* 0x3FE45F30, 0x6DC9C883 */
  
  static double common(uint32_t ix, double x, int y1, int sign) {
      double z, s, c, ss, cc;
  
      /*
       * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
       * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
       *
       * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
       * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
       * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
       */
      s = sin(x);
      if (y1)
          s = -s;
      c = cos(x);
      cc = s - c;
      if (ix < 0x7fe00000) {
          /* avoid overflow in 2*x */
          ss = -s - c;
          z = cos(2 * x);
          if (s * c > 0)
              cc = z / ss;
          else
              ss = z / cc;
          if (ix < 0x48000000) {
              if (y1)
                  ss = -ss;
              cc = pone(x) * cc - qone(x) * ss;
          }
      }
      if (sign)
          cc = -cc;
      return invsqrtpi * cc / sqrt(x);
  }
  
  /* R0/S0 on [0,2] */
  static const double r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
      r01 = 1.40705666955189706048e-03,                  /* 0x3F570D9F, 0x98472C61 */
      r02 = -1.59955631084035597520e-05,                 /* 0xBEF0C5C6, 0xBA169668 */
      r03 = 4.96727999609584448412e-08,                  /* 0x3E6AAAFA, 0x46CA0BD9 */
      s01 = 1.91537599538363460805e-02,                  /* 0x3F939D0B, 0x12637E53 */
      s02 = 1.85946785588630915560e-04,                  /* 0x3F285F56, 0xB9CDF664 */
      s03 = 1.17718464042623683263e-06,                  /* 0x3EB3BFF8, 0x333F8498 */
      s04 = 5.04636257076217042715e-09,                  /* 0x3E35AC88, 0xC97DFF2C */
      s05 = 1.23542274426137913908e-11;                  /* 0x3DAB2ACF, 0xCFB97ED8 */
  
  double j1(double x) {
      double z, r, s;
      uint32_t ix;
      int sign;
  
      GET_HIGH_WORD(ix, x);
      sign = ix >> 31;
      ix &= 0x7fffffff;
      if (ix >= 0x7ff00000)
          return 1 / (x * x);
      if (ix >= 0x40000000) /* |x| >= 2 */
          return common(ix, fabs(x), 0, sign);
      if (ix >= 0x38000000) { /* |x| >= 2**-127 */
          z = x * x;
          r = z * (r00 + z * (r01 + z * (r02 + z * r03)));
          s = 1 + z * (s01 + z * (s02 + z * (s03 + z * (s04 + z * s05))));
          z = r / s;
      } else
          /* avoid underflow, raise inexact if x!=0 */
          z = x;
      return (0.5 + z) * x;
  }
  
  static const double U0[5] = {
      -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
      5.04438716639811282616e-02,  /* 0x3FA9D3C7, 0x76292CD1 */
      -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
      2.35252600561610495928e-05,  /* 0x3EF8AB03, 0x8FA6B88E */
      -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
  };
  static const double V0[5] = {
      1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
      2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
      1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
      6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
      1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
  };
  
  double y1(double x) {
      double z, u, v;
      uint32_t ix, lx;
  
      EXTRACT_WORDS(ix, lx, x);
      /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */
      if ((ix << 1 | lx) == 0)
          return -1 / 0.0;
      if (ix >> 31)
          return 0 / 0.0;
      if (ix >= 0x7ff00000)
          return 1 / x;
  
      if (ix >= 0x40000000) /* x >= 2 */
          return common(ix, x, 1, 0);
      if (ix < 0x3c900000) /* x < 2**-54 */
          return -tpi / x;
      z = x * x;
      u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
      v = 1 + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
      return x * (u / v) + tpi * (j1(x) * log(x) - 1 / 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)
   * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
   *        S = 1 + ps0*s^2 + ... + ps4*s^10
   * and
   *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
   */
  
  static const double pr8[6] = {
      /* for x in [inf, 8]=1/[0,0.125] */
      0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
      1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
      1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
      4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
      3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
      7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
  };
  static const double ps8[5] = {
      1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
      3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
      3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
      9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
      3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
  };
  
  static const double pr5[6] = {
      /* for x in [8,4.5454]=1/[0.125,0.22001] */
      1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
      1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
      6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
      1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
      5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
      5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
  };
  static const double ps5[5] = {
      5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
      9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
      5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
      7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
      1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
  };
  
  static const double pr3[6] = {
      3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
      1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
      3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
      3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
      9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
      4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
  };
  static const double ps3[5] = {
      3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
      3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
      1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
      8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
      1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
  };
  
  static const double pr2[6] = {
      /* for x in [2.8570,2]=1/[0.3499,0.5] */
      1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
      1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
      2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
      1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
      1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
      5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
  };
  static const double ps2[5] = {
      2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
      1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
      2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
      1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
      8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
  };
  
  static double pone(double x) {
      const double *p, *q;
      double_t z, r, s;
      uint32_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 /*ix >= 0x40000000*/ {
          p = pr2;
          q = ps2;
      }
      z = 1.0 / (x * x);
      r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
      s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
      return 1.0 + 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))
   * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
   *        S = 1 + qs1*s^2 + ... + qs6*s^12
   * and
   *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
   */
  
  static const double qr8[6] = {
      /* for x in [inf, 8]=1/[0,0.125] */
      0.00000000000000000000e+00,  /* 0x00000000, 0x00000000 */
      -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
      -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
      -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
      -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
      -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
  };
  static const double qs8[6] = {
      1.61395369700722909556e+02,  /* 0x40642CA6, 0xDE5BCDE5 */
      7.82538599923348465381e+03,  /* 0x40BE9162, 0xD0D88419 */
      1.33875336287249578163e+05,  /* 0x4100579A, 0xB0B75E98 */
      7.19657723683240939863e+05,  /* 0x4125F653, 0x72869C19 */
      6.66601232617776375264e+05,  /* 0x412457D2, 0x7719AD5C */
      -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
  };
  
  static const double qr5[6] = {
      /* for x in [8,4.5454]=1/[0.125,0.22001] */
      -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
      -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
      -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
      -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
      -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
      -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
  };
  static const double qs5[6] = {
      8.12765501384335777857e+01,  /* 0x405451B2, 0xFF5A11B2 */
      1.99179873460485964642e+03,  /* 0x409F1F31, 0xE77BF839 */
      1.74684851924908907677e+04,  /* 0x40D10F1F, 0x0D64CE29 */
      4.98514270910352279316e+04,  /* 0x40E8576D, 0xAABAD197 */
      2.79480751638918118260e+04,  /* 0x40DB4B04, 0xCF7C364B */
      -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
  };
  
  static const double qr3[6] = {
      -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
      -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
      -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
      -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
      -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
      -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
  };
  static const double qs3[6] = {
      4.76651550323729509273e+01,  /* 0x4047D523, 0xCCD367E4 */
      6.73865112676699709482e+02,  /* 0x40850EEB, 0xC031EE3E */
      3.38015286679526343505e+03,  /* 0x40AA684E, 0x448E7C9A */
      5.54772909720722782367e+03,  /* 0x40B5ABBA, 0xA61D54A6 */
      1.90311919338810798763e+03,  /* 0x409DBC7A, 0x0DD4DF4B */
      -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
  };
  
  static const double qr2[6] = {
      /* for x in [2.8570,2]=1/[0.3499,0.5] */
      -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
      -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
      -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
      -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
      -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
      -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
  };
  static const double qs2[6] = {
      2.95333629060523854548e+01,  /* 0x403D888A, 0x78AE64FF */
      2.52981549982190529136e+02,  /* 0x406F9F68, 0xDB821CBA */
      7.57502834868645436472e+02,  /* 0x4087AC05, 0xCE49A0F7 */
      7.39393205320467245656e+02,  /* 0x40871B25, 0x48D4C029 */
      1.55949003336666123687e+02,  /* 0x40637E5E, 0x3C3ED8D4 */
      -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
  };
  
  static double qone(double x) {
      const double *p, *q;
      double_t s, r, z;
      uint32_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 /*ix >= 0x40000000*/ {
          p = qr2;
          q = qs2;
      }
      z = 1.0 / (x * x);
      r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
      s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
      return (.375 + r / s) / x;
  }
  