/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.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.
 * ====================================================
 */
/*
 * jn(n, x), yn(n, x)
 * floating point Bessel's function of the 1st and 2nd kind
 * of order n
 *
 * Special cases:
 *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
 *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
 * Note 2. About jn(n,x), yn(n,x)
 *      For n=0, j0(x) is called,
 *      for n=1, j1(x) is called,
 *      for n<=x, forward recursion is used starting
 *      from values of j0(x) and j1(x).
 *      for n>x, a continued fraction approximation to
 *      j(n,x)/j(n-1,x) is evaluated and then backward
 *      recursion is used starting from a supposed value
 *      for j(n,x). The resulting value of j(0,x) is
 *      compared with the actual value to correct the
 *      supposed value of j(n,x).
 *
 *      yn(n,x) is similar in all respects, except
 *      that forward recursion is used for all
 *      values of n>1.
 */

#include "libm.h"

static const double invsqrtpi =
    5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */

double jn(int n, double x) {
  uint32_t ix, lx;
  int nm1, i, sign;
  double a, b, temp;

  EXTRACT_WORDS(ix, lx, x);
  sign = ix >> 31;
  ix &= 0x7fffffff;

  if ((ix | (lx | -lx) >> 31) > 0x7ff00000) /* nan */
    return x;

  /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
   * Thus, J(-n,x) = J(n,-x)
   */
  /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
  if (n == 0)
    return j0(x);
  if (n < 0) {
    nm1 = -(n + 1);
    x = -x;
    sign ^= 1;
  } else
    nm1 = n - 1;
  if (nm1 == 0)
    return j1(x);

  sign &= n; /* even n: 0, odd n: signbit(x) */
  x = fabs(x);
  if ((ix | lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */
    b = 0.0;
  else if (nm1 < x) {
    /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
    if (ix >= 0x52d00000) { /* x > 2**302 */
                            /* (x >> n**2)
                             *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
                             *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
                             *      Let s=sin(x), c=cos(x),
                             *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
                             *
                             *             n    sin(xn)*sqt2    cos(xn)*sqt2
                             *          ----------------------------------
                             *             0     s-c             c+s
                             *             1    -s-c            -c+s
                             *             2    -s+c            -c-s
                             *             3     s+c             c-s
                             */
      switch (nm1 & 3) {
        case 0:
          temp = -cos(x) + sin(x);
          break;
        case 1:
          temp = -cos(x) - sin(x);
          break;
        case 2:
          temp = cos(x) - sin(x);
          break;
        default:
        case 3:
          temp = cos(x) + sin(x);
          break;
      }
      b = invsqrtpi * temp / sqrt(x);
    } else {
      a = j0(x);
      b = j1(x);
      for (i = 0; i < nm1;) {
        i++;
        temp = b;
        b = b * (2.0 * i / x) - a; /* avoid underflow */
        a = temp;
      }
    }
  } else {
    if (ix < 0x3e100000) { /* x < 2**-29 */
      /* x is tiny, return the first Taylor expansion of J(n,x)
       * J(n,x) = 1/n!*(x/2)^n  - ...
       */
      if (nm1 > 32) /* underflow */
        b = 0.0;
      else {
        temp = x * 0.5;
        b = temp;
        a = 1.0;
        for (i = 2; i <= nm1 + 1; i++) {
          a *= (double)i; /* a = n! */
          b *= temp;      /* b = (x/2)^n */
        }
        b = b / a;
      }
    } else {
      /* use backward recurrence */
      /*                      x      x^2      x^2
       *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
       *                      2n  - 2(n+1) - 2(n+2)
       *
       *                      1      1        1
       *  (for large x)   =  ----  ------   ------   .....
       *                      2n   2(n+1)   2(n+2)
       *                      -- - ------ - ------ -
       *                       x     x         x
       *
       * Let w = 2n/x and h=2/x, then the above quotient
       * is equal to the continued fraction:
       *                  1
       *      = -----------------------
       *                     1
       *         w - -----------------
       *                        1
       *              w+h - ---------
       *                     w+2h - ...
       *
       * To determine how many terms needed, let
       * Q(0) = w, Q(1) = w(w+h) - 1,
       * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
       * When Q(k) > 1e4      good for single
       * When Q(k) > 1e9      good for double
       * When Q(k) > 1e17     good for quadruple
       */
      /* determine k */
      double t, q0, q1, w, h, z, tmp, nf;
      int k;

      nf = nm1 + 1.0;
      w = 2 * nf / x;
      h = 2 / x;
      z = w + h;
      q0 = w;
      q1 = w * z - 1.0;
      k = 1;
      while (q1 < 1.0e9) {
        k += 1;
        z += h;
        tmp = z * q1 - q0;
        q0 = q1;
        q1 = tmp;
      }
      for (t = 0.0, i = k; i >= 0; i--)
        t = 1 / (2 * (i + nf) / x - t);
      a = t;
      b = 1.0;
      /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
       *  Hence, if n*(log(2n/x)) > ...
       *  single 8.8722839355e+01
       *  double 7.09782712893383973096e+02
       *  long double 1.1356523406294143949491931077970765006170e+04
       *  then recurrent value may overflow and the result is
       *  likely underflow to zero
       */
      tmp = nf * log(fabs(w));
      if (tmp < 7.09782712893383973096e+02) {
        for (i = nm1; i > 0; i--) {
          temp = b;
          b = b * (2.0 * i) / x - a;
          a = temp;
        }
      } else {
        for (i = nm1; i > 0; i--) {
          temp = b;
          b = b * (2.0 * i) / x - a;
          a = temp;
          /* scale b to avoid spurious overflow */
          if (b > 0x1p500) {
            a /= b;
            t /= b;
            b = 1.0;
          }
        }
      }
      z = j0(x);
      w = j1(x);
      if (fabs(z) >= fabs(w))
        b = t * z / b;
      else
        b = t * w / a;
    }
  }
  return sign ? -b : b;
}

double yn(int n, double x) {
  uint32_t ix, lx, ib;
  int nm1, sign, i;
  double a, b, temp;

  EXTRACT_WORDS(ix, lx, x);
  sign = ix >> 31;
  ix &= 0x7fffffff;

  if ((ix | (lx | -lx) >> 31) > 0x7ff00000) /* nan */
    return x;
  if (sign && (ix | lx) != 0) /* x < 0 */
    return 0 / 0.0;
  if (ix == 0x7ff00000)
    return 0.0;

  if (n == 0)
    return y0(x);
  if (n < 0) {
    nm1 = -(n + 1);
    sign = n & 1;
  } else {
    nm1 = n - 1;
    sign = 0;
  }
  if (nm1 == 0)
    return sign ? -y1(x) : y1(x);

  if (ix >= 0x52d00000) { /* x > 2**302 */
                          /* (x >> n**2)
                           *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
                           *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
                           *      Let s=sin(x), c=cos(x),
                           *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
                           *
                           *             n    sin(xn)*sqt2    cos(xn)*sqt2
                           *          ----------------------------------
                           *             0     s-c             c+s
                           *             1    -s-c            -c+s
                           *             2    -s+c            -c-s
                           *             3     s+c             c-s
                           */
    switch (nm1 & 3) {
      case 0:
        temp = -sin(x) - cos(x);
        break;
      case 1:
        temp = -sin(x) + cos(x);
        break;
      case 2:
        temp = sin(x) + cos(x);
        break;
      default:
      case 3:
        temp = sin(x) - cos(x);
        break;
    }
    b = invsqrtpi * temp / sqrt(x);
  } else {
    a = y0(x);
    b = y1(x);
    /* quit if b is -inf */
    GET_HIGH_WORD(ib, b);
    for (i = 0; i < nm1 && ib != 0xfff00000;) {
      i++;
      temp = b;
      b = (2.0 * i / x) * b - a;
      GET_HIGH_WORD(ib, b);
      a = temp;
    }
  }
  return sign ? -b : b;
}