/* @(#)e_jn.c 5.1 93/09/24 */

/*

 * ====================================================

 * 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.

 * ====================================================

 */

/*

 * __ieee754_jn(n, x), __ieee754_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 overflow 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

 *	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 <errno.h>

#include <float.h>

#include <math.h>

#include <math-narrow-eval.h>

#include <math_private.h>

#include <math-underflow.h>

static const double

  invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */

  two = 2.00000000000000000000e+00,  /* 0x40000000, 0x00000000 */

  one = 1.00000000000000000000e+00;  /* 0x3FF00000, 0x00000000 */

static const double zero = 0.00000000000000000000e+00;

double

__ieee754_jn (int n, double x)

{

  int32_t i, hx, ix, lx, sgn;

  double a, b, temp, di, ret;

  double z, w;

  /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)

   * Thus, J(-n,x) = J(n,-x)

   */

  EXTRACT_WORDS (hx, lx, x);

  ix = 0x7fffffff & hx;

  /* if J(n,NaN) is NaN */

  if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))

    return x + x;

  if (n < 0)

    {

      n = -n;

      x = -x;

      hx ^= 0x80000000;

    }

  if (n == 0)

    return (__ieee754_j0 (x));

  if (n == 1)

    return (__ieee754_j1 (x));

  sgn = (n & 1) & (hx >> 31);   /* even n -- 0, odd n -- sign(x) */

  x = fabs (x);

  {

    SET_RESTORE_ROUND (FE_TONEAREST);

    if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))

      /* if x is 0 or inf */

      return sgn == 1 ? -zero : zero;

    else if ((double) n <= 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

			 */

	    double s;

	    double c;

	    __sincos (x, &s, &c);

	    switch (n & 3)

	      {

	      case 0: temp = c + s; break;

	      case 1: temp = -c + s; break;

	      case 2: temp = -c - s; break;

	      case 3: temp = c - s; break;

	      }

	    b = invsqrtpi * temp / sqrt (x);

	  }

	else

	  {

	    a = __ieee754_j0 (x);

	    b = __ieee754_j1 (x);

	    for (i = 1; i < n; i++)

	      {

		temp = b;

		b = b * ((double) (i + 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 (n > 33)           /* underflow */

	      b = zero;

	    else

	      {

		temp = x * 0.5; b = temp;

		for (a = one, i = 2; i <= n; 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, v;

	    double q0, q1, h, tmp; int32_t k, m;

	    w = (n + n) / (double) x; h = 2.0 / (double) x;

	    q0 = w;  z = w + h; q1 = w * z - 1.0; k = 1;

	    while (q1 < 1.0e9)

	      {

		k += 1; z += h;

		tmp = z * q1 - q0;

		q0 = q1;

		q1 = tmp;

	      }

	    m = n + n;

	    for (t = zero, i = 2 * (n + k); i >= m; i -= 2)

	      t = one / (i / x - t);

	    a = t;

	    b = one;

	    /*  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 = n;

	    v = two / x;

	    tmp = tmp * __ieee754_log (fabs (v * tmp));

	    if (tmp < 7.09782712893383973096e+02)

	      {

		for (i = n - 1, di = (double) (i + i); i > 0; i--)

		  {

		    temp = b;

		    b *= di;

		    b = b / x - a;

		    a = temp;

		    di -= two;

		  }

	      }

	    else

	      {

		for (i = n - 1, di = (double) (i + i); i > 0; i--)

		  {

		    temp = b;

		    b *= di;

		    b = b / x - a;

		    a = temp;

		    di -= two;

		    /* scale b to avoid spurious overflow */

		    if (b > 1e100)

		      {

			a /= b;

			t /= b;

			b = one;

		      }

		  }

	      }

	    /* j0() and j1() suffer enormous loss of precision at and

	     * near zero; however, we know that their zero points never

	     * coincide, so just choose the one further away from zero.

	     */

	    z = __ieee754_j0 (x);

	    w = __ieee754_j1 (x);

	    if (fabs (z) >= fabs (w))

	      b = (t * z / b);

	    else

	      b = (t * w / a);

	  }

      }

    if (sgn == 1)

      ret = -b;

    else

      ret = b;

    ret = math_narrow_eval (ret);

  }

  if (ret == 0)

    {

      ret = math_narrow_eval (__copysign (DBL_MIN, ret) * DBL_MIN);

      __set_errno (ERANGE);

    }

  else

    math_check_force_underflow (ret);

  return ret;

}

strong_alias (__ieee754_jn, __jn_finite)

double

__ieee754_yn (int n, double x)

{

  int32_t i, hx, ix, lx;

  int32_t sign;

  double a, b, temp, ret;

  EXTRACT_WORDS (hx, lx, x);

  ix = 0x7fffffff & hx;

  /* if Y(n,NaN) is NaN */

  if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))

    return x + x;

  sign = 1;

  if (n < 0)

    {

      n = -n;

      sign = 1 - ((n & 1) << 1);

    }

  if (n == 0)

    return (__ieee754_y0 (x));

  if (__glibc_unlikely ((ix | lx) == 0))

    return -sign / zero;

  /* -inf and overflow exception.  */;

  if (__glibc_unlikely (hx < 0))

    return zero / (zero * x);

  {

    SET_RESTORE_ROUND (FE_TONEAREST);

    if (n == 1)

      {

	ret = sign * __ieee754_y1 (x);

	goto out;

      }

    if (__glibc_unlikely (ix == 0x7ff00000))

      return zero;

    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

	 */

	double c;

	double s;

	__sincos (x, &s, &c);

	switch (n & 3)

	  {

	  case 0: temp = s - c; break;

	  case 1: temp = -s - c; break;

	  case 2: temp = -s + c; break;

	  case 3: temp = s + c; break;

	  }

	b = invsqrtpi * temp / sqrt (x);

      }

    else

      {

	uint32_t high;

	a = __ieee754_y0 (x);

	b = __ieee754_y1 (x);

	/* quit if b is -inf */

	GET_HIGH_WORD (high, b);

	for (i = 1; i < n && high != 0xfff00000; i++)

	  {

	    temp = b;

	    b = ((double) (i + i) / x) * b - a;

	    GET_HIGH_WORD (high, b);

	    a = temp;

	  }

	/* If B is +-Inf, set up errno accordingly.  */

	if (!isfinite (b))

	  __set_errno (ERANGE);

      }

    if (sign > 0)

      ret = b;

    else

      ret = -b;

  }

 out:

  if (isinf (ret))

    ret = __copysign (DBL_MAX, ret) * DBL_MAX;

  return ret;

}

strong_alias (__ieee754_yn, __yn_finite)