glibc/sysdeps/ieee754/dbl-64/e_jn.c

/* @(#)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<x, forward recursion us 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 <errno.h>
#include <float.h>
#include <math.h>
#include <math-narrow-eval.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-finite.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;
	      default: __builtin_unreachable ();
	      }
	    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;
}
libm_alias_finite (__ieee754_jn, __jn)

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;
	  default: __builtin_unreachable ();
	  }
	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;
}
libm_alias_finite (__ieee754_yn, __yn)