/* @(#)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 <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)