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| | /* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
#define _GNU_SOURCE
#include <fenv.h>
#include "libm.h"
float jnf(int n, float x)
{
uint32_t ix;
int nm1, sign, i;
float a, b, temp;
GET_FLOAT_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix > 0x7f800000) /* nan */
return x;
/* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
if (n == 0)
return j0f(x);
if (n < 0) {
nm1 = -(n+1);
x = -x;
sign ^= 1;
} else
nm1 = n-1;
if (nm1 == 0)
return j1f(x);
sign &= n; /* even n: 0, odd n: signbit(x) */
x = fabsf(x);
if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */
b = 0.0f;
else if (nm1 < x) {
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
a = j0f(x);
b = j1f(x);
for (i=0; i<nm1; ){
i++;
temp = b;
b = b*(2.0f*i/x) - a;
a = temp;
}
} else {
if (ix < 0x35800000) { /* x < 2**-20 */
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
if (nm1 > 8) /* underflow */
nm1 = 8;
temp = 0.5f * x;
b = temp;
a = 1.0f;
for (i=2; i<=nm1+1; i++) {
a *= (float)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 */
float t,q0,q1,w,h,z,tmp,nf;
int k;
nf = nm1+1.0f;
w = 2*nf/x;
h = 2/x;
z = w+h;
q0 = w;
q1 = w*z - 1.0f;
k = 1;
while (q1 < 1.0e4f) {
k += 1;
z += h;
tmp = z*q1 - q0;
q0 = q1;
q1 = tmp;
}
for (t=0.0f, i=k; i>=0; i--)
t = 1.0f/(2*(i+nf)/x-t);
a = t;
b = 1.0f;
/* 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*logf(fabsf(w));
if (tmp < 88.721679688f) {
for (i=nm1; i>0; i--) {
temp = b;
b = 2.0f*i*b/x - a;
a = temp;
}
} else {
for (i=nm1; i>0; i--){
temp = b;
b = 2.0f*i*b/x - a;
a = temp;
/* scale b to avoid spurious overflow */
if (b > 0x1p60f) {
a /= b;
t /= b;
b = 1.0f;
}
}
}
z = j0f(x);
w = j1f(x);
if (fabsf(z) >= fabsf(w))
b = t*z/b;
else
b = t*w/a;
}
}
return sign ? -b : b;
}
float ynf(int n, float x)
{
uint32_t ix, ib;
int nm1, sign, i;
float a, b, temp;
GET_FLOAT_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix > 0x7f800000) /* nan */
return x;
if (sign && ix != 0) { /* x < 0 */
feraiseexcept(FE_INVALID);
return 0/0.0f;
}
if (ix == 0x7f800000)
return 0.0f;
if (n == 0)
return y0f(x);
if (n < 0) {
nm1 = -(n+1);
sign = n&1;
} else {
nm1 = n-1;
sign = 0;
}
if (nm1 == 0)
return sign ? -y1f(x) : y1f(x);
a = y0f(x);
b = y1f(x);
/* quit if b is -inf */
GET_FLOAT_WORD(ib,b);
for (i = 0; i < nm1 && ib != 0xff800000; ) {
i++;
temp = b;
b = (2.0f*i/x)*b - a;
GET_FLOAT_WORD(ib, b);
a = temp;
}
return sign ? -b : b;
}
|