a0ee07829a
This brings us up to date with FreeBSD HEAD, fixes various bugs, unifies the set of functions we support on ARM, MIPS, and x86, fixes "long double", adds ISO C99 support, and adds basic unit tests. It turns out that our "long double" functions have always been broken for non-normal numbers. This patch fixes that by not using the upstream implementations and just forwarding to the regular "double" implementation instead (since "long double" on Android is just "double" anyway, which is what BSD doesn't support). All the tests pass on ARM, MIPS, and x86, plus glibc on x86-64. Bug: 3169850 Bug: 8012787 Bug: https://code.google.com/p/android/issues/detail?id=6697 Change-Id: If0c343030959c24bfc50d4d21c9530052c581837
271 lines
7.0 KiB
C
271 lines
7.0 KiB
C
|
|
/* @(#)e_jn.c 1.4 95/01/18 */
|
|
/*
|
|
* ====================================================
|
|
* 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.
|
|
* ====================================================
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
/*
|
|
* __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 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 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 "math.h"
|
|
#include "math_private.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;
|
|
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((ix|((u_int32_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);
|
|
if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
|
|
b = 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
|
|
*/
|
|
switch(n&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;
|
|
case 3: temp = cos(x)-sin(x); 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;
|
|
}
|
|
}
|
|
}
|
|
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) return -b; else return b;
|
|
}
|
|
|
|
double
|
|
__ieee754_yn(int n, double x)
|
|
{
|
|
int32_t i,hx,ix,lx;
|
|
int32_t sign;
|
|
double a, b, temp;
|
|
|
|
EXTRACT_WORDS(hx,lx,x);
|
|
ix = 0x7fffffff&hx;
|
|
/* if Y(n,NaN) is NaN */
|
|
if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
|
|
if((ix|lx)==0) return -one/zero;
|
|
if(hx<0) return zero/zero;
|
|
sign = 1;
|
|
if(n<0){
|
|
n = -n;
|
|
sign = 1 - ((n&1)<<1);
|
|
}
|
|
if(n==0) return(__ieee754_y0(x));
|
|
if(n==1) return(sign*__ieee754_y1(x));
|
|
if(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
|
|
*/
|
|
switch(n&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;
|
|
case 3: temp = sin(x)+cos(x); break;
|
|
}
|
|
b = invsqrtpi*temp/sqrt(x);
|
|
} else {
|
|
u_int32_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(sign>0) return b; else return -b;
|
|
}
|