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add fdlibm
git-svn-id: http://svn2.nishi.boats/svn/milsko/trunk@557 b9cfdab3-6d41-4d17-bbe4-086880011989
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NishiOwO
283
external/fdlibm/e_jn.c
vendored
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283
external/fdlibm/e_jn.c
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/* @(#)e_jn.c 1.4 95/01/18 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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* __fdlibm_jn(n, x), __fdlibm_yn(n, x)
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* floating point Bessel's function of the 1st and 2nd kind
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* of order n
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*
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* Special cases:
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* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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* Note 2. About jn(n,x), yn(n,x)
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* For n=0, j0(x) is called,
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* for n=1, j1(x) is called,
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* for n<x, forward recursion us used starting
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* from values of j0(x) and j1(x).
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* for n>x, a continued fraction approximation to
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* j(n,x)/j(n-1,x) is evaluated and then backward
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* recursion is used starting from a supposed value
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* for j(n,x). The resulting value of j(0,x) is
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* compared with the actual value to correct the
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* supposed value of j(n,x).
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*
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* yn(n,x) is similar in all respects, except
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* that forward recursion is used for all
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* values of n>1.
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*
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*/
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#include "math.h"
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static const double
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invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
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one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
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static double zero = 0.00000000000000000000e+00;
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double __fdlibm_jn(int n, double x) {
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int i, hx, ix, lx, sgn;
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double a, b, temp, di;
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double z, w;
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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* Thus, J(-n,x) = J(n,-x)
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*/
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hx = __HI(x);
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ix = 0x7fffffff & hx;
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lx = __LO(x);
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/* if J(n,NaN) is NaN */
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if((ix | ((unsigned)(lx | -lx)) >> 31) > 0x7ff00000) return x + x;
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if(n < 0) {
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n = -n;
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x = -x;
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hx ^= 0x80000000;
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}
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if(n == 0) return (__fdlibm_j0(x));
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if(n == 1) return (__fdlibm_j1(x));
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sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */
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x = fabs(x);
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if((ix | lx) == 0 || ix >= 0x7ff00000) /* if x is 0 or inf */
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b = zero;
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else if((double)n <= x) {
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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if(ix >= 0x52D00000) { /* x > 2**302 */
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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switch(n & 3) {
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case 0:
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temp = cos(x) + sin(x);
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break;
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case 1:
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temp = -cos(x) + sin(x);
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break;
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case 2:
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temp = -cos(x) - sin(x);
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break;
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case 3:
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temp = cos(x) - sin(x);
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break;
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}
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b = invsqrtpi * temp / sqrt(x);
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} else {
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a = __fdlibm_j0(x);
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b = __fdlibm_j1(x);
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for(i = 1; i < n; i++) {
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temp = b;
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b = b * ((double)(i + i) / x) - a; /* avoid underflow */
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a = temp;
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}
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}
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} else {
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if(ix < 0x3e100000) { /* x < 2**-29 */
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/* x is tinyv, return the first Taylor expansion of J(n,x)
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* J(n,x) = 1/n!*(x/2)^n - ...
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*/
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if(n > 33) /* underflow */
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b = zero;
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else {
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temp = x * 0.5;
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b = temp;
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for(a = one, i = 2; i <= n; i++) {
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a *= (double)i; /* a = n! */
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b *= temp; /* b = (x/2)^n */
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}
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b = b / a;
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}
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} else {
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/* use backward recurrence */
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/* x x^2 x^2
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* J(n,x)/J(n-1,x) = ---- ------ ------ .....
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* 2n - 2(n+1) - 2(n+2)
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*
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* 1 1 1
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* (for large x) = ---- ------ ------ .....
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* 2n 2(n+1) 2(n+2)
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* -- - ------ - ------ -
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* x x x
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*
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* Let w = 2n/x and h=2/x, then the above quotient
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* is equal to the continued fraction:
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* 1
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* = -----------------------
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* 1
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* w - -----------------
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* 1
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* w+h - ---------
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* w+2h - ...
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*
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* To determine how many terms needed, let
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* Q(0) = w, Q(1) = w(w+h) - 1,
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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* When Q(k) > 1e4 good for single
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* When Q(k) > 1e9 good for double
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* When Q(k) > 1e17 good for quadruple
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*/
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/* determine k */
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double t, v;
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double q0, q1, h, tmp;
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int k, m;
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w = (n + n) / (double)x;
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h = 2.0 / (double)x;
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q0 = w;
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z = w + h;
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q1 = w * z - 1.0;
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k = 1;
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while(q1 < 1.0e9) {
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k += 1;
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z += h;
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tmp = z * q1 - q0;
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q0 = q1;
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q1 = tmp;
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}
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m = n + n;
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for(t = zero, i = 2 * (n + k); i >= m; i -= 2) t = one / (i / x - t);
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a = t;
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b = one;
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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* Hence, if n*(log(2n/x)) > ...
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* single 8.8722839355e+01
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* double 7.09782712893383973096e+02
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* long double 1.1356523406294143949491931077970765006170e+04
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* then recurrent value may overflow and the result is
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* likely underflow to zero
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*/
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tmp = n;
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v = two / x;
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tmp = tmp * __fdlibm_log(fabs(v * tmp));
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if(tmp < 7.09782712893383973096e+02) {
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for(i = n - 1, di = (double)(i + i); i > 0; i--) {
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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}
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} else {
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for(i = n - 1, di = (double)(i + i); i > 0; i--) {
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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/* scale b to avoid spurious overflow */
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if(b > 1e100) {
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a /= b;
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t /= b;
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b = one;
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}
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}
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}
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b = (t * __fdlibm_j0(x) / b);
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}
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}
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if(sgn == 1) return -b;
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else
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return b;
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}
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double __fdlibm_yn(int n, double x) {
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int i, hx, ix, lx;
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int sign;
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double a, b, temp;
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hx = __HI(x);
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ix = 0x7fffffff & hx;
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lx = __LO(x);
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/* if Y(n,NaN) is NaN */
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if((ix | ((unsigned)(lx | -lx)) >> 31) > 0x7ff00000) return x + x;
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if((ix | lx) == 0) return -one / zero;
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if(hx < 0) return zero / zero;
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sign = 1;
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if(n < 0) {
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n = -n;
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sign = 1 - ((n & 1) << 1);
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}
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if(n == 0) return (__fdlibm_y0(x));
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if(n == 1) return (sign * __fdlibm_y1(x));
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if(ix == 0x7ff00000) return zero;
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if(ix >= 0x52D00000) { /* x > 2**302 */
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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switch(n & 3) {
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case 0:
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temp = sin(x) - cos(x);
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break;
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case 1:
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temp = -sin(x) - cos(x);
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break;
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case 2:
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temp = -sin(x) + cos(x);
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break;
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case 3:
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temp = sin(x) + cos(x);
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break;
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}
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b = invsqrtpi * temp / sqrt(x);
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} else {
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a = __fdlibm_y0(x);
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b = __fdlibm_y1(x);
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/* quit if b is -inf */
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for(i = 1; i < n && (__HI(b) != 0xfff00000); i++) {
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temp = b;
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b = ((double)(i + i) / x) * b - a;
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a = temp;
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}
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}
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if(sign > 0) return b;
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else
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return -b;
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}
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