1/* @(#)e_jn.c 1.4 95/01/18 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13#include <sys/cdefs.h>
14__FBSDID("$FreeBSD: stable/11/lib/msun/src/e_jn.c 347068 2019-05-03 22:56:50Z peterj $");
15
16/*
17 * __ieee754_jn(n, x), __ieee754_yn(n, x)
18 * floating point Bessel's function of the 1st and 2nd kind
19 * of order n
20 *
21 * Special cases:
22 *	y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
23 *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
24 * Note 2. About jn(n,x), yn(n,x)
25 *	For n=0, j0(x) is called,
26 *	for n=1, j1(x) is called,
27 *	for n<x, forward recursion us used starting
28 *	from values of j0(x) and j1(x).
29 *	for n>x, a continued fraction approximation to
30 *	j(n,x)/j(n-1,x) is evaluated and then backward
31 *	recursion is used starting from a supposed value
32 *	for j(n,x). The resulting value of j(0,x) is
33 *	compared with the actual value to correct the
34 *	supposed value of j(n,x).
35 *
36 *	yn(n,x) is similar in all respects, except
37 *	that forward recursion is used for all
38 *	values of n>1.
39 */
40
41#include "math.h"
42#include "math_private.h"
43
44static const volatile double vone = 1, vzero = 0;
45
46static const double
47invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
48two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
49one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
50
51static const double zero  =  0.00000000000000000000e+00;
52
53double
54__ieee754_jn(int n, double x)
55{
56	int32_t i,hx,ix,lx, sgn;
57	double a, b, c, s, temp, di;
58	double z, w;
59
60    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
61     * Thus, J(-n,x) = J(n,-x)
62     */
63	EXTRACT_WORDS(hx,lx,x);
64	ix = 0x7fffffff&hx;
65    /* if J(n,NaN) is NaN */
66	if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
67	if(n<0){
68		n = -n;
69		x = -x;
70		hx ^= 0x80000000;
71	}
72	if(n==0) return(__ieee754_j0(x));
73	if(n==1) return(__ieee754_j1(x));
74	sgn = (n&1)&(hx>>31);	/* even n -- 0, odd n -- sign(x) */
75	x = fabs(x);
76	if((ix|lx)==0||ix>=0x7ff00000) 	/* if x is 0 or inf */
77	    b = zero;
78	else if((double)n<=x) {
79		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
80	    if(ix>=0x52D00000) { /* x > 2**302 */
81    /* (x >> n**2)
82     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
83     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
84     *	    Let s=sin(x), c=cos(x),
85     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
86     *
87     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
88     *		----------------------------------
89     *		   0	 s-c		 c+s
90     *		   1	-s-c 		-c+s
91     *		   2	-s+c		-c-s
92     *		   3	 s+c		 c-s
93     */
94		sincos(x, &s, &c);
95		switch(n&3) {
96		    case 0: temp =  c+s; break;
97		    case 1: temp = -c+s; break;
98		    case 2: temp = -c-s; break;
99		    case 3: temp =  c-s; break;
100		}
101		b = invsqrtpi*temp/sqrt(x);
102	    } else {
103	        a = __ieee754_j0(x);
104	        b = __ieee754_j1(x);
105	        for(i=1;i<n;i++){
106		    temp = b;
107		    b = b*((double)(i+i)/x) - a; /* avoid underflow */
108		    a = temp;
109	        }
110	    }
111	} else {
112	    if(ix<0x3e100000) {	/* x < 2**-29 */
113    /* x is tiny, return the first Taylor expansion of J(n,x)
114     * J(n,x) = 1/n!*(x/2)^n  - ...
115     */
116		if(n>33)	/* underflow */
117		    b = zero;
118		else {
119		    temp = x*0.5; b = temp;
120		    for (a=one,i=2;i<=n;i++) {
121			a *= (double)i;		/* a = n! */
122			b *= temp;		/* b = (x/2)^n */
123		    }
124		    b = b/a;
125		}
126	    } else {
127		/* use backward recurrence */
128		/* 			x      x^2      x^2
129		 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
130		 *			2n  - 2(n+1) - 2(n+2)
131		 *
132		 * 			1      1        1
133		 *  (for large x)   =  ----  ------   ------   .....
134		 *			2n   2(n+1)   2(n+2)
135		 *			-- - ------ - ------ -
136		 *			 x     x         x
137		 *
138		 * Let w = 2n/x and h=2/x, then the above quotient
139		 * is equal to the continued fraction:
140		 *		    1
141		 *	= -----------------------
142		 *		       1
143		 *	   w - -----------------
144		 *			  1
145		 * 	        w+h - ---------
146		 *		       w+2h - ...
147		 *
148		 * To determine how many terms needed, let
149		 * Q(0) = w, Q(1) = w(w+h) - 1,
150		 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
151		 * When Q(k) > 1e4	good for single
152		 * When Q(k) > 1e9	good for double
153		 * When Q(k) > 1e17	good for quadruple
154		 */
155	    /* determine k */
156		double t,v;
157		double q0,q1,h,tmp; int32_t k,m;
158		w  = (n+n)/(double)x; h = 2.0/(double)x;
159		q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
160		while(q1<1.0e9) {
161			k += 1; z += h;
162			tmp = z*q1 - q0;
163			q0 = q1;
164			q1 = tmp;
165		}
166		m = n+n;
167		for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
168		a = t;
169		b = one;
170		/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
171		 *  Hence, if n*(log(2n/x)) > ...
172		 *  single 8.8722839355e+01
173		 *  double 7.09782712893383973096e+02
174		 *  long double 1.1356523406294143949491931077970765006170e+04
175		 *  then recurrent value may overflow and the result is
176		 *  likely underflow to zero
177		 */
178		tmp = n;
179		v = two/x;
180		tmp = tmp*__ieee754_log(fabs(v*tmp));
181		if(tmp<7.09782712893383973096e+02) {
182	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
183		        temp = b;
184			b *= di;
185			b  = b/x - a;
186		        a = temp;
187			di -= two;
188	     	    }
189		} else {
190	    	    for(i=n-1,di=(double)(i+i);i>0;i--){
191		        temp = b;
192			b *= di;
193			b  = b/x - a;
194		        a = temp;
195			di -= two;
196		    /* scale b to avoid spurious overflow */
197			if(b>1e100) {
198			    a /= b;
199			    t /= b;
200			    b  = one;
201			}
202	     	    }
203		}
204		z = __ieee754_j0(x);
205		w = __ieee754_j1(x);
206		if (fabs(z) >= fabs(w))
207		    b = (t*z/b);
208		else
209		    b = (t*w/a);
210	    }
211	}
212	if(sgn==1) return -b; else return b;
213}
214
215double
216__ieee754_yn(int n, double x)
217{
218	int32_t i,hx,ix,lx;
219	int32_t sign;
220	double a, b, c, s, temp;
221
222	EXTRACT_WORDS(hx,lx,x);
223	ix = 0x7fffffff&hx;
224	/* yn(n,NaN) = NaN */
225	if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
226	/* yn(n,+-0) = -inf and raise divide-by-zero exception. */
227	if((ix|lx)==0) return -one/vzero;
228	/* yn(n,x<0) = NaN and raise invalid exception. */
229	if(hx<0) return vzero/vzero;
230	sign = 1;
231	if(n<0){
232		n = -n;
233		sign = 1 - ((n&1)<<1);
234	}
235	if(n==0) return(__ieee754_y0(x));
236	if(n==1) return(sign*__ieee754_y1(x));
237	if(ix==0x7ff00000) return zero;
238	if(ix>=0x52D00000) { /* x > 2**302 */
239    /* (x >> n**2)
240     *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
241     *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
242     *	    Let s=sin(x), c=cos(x),
243     *		xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
244     *
245     *		   n	sin(xn)*sqt2	cos(xn)*sqt2
246     *		----------------------------------
247     *		   0	 s-c		 c+s
248     *		   1	-s-c 		-c+s
249     *		   2	-s+c		-c-s
250     *		   3	 s+c		 c-s
251     */
252		sincos(x, &s, &c);
253		switch(n&3) {
254		    case 0: temp =  s-c; break;
255		    case 1: temp = -s-c; break;
256		    case 2: temp = -s+c; break;
257		    case 3: temp =  s+c; break;
258		}
259		b = invsqrtpi*temp/sqrt(x);
260	} else {
261	    u_int32_t high;
262	    a = __ieee754_y0(x);
263	    b = __ieee754_y1(x);
264	/* quit if b is -inf */
265	    GET_HIGH_WORD(high,b);
266	    for(i=1;i<n&&high!=0xfff00000;i++){
267		temp = b;
268		b = ((double)(i+i)/x)*b - a;
269		GET_HIGH_WORD(high,b);
270		a = temp;
271	    }
272	}
273	if(sign>0) return b; else return -b;
274}
275