1/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
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 * jn(n, x), yn(n, x)
14 * floating point Bessel's function of the 1st and 2nd kind
15 * of order n
16 *
17 * Special cases:
18 *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
19 *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
20 * Note 2. About jn(n,x), yn(n,x)
21 *      For n=0, j0(x) is called,
22 *      for n=1, j1(x) is called,
23 *      for n<=x, forward recursion is used starting
24 *      from values of j0(x) and j1(x).
25 *      for n>x, a continued fraction approximation to
26 *      j(n,x)/j(n-1,x) is evaluated and then backward
27 *      recursion is used starting from a supposed value
28 *      for j(n,x). The resulting value of j(0,x) is
29 *      compared with the actual value to correct the
30 *      supposed value of j(n,x).
31 *
32 *      yn(n,x) is similar in all respects, except
33 *      that forward recursion is used for all
34 *      values of n>1.
35 */
36
37#include "libm.h"
38
39static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
40
41double jn(int n, double x)
42{
43	uint32_t ix, lx;
44	int nm1, i, sign;
45	double a, b, temp;
46
47	EXTRACT_WORDS(ix, lx, x);
48	sign = ix>>31;
49	ix &= 0x7fffffff;
50
51	if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
52		return x;
53
54	/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
55	 * Thus, J(-n,x) = J(n,-x)
56	 */
57	/* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
58	if (n == 0)
59		return j0(x);
60	if (n < 0) {
61		nm1 = -(n+1);
62		x = -x;
63		sign ^= 1;
64	} else
65		nm1 = n-1;
66	if (nm1 == 0)
67		return j1(x);
68
69	sign &= n;  /* even n: 0, odd n: signbit(x) */
70	x = fabs(x);
71	if ((ix|lx) == 0 || ix == 0x7ff00000)  /* if x is 0 or inf */
72		b = 0.0;
73	else if (nm1 < x) {
74		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
75		if (ix >= 0x52d00000) { /* x > 2**302 */
76			/* (x >> n**2)
77			 *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
78			 *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
79			 *      Let s=sin(x), c=cos(x),
80			 *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
81			 *
82			 *             n    sin(xn)*sqt2    cos(xn)*sqt2
83			 *          ----------------------------------
84			 *             0     s-c             c+s
85			 *             1    -s-c            -c+s
86			 *             2    -s+c            -c-s
87			 *             3     s+c             c-s
88			 */
89			switch(nm1&3) {
90			case 0: temp = -cos(x)+sin(x); break;
91			case 1: temp = -cos(x)-sin(x); break;
92			case 2: temp =  cos(x)-sin(x); break;
93			default:
94			case 3: temp =  cos(x)+sin(x); break;
95			}
96			b = invsqrtpi*temp/sqrt(x);
97		} else {
98			a = j0(x);
99			b = j1(x);
100			for (i=0; i<nm1; ) {
101				i++;
102				temp = b;
103				b = b*(2.0*i/x) - a; /* avoid underflow */
104				a = temp;
105			}
106		}
107	} else {
108		if (ix < 0x3e100000) { /* x < 2**-29 */
109			/* x is tiny, return the first Taylor expansion of J(n,x)
110			 * J(n,x) = 1/n!*(x/2)^n  - ...
111			 */
112			if (nm1 > 32)  /* underflow */
113				b = 0.0;
114			else {
115				temp = x*0.5;
116				b = temp;
117				a = 1.0;
118				for (i=2; i<=nm1+1; i++) {
119					a *= (double)i; /* a = n! */
120					b *= temp;      /* b = (x/2)^n */
121				}
122				b = b/a;
123			}
124		} else {
125			/* use backward recurrence */
126			/*                      x      x^2      x^2
127			 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
128			 *                      2n  - 2(n+1) - 2(n+2)
129			 *
130			 *                      1      1        1
131			 *  (for large x)   =  ----  ------   ------   .....
132			 *                      2n   2(n+1)   2(n+2)
133			 *                      -- - ------ - ------ -
134			 *                       x     x         x
135			 *
136			 * Let w = 2n/x and h=2/x, then the above quotient
137			 * is equal to the continued fraction:
138			 *                  1
139			 *      = -----------------------
140			 *                     1
141			 *         w - -----------------
142			 *                        1
143			 *              w+h - ---------
144			 *                     w+2h - ...
145			 *
146			 * To determine how many terms needed, let
147			 * Q(0) = w, Q(1) = w(w+h) - 1,
148			 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
149			 * When Q(k) > 1e4      good for single
150			 * When Q(k) > 1e9      good for double
151			 * When Q(k) > 1e17     good for quadruple
152			 */
153			/* determine k */
154			double t,q0,q1,w,h,z,tmp,nf;
155			int k;
156
157			nf = nm1 + 1.0;
158			w = 2*nf/x;
159			h = 2/x;
160			z = w+h;
161			q0 = w;
162			q1 = w*z - 1.0;
163			k = 1;
164			while (q1 < 1.0e9) {
165				k += 1;
166				z += h;
167				tmp = z*q1 - q0;
168				q0 = q1;
169				q1 = tmp;
170			}
171			for (t=0.0, i=k; i>=0; i--)
172				t = 1/(2*(i+nf)/x - t);
173			a = t;
174			b = 1.0;
175			/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
176			 *  Hence, if n*(log(2n/x)) > ...
177			 *  single 8.8722839355e+01
178			 *  double 7.09782712893383973096e+02
179			 *  long double 1.1356523406294143949491931077970765006170e+04
180			 *  then recurrent value may overflow and the result is
181			 *  likely underflow to zero
182			 */
183			tmp = nf*log(fabs(w));
184			if (tmp < 7.09782712893383973096e+02) {
185				for (i=nm1; i>0; i--) {
186					temp = b;
187					b = b*(2.0*i)/x - a;
188					a = temp;
189				}
190			} else {
191				for (i=nm1; i>0; i--) {
192					temp = b;
193					b = b*(2.0*i)/x - a;
194					a = temp;
195					/* scale b to avoid spurious overflow */
196					if (b > 0x1p500) {
197						a /= b;
198						t /= b;
199						b  = 1.0;
200					}
201				}
202			}
203			z = j0(x);
204			w = j1(x);
205			if (fabs(z) >= fabs(w))
206				b = t*z/b;
207			else
208				b = t*w/a;
209		}
210	}
211	return sign ? -b : b;
212}
213
214
215double yn(int n, double x)
216{
217	uint32_t ix, lx, ib;
218	int nm1, sign, i;
219	double a, b, temp;
220
221	EXTRACT_WORDS(ix, lx, x);
222	sign = ix>>31;
223	ix &= 0x7fffffff;
224
225	if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
226		return x;
227	if (sign && (ix|lx)!=0) /* x < 0 */
228		return 0/0.0;
229	if (ix == 0x7ff00000)
230		return 0.0;
231
232	if (n == 0)
233		return y0(x);
234	if (n < 0) {
235		nm1 = -(n+1);
236		sign = n&1;
237	} else {
238		nm1 = n-1;
239		sign = 0;
240	}
241	if (nm1 == 0)
242		return sign ? -y1(x) : y1(x);
243
244	if (ix >= 0x52d00000) { /* x > 2**302 */
245		/* (x >> n**2)
246		 *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
247		 *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
248		 *      Let s=sin(x), c=cos(x),
249		 *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
250		 *
251		 *             n    sin(xn)*sqt2    cos(xn)*sqt2
252		 *          ----------------------------------
253		 *             0     s-c             c+s
254		 *             1    -s-c            -c+s
255		 *             2    -s+c            -c-s
256		 *             3     s+c             c-s
257		 */
258		switch(nm1&3) {
259		case 0: temp = -sin(x)-cos(x); break;
260		case 1: temp = -sin(x)+cos(x); break;
261		case 2: temp =  sin(x)+cos(x); break;
262		default:
263		case 3: temp =  sin(x)-cos(x); break;
264		}
265		b = invsqrtpi*temp/sqrt(x);
266	} else {
267		a = y0(x);
268		b = y1(x);
269		/* quit if b is -inf */
270		GET_HIGH_WORD(ib, b);
271		for (i=0; i<nm1 && ib!=0xfff00000; ){
272			i++;
273			temp = b;
274			b = (2.0*i/x)*b - a;
275			GET_HIGH_WORD(ib, b);
276			a = temp;
277		}
278	}
279	return sign ? -b : b;
280}
281