1/* @(#)e_jn.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, 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/*
14 * __ieee754_jn(n, x), __ieee754_yn(n, x)
15 * floating point Bessel's function of the 1st and 2nd kind
16 * of order n
17 *
18 * Special cases:
19 * y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
20 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
21 * Note 2. About jn(n,x), yn(n,x)
22 * For n=0, j0(x) is called,
23 * for n=1, j1(x) is called,
24 * for n<x, forward recursion us used starting
25 * from values of j0(x) and j1(x).
26 * for n>x, a continued fraction approximation to
27 * j(n,x)/j(n-1,x) is evaluated and then backward
28 * recursion is used starting from a supposed value
29 * for j(n,x). The resulting value of j(0,x) is
30 * compared with the actual value to correct the
31 * supposed value of j(n,x).
32 *
33 * yn(n,x) is similar in all respects, except
34 * that forward recursion is used for all
35 * values of n>1.
36 *
37 */
38
39#include <errno.h>
40#include <float.h>
41#include <math.h>
42#include <math_private.h>
43
44static const double
45 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
46 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
47 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
48
49static const double zero = 0.00000000000000000000e+00;
50
51double
52__ieee754_jn (int n, double x)
53{
54 int32_t i, hx, ix, lx, sgn;
55 double a, b, temp, di, ret;
56 double z, w;
57
58 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
59 * Thus, J(-n,x) = J(n,-x)
60 */
61 EXTRACT_WORDS (hx, lx, x);
62 ix = 0x7fffffff & hx;
63 /* if J(n,NaN) is NaN */
64 if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))
65 return x + x;
66 if (n < 0)
67 {
68 n = -n;
69 x = -x;
70 hx ^= 0x80000000;
71 }
72 if (n == 0)
73 return (__ieee754_j0 (x));
74 if (n == 1)
75 return (__ieee754_j1 (x));
76 sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */
77 x = fabs (x);
78 {
79 SET_RESTORE_ROUND (FE_TONEAREST);
80 if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
81 /* if x is 0 or inf */
82 return sgn == 1 ? -zero : zero;
83 else if ((double) n <= x)
84 {
85 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
86 if (ix >= 0x52D00000) /* x > 2**302 */
87 { /* (x >> n**2)
88 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
89 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
90 * Let s=sin(x), c=cos(x),
91 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
92 *
93 * n sin(xn)*sqt2 cos(xn)*sqt2
94 * ----------------------------------
95 * 0 s-c c+s
96 * 1 -s-c -c+s
97 * 2 -s+c -c-s
98 * 3 s+c c-s
99 */
100 double s;
101 double c;
102 __sincos (x, &s, &c);
103 switch (n & 3)
104 {
105 case 0: temp = c + s; break;
106 case 1: temp = -c + s; break;
107 case 2: temp = -c - s; break;
108 case 3: temp = c - s; break;
109 }
110 b = invsqrtpi * temp / __ieee754_sqrt (x);
111 }
112 else
113 {
114 a = __ieee754_j0 (x);
115 b = __ieee754_j1 (x);
116 for (i = 1; i < n; i++)
117 {
118 temp = b;
119 b = b * ((double) (i + i) / x) - a; /* avoid underflow */
120 a = temp;
121 }
122 }
123 }
124 else
125 {
126 if (ix < 0x3e100000) /* x < 2**-29 */
127 { /* x is tiny, return the first Taylor expansion of J(n,x)
128 * J(n,x) = 1/n!*(x/2)^n - ...
129 */
130 if (n > 33) /* underflow */
131 b = zero;
132 else
133 {
134 temp = x * 0.5; b = temp;
135 for (a = one, i = 2; i <= n; i++)
136 {
137 a *= (double) i; /* a = n! */
138 b *= temp; /* b = (x/2)^n */
139 }
140 b = b / a;
141 }
142 }
143 else
144 {
145 /* use backward recurrence */
146 /* x x^2 x^2
147 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
148 * 2n - 2(n+1) - 2(n+2)
149 *
150 * 1 1 1
151 * (for large x) = ---- ------ ------ .....
152 * 2n 2(n+1) 2(n+2)
153 * -- - ------ - ------ -
154 * x x x
155 *
156 * Let w = 2n/x and h=2/x, then the above quotient
157 * is equal to the continued fraction:
158 * 1
159 * = -----------------------
160 * 1
161 * w - -----------------
162 * 1
163 * w+h - ---------
164 * w+2h - ...
165 *
166 * To determine how many terms needed, let
167 * Q(0) = w, Q(1) = w(w+h) - 1,
168 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
169 * When Q(k) > 1e4 good for single
170 * When Q(k) > 1e9 good for double
171 * When Q(k) > 1e17 good for quadruple
172 */
173 /* determine k */
174 double t, v;
175 double q0, q1, h, tmp; int32_t k, m;
176 w = (n + n) / (double) x; h = 2.0 / (double) x;
177 q0 = w; z = w + h; q1 = w * z - 1.0; k = 1;
178 while (q1 < 1.0e9)
179 {
180 k += 1; z += h;
181 tmp = z * q1 - q0;
182 q0 = q1;
183 q1 = tmp;
184 }
185 m = n + n;
186 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
187 t = one / (i / x - t);
188 a = t;
189 b = one;
190 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
191 * Hence, if n*(log(2n/x)) > ...
192 * single 8.8722839355e+01
193 * double 7.09782712893383973096e+02
194 * long double 1.1356523406294143949491931077970765006170e+04
195 * then recurrent value may overflow and the result is
196 * likely underflow to zero
197 */
198 tmp = n;
199 v = two / x;
200 tmp = tmp * __ieee754_log (fabs (v * tmp));
201 if (tmp < 7.09782712893383973096e+02)
202 {
203 for (i = n - 1, di = (double) (i + i); i > 0; i--)
204 {
205 temp = b;
206 b *= di;
207 b = b / x - a;
208 a = temp;
209 di -= two;
210 }
211 }
212 else
213 {
214 for (i = n - 1, di = (double) (i + i); i > 0; i--)
215 {
216 temp = b;
217 b *= di;
218 b = b / x - a;
219 a = temp;
220 di -= two;
221 /* scale b to avoid spurious overflow */
222 if (b > 1e100)
223 {
224 a /= b;
225 t /= b;
226 b = one;
227 }
228 }
229 }
230 /* j0() and j1() suffer enormous loss of precision at and
231 * near zero; however, we know that their zero points never
232 * coincide, so just choose the one further away from zero.
233 */
234 z = __ieee754_j0 (x);
235 w = __ieee754_j1 (x);
236 if (fabs (z) >= fabs (w))
237 b = (t * z / b);
238 else
239 b = (t * w / a);
240 }
241 }
242 if (sgn == 1)
243 ret = -b;
244 else
245 ret = b;
246 ret = math_narrow_eval (ret);
247 }
248 if (ret == 0)
249 {
250 ret = math_narrow_eval (__copysign (DBL_MIN, ret) * DBL_MIN);
251 __set_errno (ERANGE);
252 }
253 else
254 math_check_force_underflow (ret);
255 return ret;
256}
257strong_alias (__ieee754_jn, __jn_finite)
258
259double
260__ieee754_yn (int n, double x)
261{
262 int32_t i, hx, ix, lx;
263 int32_t sign;
264 double a, b, temp, ret;
265
266 EXTRACT_WORDS (hx, lx, x);
267 ix = 0x7fffffff & hx;
268 /* if Y(n,NaN) is NaN */
269 if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))
270 return x + x;
271 sign = 1;
272 if (n < 0)
273 {
274 n = -n;
275 sign = 1 - ((n & 1) << 1);
276 }
277 if (n == 0)
278 return (__ieee754_y0 (x));
279 if (__glibc_unlikely ((ix | lx) == 0))
280 return -sign / zero;
281 /* -inf and overflow exception. */;
282 if (__glibc_unlikely (hx < 0))
283 return zero / (zero * x);
284 {
285 SET_RESTORE_ROUND (FE_TONEAREST);
286 if (n == 1)
287 {
288 ret = sign * __ieee754_y1 (x);
289 goto out;
290 }
291 if (__glibc_unlikely (ix == 0x7ff00000))
292 return zero;
293 if (ix >= 0x52D00000) /* x > 2**302 */
294 { /* (x >> n**2)
295 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
296 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
297 * Let s=sin(x), c=cos(x),
298 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
299 *
300 * n sin(xn)*sqt2 cos(xn)*sqt2
301 * ----------------------------------
302 * 0 s-c c+s
303 * 1 -s-c -c+s
304 * 2 -s+c -c-s
305 * 3 s+c c-s
306 */
307 double c;
308 double s;
309 __sincos (x, &s, &c);
310 switch (n & 3)
311 {
312 case 0: temp = s - c; break;
313 case 1: temp = -s - c; break;
314 case 2: temp = -s + c; break;
315 case 3: temp = s + c; break;
316 }
317 b = invsqrtpi * temp / __ieee754_sqrt (x);
318 }
319 else
320 {
321 uint32_t high;
322 a = __ieee754_y0 (x);
323 b = __ieee754_y1 (x);
324 /* quit if b is -inf */
325 GET_HIGH_WORD (high, b);
326 for (i = 1; i < n && high != 0xfff00000; i++)
327 {
328 temp = b;
329 b = ((double) (i + i) / x) * b - a;
330 GET_HIGH_WORD (high, b);
331 a = temp;
332 }
333 /* If B is +-Inf, set up errno accordingly. */
334 if (!isfinite (b))
335 __set_errno (ERANGE);
336 }
337 if (sign > 0)
338 ret = b;
339 else
340 ret = -b;
341 }
342 out:
343 if (isinf (ret))
344 ret = __copysign (DBL_MAX, ret) * DBL_MAX;
345 return ret;
346}
347strong_alias (__ieee754_yn, __yn_finite)
348