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-narrow-eval.h>
43 #include <math_private.h>
44 #include <fenv_private.h>
45 #include <math-underflow.h>
46 #include <libm-alias-finite.h>
47
48 static const double
49 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
50 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
51 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
52
53 static const double zero = 0.00000000000000000000e+00;
54
55 double
__ieee754_jn(int n,double x)56 __ieee754_jn (int n, double x)
57 {
58 int32_t i, hx, ix, lx, sgn;
59 double a, b, temp, di, ret;
60 double z, w;
61
62 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
63 * Thus, J(-n,x) = J(n,-x)
64 */
65 EXTRACT_WORDS (hx, lx, x);
66 ix = 0x7fffffff & hx;
67 /* if J(n,NaN) is NaN */
68 if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))
69 return x + x;
70 if (n < 0)
71 {
72 n = -n;
73 x = -x;
74 hx ^= 0x80000000;
75 }
76 if (n == 0)
77 return (__ieee754_j0 (x));
78 if (n == 1)
79 return (__ieee754_j1 (x));
80 sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */
81 x = fabs (x);
82 {
83 SET_RESTORE_ROUND (FE_TONEAREST);
84 if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
85 /* if x is 0 or inf */
86 return sgn == 1 ? -zero : zero;
87 else if ((double) n <= x)
88 {
89 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
90 if (ix >= 0x52D00000) /* x > 2**302 */
91 { /* (x >> n**2)
92 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
93 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
94 * Let s=sin(x), c=cos(x),
95 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
96 *
97 * n sin(xn)*sqt2 cos(xn)*sqt2
98 * ----------------------------------
99 * 0 s-c c+s
100 * 1 -s-c -c+s
101 * 2 -s+c -c-s
102 * 3 s+c c-s
103 */
104 double s;
105 double c;
106 __sincos (x, &s, &c);
107 switch (n & 3)
108 {
109 case 0: temp = c + s; break;
110 case 1: temp = -c + s; break;
111 case 2: temp = -c - s; break;
112 case 3: temp = c - s; break;
113 default: __builtin_unreachable ();
114 }
115 b = invsqrtpi * temp / sqrt (x);
116 }
117 else
118 {
119 a = __ieee754_j0 (x);
120 b = __ieee754_j1 (x);
121 for (i = 1; i < n; i++)
122 {
123 temp = b;
124 b = b * ((double) (i + i) / x) - a; /* avoid underflow */
125 a = temp;
126 }
127 }
128 }
129 else
130 {
131 if (ix < 0x3e100000) /* x < 2**-29 */
132 { /* x is tiny, return the first Taylor expansion of J(n,x)
133 * J(n,x) = 1/n!*(x/2)^n - ...
134 */
135 if (n > 33) /* underflow */
136 b = zero;
137 else
138 {
139 temp = x * 0.5; b = temp;
140 for (a = one, i = 2; i <= n; i++)
141 {
142 a *= (double) i; /* a = n! */
143 b *= temp; /* b = (x/2)^n */
144 }
145 b = b / a;
146 }
147 }
148 else
149 {
150 /* use backward recurrence */
151 /* x x^2 x^2
152 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
153 * 2n - 2(n+1) - 2(n+2)
154 *
155 * 1 1 1
156 * (for large x) = ---- ------ ------ .....
157 * 2n 2(n+1) 2(n+2)
158 * -- - ------ - ------ -
159 * x x x
160 *
161 * Let w = 2n/x and h=2/x, then the above quotient
162 * is equal to the continued fraction:
163 * 1
164 * = -----------------------
165 * 1
166 * w - -----------------
167 * 1
168 * w+h - ---------
169 * w+2h - ...
170 *
171 * To determine how many terms needed, let
172 * Q(0) = w, Q(1) = w(w+h) - 1,
173 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
174 * When Q(k) > 1e4 good for single
175 * When Q(k) > 1e9 good for double
176 * When Q(k) > 1e17 good for quadruple
177 */
178 /* determine k */
179 double t, v;
180 double q0, q1, h, tmp; int32_t k, m;
181 w = (n + n) / (double) x; h = 2.0 / (double) x;
182 q0 = w; z = w + h; q1 = w * z - 1.0; k = 1;
183 while (q1 < 1.0e9)
184 {
185 k += 1; z += h;
186 tmp = z * q1 - q0;
187 q0 = q1;
188 q1 = tmp;
189 }
190 m = n + n;
191 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
192 t = one / (i / x - t);
193 a = t;
194 b = one;
195 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
196 * Hence, if n*(log(2n/x)) > ...
197 * single 8.8722839355e+01
198 * double 7.09782712893383973096e+02
199 * long double 1.1356523406294143949491931077970765006170e+04
200 * then recurrent value may overflow and the result is
201 * likely underflow to zero
202 */
203 tmp = n;
204 v = two / x;
205 tmp = tmp * __ieee754_log (fabs (v * tmp));
206 if (tmp < 7.09782712893383973096e+02)
207 {
208 for (i = n - 1, di = (double) (i + i); i > 0; i--)
209 {
210 temp = b;
211 b *= di;
212 b = b / x - a;
213 a = temp;
214 di -= two;
215 }
216 }
217 else
218 {
219 for (i = n - 1, di = (double) (i + i); i > 0; i--)
220 {
221 temp = b;
222 b *= di;
223 b = b / x - a;
224 a = temp;
225 di -= two;
226 /* scale b to avoid spurious overflow */
227 if (b > 1e100)
228 {
229 a /= b;
230 t /= b;
231 b = one;
232 }
233 }
234 }
235 /* j0() and j1() suffer enormous loss of precision at and
236 * near zero; however, we know that their zero points never
237 * coincide, so just choose the one further away from zero.
238 */
239 z = __ieee754_j0 (x);
240 w = __ieee754_j1 (x);
241 if (fabs (z) >= fabs (w))
242 b = (t * z / b);
243 else
244 b = (t * w / a);
245 }
246 }
247 if (sgn == 1)
248 ret = -b;
249 else
250 ret = b;
251 ret = math_narrow_eval (ret);
252 }
253 if (ret == 0)
254 {
255 ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
256 __set_errno (ERANGE);
257 }
258 else
259 math_check_force_underflow (ret);
260 return ret;
261 }
libm_alias_finite(__ieee754_jn,__jn)262 libm_alias_finite (__ieee754_jn, __jn)
263
264 double
265 __ieee754_yn (int n, double x)
266 {
267 int32_t i, hx, ix, lx;
268 int32_t sign;
269 double a, b, temp, ret;
270
271 EXTRACT_WORDS (hx, lx, x);
272 ix = 0x7fffffff & hx;
273 /* if Y(n,NaN) is NaN */
274 if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))
275 return x + x;
276 sign = 1;
277 if (n < 0)
278 {
279 n = -n;
280 sign = 1 - ((n & 1) << 1);
281 }
282 if (n == 0)
283 return (__ieee754_y0 (x));
284 if (__glibc_unlikely ((ix | lx) == 0))
285 return -sign / zero;
286 /* -inf and overflow exception. */;
287 if (__glibc_unlikely (hx < 0))
288 return zero / (zero * x);
289 {
290 SET_RESTORE_ROUND (FE_TONEAREST);
291 if (n == 1)
292 {
293 ret = sign * __ieee754_y1 (x);
294 goto out;
295 }
296 if (__glibc_unlikely (ix == 0x7ff00000))
297 return zero;
298 if (ix >= 0x52D00000) /* x > 2**302 */
299 { /* (x >> n**2)
300 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
301 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
302 * Let s=sin(x), c=cos(x),
303 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
304 *
305 * n sin(xn)*sqt2 cos(xn)*sqt2
306 * ----------------------------------
307 * 0 s-c c+s
308 * 1 -s-c -c+s
309 * 2 -s+c -c-s
310 * 3 s+c c-s
311 */
312 double c;
313 double s;
314 __sincos (x, &s, &c);
315 switch (n & 3)
316 {
317 case 0: temp = s - c; break;
318 case 1: temp = -s - c; break;
319 case 2: temp = -s + c; break;
320 case 3: temp = s + c; break;
321 default: __builtin_unreachable ();
322 }
323 b = invsqrtpi * temp / sqrt (x);
324 }
325 else
326 {
327 uint32_t high;
328 a = __ieee754_y0 (x);
329 b = __ieee754_y1 (x);
330 /* quit if b is -inf */
331 GET_HIGH_WORD (high, b);
332 for (i = 1; i < n && high != 0xfff00000; i++)
333 {
334 temp = b;
335 b = ((double) (i + i) / x) * b - a;
336 GET_HIGH_WORD (high, b);
337 a = temp;
338 }
339 /* If B is +-Inf, set up errno accordingly. */
340 if (!isfinite (b))
341 __set_errno (ERANGE);
342 }
343 if (sign > 0)
344 ret = b;
345 else
346 ret = -b;
347 }
348 out:
349 if (isinf (ret))
350 ret = copysign (DBL_MAX, ret) * DBL_MAX;
351 return ret;
352 }
353 libm_alias_finite (__ieee754_yn, __yn)
354