1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 /*
31 * floating point Bessel's function of the first and second kinds
32 * of order zero: j1(x),y1(x);
33 *
34 * Special cases:
35 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
36 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
37 */
38
39 #pragma weak j1 = __j1
40 #pragma weak y1 = __y1
41
42 #include "libm.h"
43 #include "libm_synonyms.h"
44 #include "libm_protos.h"
45 #include <math.h>
46 #include <values.h>
47
48 #define GENERIC double
49 static const GENERIC
50 zero = 0.0,
51 small = 1.0e-5,
52 tiny = 1.0e-20,
53 one = 1.0,
54 invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
55 tpi = 0.636619772367581343075535053490057448;
56
57 static GENERIC pone(GENERIC), qone(GENERIC);
58 static const GENERIC r0[4] = {
59 -6.250000000000002203053200981413218949548e-0002,
60 1.600998455640072901321605101981501263762e-0003,
61 -1.963888815948313758552511884390162864930e-0005,
62 8.263917341093549759781339713418201620998e-0008,
63 };
64 static const GENERIC s0[7] = {
65 1.0e0,
66 1.605069137643004242395356851797873766927e-0002,
67 1.149454623251299996428500249509098499383e-0004,
68 3.849701673735260970379681807910852327825e-0007,
69 };
70 static const GENERIC r1[12] = {
71 4.999999999999999995517408894340485471724e-0001,
72 -6.003825028120475684835384519945468075423e-0002,
73 2.301719899263321828388344461995355419832e-0003,
74 -4.208494869238892934859525221654040304068e-0005,
75 4.377745135188837783031540029700282443388e-0007,
76 -2.854106755678624335145364226735677754179e-0009,
77 1.234002865443952024332943901323798413689e-0011,
78 -3.645498437039791058951273508838177134310e-0014,
79 7.404320596071797459925377103787837414422e-0017,
80 -1.009457448277522275262808398517024439084e-0019,
81 8.520158355824819796968771418801019930585e-0023,
82 -3.458159926081163274483854614601091361424e-0026,
83 };
84 static const GENERIC s1[5] = {
85 1.0e0,
86 4.923499437590484879081138588998986303306e-0003,
87 1.054389489212184156499666953501976688452e-0005,
88 1.180768373106166527048240364872043816050e-0008,
89 5.942665743476099355323245707680648588540e-0012,
90 };
91
92 GENERIC
93 j1(GENERIC x) {
94 GENERIC z, d, s, c, ss, cc, r;
95 int i, sgn;
96
97 if (!finite(x))
98 return (one/x);
99 sgn = signbit(x);
100 x = fabs(x);
101 if (x > 8.00) {
102 s = sin(x);
103 c = cos(x);
104 /*
105 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
106 * where x0 = x-3pi/4
107 * Better formula:
108 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
109 * = 1/sqrt(2) * (sin(x) - cos(x))
110 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
111 * = -1/sqrt(2) * (cos(x) + sin(x))
112 * To avoid cancellation, use
113 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
114 * to compute the worse one.
115 */
116 if (x > 8.9e307) { /* x+x may overflow */
117 ss = -s-c;
118 cc = s-c;
119 } else if (signbit(s) != signbit(c)) {
120 cc = s - c;
121 ss = cos(x+x)/cc;
122 } else {
123 ss = -s-c;
124 cc = cos(x+x)/ss;
125 }
126 /*
127 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
128 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
129 */
130 if (x > 1.0e40)
131 d = (invsqrtpi*cc)/sqrt(x);
132 else
133 d = invsqrtpi*(pone(x)*cc-qone(x)*ss)/sqrt(x);
134
135 if (x > X_TLOSS) {
136 if (sgn != 0) { d = -d; x = -x; }
137 return (_SVID_libm_err(x, d, 36));
138 } else
139 if (sgn == 0)
140 return (d);
141 else
142 return (-d);
143 }
144 if (x <= small) {
145 if (x <= tiny)
146 d = 0.5*x;
147 else
148 d = x*(0.5-x*x*0.125);
149 if (sgn == 0)
150 return (d);
151 else
152 return (-d);
153 }
154 z = x*x;
155 if (x < 1.28) {
156 r = r0[3];
157 s = s0[3];
158 for (i = 2; i >= 0; i--) {
159 r = r*z + r0[i];
160 s = s*z + s0[i];
161 }
162 d = x*0.5+x*(z*(r/s));
163 } else {
164 r = r1[11];
165 for (i = 10; i >= 0; i--) r = r*z + r1[i];
166 s = s1[0]+z*(s1[1]+z*(s1[2]+z*(s1[3]+z*s1[4])));
167 d = x*(r/s);
168 }
169 if (sgn == 0)
170 return (d);
171 else
172 return (-d);
173 }
174
175 static const GENERIC u0[4] = {
176 -1.960570906462389461018983259589655961560e-0001,
177 4.931824118350661953459180060007970291139e-0002,
178 -1.626975871565393656845930125424683008677e-0003,
179 1.359657517926394132692884168082224258360e-0005,
180 };
181 static const GENERIC v0[5] = {
182 1.0e0,
183 2.565807214838390835108224713630901653793e-0002,
184 3.374175208978404268650522752520906231508e-0004,
185 2.840368571306070719539936935220728843177e-0006,
186 1.396387402048998277638900944415752207592e-0008,
187 };
188 static const GENERIC u1[12] = {
189 -1.960570906462389473336339614647555351626e-0001,
190 5.336268030335074494231369159933012844735e-0002,
191 -2.684137504382748094149184541866332033280e-0003,
192 5.737671618979185736981543498580051903060e-0005,
193 -6.642696350686335339171171785557663224892e-0007,
194 4.692417922568160354012347591960362101664e-0009,
195 -2.161728635907789319335231338621412258355e-0011,
196 6.727353419738316107197644431844194668702e-0014,
197 -1.427502986803861372125234355906790573422e-0016,
198 2.020392498726806769468143219616642940371e-0019,
199 -1.761371948595104156753045457888272716340e-0022,
200 7.352828391941157905175042420249225115816e-0026,
201 };
202 static const GENERIC v1[5] = {
203 1.0e0,
204 5.029187436727947764916247076102283399442e-0003,
205 1.102693095808242775074856548927801750627e-0005,
206 1.268035774543174837829534603830227216291e-0008,
207 6.579416271766610825192542295821308730206e-0012,
208 };
209
210
211 GENERIC
212 y1(GENERIC x) {
213 GENERIC z, d, s, c, ss, cc, u, v;
214 int i;
215
216 if (isnan(x))
217 return (x*x); /* + -> * for Cheetah */
218 if (x <= zero) {
219 if (x == zero)
220 /* return -one/zero; */
221 return (_SVID_libm_err(x, x, 10));
222 else
223 /* return zero/zero; */
224 return (_SVID_libm_err(x, x, 11));
225 }
226 if (x > 8.0) {
227 if (!finite(x))
228 return (zero);
229 s = sin(x);
230 c = cos(x);
231 /*
232 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
233 * where x0 = x-3pi/4
234 * Better formula:
235 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
236 * = 1/sqrt(2) * (sin(x) - cos(x))
237 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
238 * = -1/sqrt(2) * (cos(x) + sin(x))
239 * To avoid cancellation, use
240 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
241 * to compute the worse one.
242 */
243 if (x > 8.9e307) { /* x+x may overflow */
244 ss = -s-c;
245 cc = s-c;
246 } else if (signbit(s) != signbit(c)) {
247 cc = s - c;
248 ss = cos(x+x)/cc;
249 } else {
250 ss = -s-c;
251 cc = cos(x+x)/ss;
252 }
253 /*
254 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
255 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
256 */
257 if (x > 1.0e91)
258 d = (invsqrtpi*ss)/sqrt(x);
259 else
260 d = invsqrtpi*(pone(x)*ss+qone(x)*cc)/sqrt(x);
261
262 if (x > X_TLOSS)
263 return (_SVID_libm_err(x, d, 37));
264 else
265 return (d);
266 }
267 if (x <= tiny) {
268 return (-tpi/x);
269 }
270 z = x*x;
271 if (x < 1.28) {
272 u = u0[3]; v = v0[3]+z*v0[4];
273 for (i = 2; i >= 0; i--) {
274 u = u*z + u0[i];
275 v = v*z + v0[i];
276 }
277 } else {
278 for (u = u1[11], i = 10; i >= 0; i--) u = u*z+u1[i];
279 v = v1[0]+z*(v1[1]+z*(v1[2]+z*(v1[3]+z*v1[4])));
280 }
281 return (x*(u/v) + tpi*(j1(x)*log(x)-one/x));
282 }
283
284 static const GENERIC pr0[6] = {
285 -.4435757816794127857114720794e7,
286 -.9942246505077641195658377899e7,
287 -.6603373248364939109255245434e7,
288 -.1523529351181137383255105722e7,
289 -.1098240554345934672737413139e6,
290 -.1611616644324610116477412898e4,
291 };
292 static const GENERIC ps0[6] = {
293 -.4435757816794127856828016962e7,
294 -.9934124389934585658967556309e7,
295 -.6585339479723087072826915069e7,
296 -.1511809506634160881644546358e7,
297 -.1072638599110382011903063867e6,
298 -.1455009440190496182453565068e4,
299 };
300 static const GENERIC huge = 1.0e10;
301
302 static GENERIC
303 pone(GENERIC x) {
304 GENERIC s, r, t, z;
305 int i;
306 /* assume x > 8 */
307 if (x > huge)
308 return (one);
309
310 t = 8.0/x; z = t*t;
311 r = pr0[5]; s = ps0[5]+z;
312 for (i = 4; i >= 0; i--) {
313 r = z*r + pr0[i];
314 s = z*s + ps0[i];
315 }
316 return (r/s);
317 }
318
319
320 static const GENERIC qr0[6] = {
321 0.3322091340985722351859704442e5,
322 0.8514516067533570196555001171e5,
323 0.6617883658127083517939992166e5,
324 0.1849426287322386679652009819e5,
325 0.1706375429020768002061283546e4,
326 0.3526513384663603218592175580e2,
327 };
328 static const GENERIC qs0[6] = {
329 0.7087128194102874357377502472e6,
330 0.1819458042243997298924553839e7,
331 0.1419460669603720892855755253e7,
332 0.4002944358226697511708610813e6,
333 0.3789022974577220264142952256e5,
334 0.8638367769604990967475517183e3,
335 };
336
337 static GENERIC
338 qone(GENERIC x) {
339 GENERIC s, r, t, z;
340 int i;
341 if (x > huge)
342 return (0.375/x);
343
344 t = 8.0/x; z = t*t;
345 /* assume x > 8 */
346 r = qr0[5]; s = qs0[5]+z;
347 for (i = 4; i >= 0; i--) {
348 r = z*r + qr0[i];
349 s = z*s + qs0[i];
350 }
351 return (t*(r/s));
352 }