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_protos.h"
44 #include <math.h>
45 #include <values.h>
46
47 #define GENERIC double
48 static const GENERIC
49 zero = 0.0,
50 small = 1.0e-5,
51 tiny = 1.0e-20,
52 one = 1.0,
53 invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
54 tpi = 0.636619772367581343075535053490057448;
55
56 static GENERIC pone(GENERIC), qone(GENERIC);
57 static const GENERIC r0[4] = {
58 -6.250000000000002203053200981413218949548e-0002,
59 1.600998455640072901321605101981501263762e-0003,
60 -1.963888815948313758552511884390162864930e-0005,
61 8.263917341093549759781339713418201620998e-0008,
62 };
63 static const GENERIC s0[7] = {
64 1.0e0,
65 1.605069137643004242395356851797873766927e-0002,
66 1.149454623251299996428500249509098499383e-0004,
67 3.849701673735260970379681807910852327825e-0007,
68 };
69 static const GENERIC r1[12] = {
70 4.999999999999999995517408894340485471724e-0001,
71 -6.003825028120475684835384519945468075423e-0002,
72 2.301719899263321828388344461995355419832e-0003,
73 -4.208494869238892934859525221654040304068e-0005,
74 4.377745135188837783031540029700282443388e-0007,
75 -2.854106755678624335145364226735677754179e-0009,
76 1.234002865443952024332943901323798413689e-0011,
77 -3.645498437039791058951273508838177134310e-0014,
78 7.404320596071797459925377103787837414422e-0017,
79 -1.009457448277522275262808398517024439084e-0019,
80 8.520158355824819796968771418801019930585e-0023,
81 -3.458159926081163274483854614601091361424e-0026,
82 };
83 static const GENERIC s1[5] = {
84 1.0e0,
85 4.923499437590484879081138588998986303306e-0003,
86 1.054389489212184156499666953501976688452e-0005,
87 1.180768373106166527048240364872043816050e-0008,
88 5.942665743476099355323245707680648588540e-0012,
89 };
90
91 GENERIC
92 j1(GENERIC x) {
93 GENERIC z, d, s, c, ss, cc, r;
94 int i, sgn;
95
96 if (!finite(x))
97 return (one/x);
98 sgn = signbit(x);
99 x = fabs(x);
100 if (x > 8.00) {
101 s = sin(x);
102 c = cos(x);
103 /*
104 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
105 * where x0 = x-3pi/4
106 * Better formula:
107 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
108 * = 1/sqrt(2) * (sin(x) - cos(x))
109 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
110 * = -1/sqrt(2) * (cos(x) + sin(x))
111 * To avoid cancellation, use
112 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
113 * to compute the worse one.
114 */
115 if (x > 8.9e307) { /* x+x may overflow */
116 ss = -s-c;
117 cc = s-c;
118 } else if (signbit(s) != signbit(c)) {
119 cc = s - c;
120 ss = cos(x+x)/cc;
121 } else {
122 ss = -s-c;
123 cc = cos(x+x)/ss;
124 }
125 /*
126 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
127 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
128 */
129 if (x > 1.0e40)
130 d = (invsqrtpi*cc)/sqrt(x);
131 else
132 d = invsqrtpi*(pone(x)*cc-qone(x)*ss)/sqrt(x);
133
134 if (x > X_TLOSS) {
135 if (sgn != 0) { d = -d; x = -x; }
136 return (_SVID_libm_err(x, d, 36));
137 } else
138 if (sgn == 0)
139 return (d);
140 else
141 return (-d);
142 }
143 if (x <= small) {
144 if (x <= tiny)
145 d = 0.5*x;
146 else
147 d = x*(0.5-x*x*0.125);
148 if (sgn == 0)
149 return (d);
150 else
151 return (-d);
152 }
153 z = x*x;
154 if (x < 1.28) {
155 r = r0[3];
156 s = s0[3];
157 for (i = 2; i >= 0; i--) {
158 r = r*z + r0[i];
159 s = s*z + s0[i];
160 }
161 d = x*0.5+x*(z*(r/s));
162 } else {
163 r = r1[11];
164 for (i = 10; i >= 0; i--) r = r*z + r1[i];
165 s = s1[0]+z*(s1[1]+z*(s1[2]+z*(s1[3]+z*s1[4])));
166 d = x*(r/s);
167 }
168 if (sgn == 0)
169 return (d);
170 else
171 return (-d);
172 }
173
174 static const GENERIC u0[4] = {
175 -1.960570906462389461018983259589655961560e-0001,
176 4.931824118350661953459180060007970291139e-0002,
177 -1.626975871565393656845930125424683008677e-0003,
178 1.359657517926394132692884168082224258360e-0005,
179 };
180 static const GENERIC v0[5] = {
181 1.0e0,
182 2.565807214838390835108224713630901653793e-0002,
183 3.374175208978404268650522752520906231508e-0004,
184 2.840368571306070719539936935220728843177e-0006,
185 1.396387402048998277638900944415752207592e-0008,
186 };
187 static const GENERIC u1[12] = {
188 -1.960570906462389473336339614647555351626e-0001,
189 5.336268030335074494231369159933012844735e-0002,
190 -2.684137504382748094149184541866332033280e-0003,
191 5.737671618979185736981543498580051903060e-0005,
192 -6.642696350686335339171171785557663224892e-0007,
193 4.692417922568160354012347591960362101664e-0009,
194 -2.161728635907789319335231338621412258355e-0011,
195 6.727353419738316107197644431844194668702e-0014,
196 -1.427502986803861372125234355906790573422e-0016,
197 2.020392498726806769468143219616642940371e-0019,
198 -1.761371948595104156753045457888272716340e-0022,
199 7.352828391941157905175042420249225115816e-0026,
200 };
201 static const GENERIC v1[5] = {
202 1.0e0,
203 5.029187436727947764916247076102283399442e-0003,
204 1.102693095808242775074856548927801750627e-0005,
205 1.268035774543174837829534603830227216291e-0008,
206 6.579416271766610825192542295821308730206e-0012,
207 };
208
209
210 GENERIC
211 y1(GENERIC x) {
212 GENERIC z, d, s, c, ss, cc, u, v;
213 int i;
214
215 if (isnan(x))
216 return (x*x); /* + -> * for Cheetah */
217 if (x <= zero) {
218 if (x == zero)
219 /* return -one/zero; */
220 return (_SVID_libm_err(x, x, 10));
221 else
222 /* return zero/zero; */
223 return (_SVID_libm_err(x, x, 11));
224 }
225 if (x > 8.0) {
226 if (!finite(x))
227 return (zero);
228 s = sin(x);
229 c = cos(x);
230 /*
231 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
232 * where x0 = x-3pi/4
233 * Better formula:
234 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
235 * = 1/sqrt(2) * (sin(x) - cos(x))
236 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
237 * = -1/sqrt(2) * (cos(x) + sin(x))
238 * To avoid cancellation, use
239 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
240 * to compute the worse one.
241 */
242 if (x > 8.9e307) { /* x+x may overflow */
243 ss = -s-c;
244 cc = s-c;
245 } else if (signbit(s) != signbit(c)) {
246 cc = s - c;
247 ss = cos(x+x)/cc;
248 } else {
249 ss = -s-c;
250 cc = cos(x+x)/ss;
251 }
252 /*
253 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
254 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
255 */
256 if (x > 1.0e91)
257 d = (invsqrtpi*ss)/sqrt(x);
258 else
259 d = invsqrtpi*(pone(x)*ss+qone(x)*cc)/sqrt(x);
260
261 if (x > X_TLOSS)
262 return (_SVID_libm_err(x, d, 37));
263 else
264 return (d);
265 }
266 if (x <= tiny) {
267 return (-tpi/x);
268 }
269 z = x*x;
270 if (x < 1.28) {
271 u = u0[3]; v = v0[3]+z*v0[4];
272 for (i = 2; i >= 0; i--) {
273 u = u*z + u0[i];
274 v = v*z + v0[i];
275 }
276 } else {
277 for (u = u1[11], i = 10; i >= 0; i--) u = u*z+u1[i];
278 v = v1[0]+z*(v1[1]+z*(v1[2]+z*(v1[3]+z*v1[4])));
279 }
280 return (x*(u/v) + tpi*(j1(x)*log(x)-one/x));
281 }
282
283 static const GENERIC pr0[6] = {
284 -.4435757816794127857114720794e7,
285 -.9942246505077641195658377899e7,
286 -.6603373248364939109255245434e7,
287 -.1523529351181137383255105722e7,
288 -.1098240554345934672737413139e6,
289 -.1611616644324610116477412898e4,
290 };
291 static const GENERIC ps0[6] = {
292 -.4435757816794127856828016962e7,
293 -.9934124389934585658967556309e7,
294 -.6585339479723087072826915069e7,
295 -.1511809506634160881644546358e7,
296 -.1072638599110382011903063867e6,
297 -.1455009440190496182453565068e4,
298 };
299 static const GENERIC huge = 1.0e10;
300
301 static GENERIC
302 pone(GENERIC x) {
303 GENERIC s, r, t, z;
304 int i;
305 /* assume x > 8 */
306 if (x > huge)
307 return (one);
308
309 t = 8.0/x; z = t*t;
310 r = pr0[5]; s = ps0[5]+z;
311 for (i = 4; i >= 0; i--) {
312 r = z*r + pr0[i];
313 s = z*s + ps0[i];
314 }
315 return (r/s);
316 }
317
318
319 static const GENERIC qr0[6] = {
320 0.3322091340985722351859704442e5,
321 0.8514516067533570196555001171e5,
322 0.6617883658127083517939992166e5,
323 0.1849426287322386679652009819e5,
324 0.1706375429020768002061283546e4,
325 0.3526513384663603218592175580e2,
326 };
327 static const GENERIC qs0[6] = {
328 0.7087128194102874357377502472e6,
329 0.1819458042243997298924553839e7,
330 0.1419460669603720892855755253e7,
331 0.4002944358226697511708610813e6,
332 0.3789022974577220264142952256e5,
333 0.8638367769604990967475517183e3,
334 };
335
336 static GENERIC
337 qone(GENERIC x) {
338 GENERIC s, r, t, z;
339 int i;
340 if (x > huge)
341 return (0.375/x);
342
343 t = 8.0/x; z = t*t;
344 /* assume x > 8 */
345 r = qr0[5]; s = qs0[5]+z;
346 for (i = 4; i >= 0; i--) {
347 r = z*r + qr0[i];
348 s = z*s + qs0[i];
349 }
350 return (t*(r/s));
351 }