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: j0(x),y0(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 j0 = __j0
40 #pragma weak y0 = __y0
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-18,
53 one = 1.0,
54 eight = 8.0,
55 invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
56 tpi = 0.636619772367581343075535053490057448;
57
58 static GENERIC pzero(GENERIC), qzero(GENERIC);
59 static const GENERIC r0[4] = { /* [1.e-5, 1.28] */
60 -2.500000000000003622131880894830476755537e-0001,
61 1.095597547334830263234433855932375353303e-0002,
62 -1.819734750463320921799187258987098087697e-0004,
63 9.977001946806131657544212501069893930846e-0007,
64 };
65 static const GENERIC s0[4] = { /* [1.e-5, 1.28] */
66 1.0,
67 1.867609810662950169966782360588199673741e-0002,
68 1.590389206181565490878430827706972074208e-0004,
69 6.520867386742583632375520147714499522721e-0007,
70 };
71 static const GENERIC r1[9] = { /* [1.28,8] */
72 9.999999999999999942156495584397047660949e-0001,
73 -2.389887722731319130476839836908143731281e-0001,
74 1.293359476138939027791270393439493640570e-0002,
75 -2.770985642343140122168852400228563364082e-0004,
76 2.905241575772067678086738389169625218912e-0006,
77 -1.636846356264052597969042009265043251279e-0008,
78 5.072306160724884775085431059052611737827e-0011,
79 -8.187060730684066824228914775146536139112e-0014,
80 5.422219326959949863954297860723723423842e-0017,
81 };
82 static const GENERIC s1[9] = { /* [1.28,8] */
83 1.0,
84 1.101122772686807702762104741932076228349e-0002,
85 6.140169310641649223411427764669143978228e-0005,
86 2.292035877515152097976946119293215705250e-0007,
87 6.356910426504644334558832036362219583789e-0010,
88 1.366626326900219555045096999553948891401e-0012,
89 2.280399586866739522891837985560481180088e-0015,
90 2.801559820648939665270492520004836611187e-0018,
91 2.073101088320349159764410261466350732968e-0021,
92 };
93
94 GENERIC
95 j0(GENERIC x) {
96 GENERIC z, s, c, ss, cc, r, u, v, ox;
97 int i;
98
99 if (isnan(x))
100 return (x*x); /* + -> * for Cheetah */
101 ox = x;
102 x = fabs(x);
103 if (x > 8.0) {
104 if (!finite(x))
105 return (zero);
106 s = sin(x);
107 c = cos(x);
108 /*
109 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
110 * where x0 = x-pi/4
111 * Better formula:
112 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
113 * = 1/sqrt(2) * (cos(x) + sin(x))
114 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
115 * = 1/sqrt(2) * (sin(x) - cos(x))
116 * To avoid cancellation, use
117 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
118 * to compute the worse one.
119 */
120 if (x > 8.9e307) { /* x+x may overflow */
121 ss = s-c;
122 cc = s+c;
123 } else if (signbit(s) != signbit(c)) {
124 ss = s - c;
125 cc = -cos(x+x)/ss;
126 } else {
127 cc = s + c;
128 ss = -cos(x+x)/cc;
129 }
130 /*
131 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
132 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
133 */
134 if (x > 1.0e40) z = (invsqrtpi*cc)/sqrt(x);
135 else {
136 u = pzero(x); v = qzero(x);
137 z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
138 }
139 /* force to pass SVR4 even the result is wrong (sign) */
140 if (x > X_TLOSS)
141 return (_SVID_libm_err(ox, z, 34));
142 else
143 return (z);
144 }
145 if (x <= small) {
146 if (x <= tiny)
147 return (one-x);
148 else
149 return (one-x*x*0.25);
150 }
151 z = x*x;
152 if (x <= 1.28) {
153 r = r0[0]+z*(r0[1]+z*(r0[2]+z*r0[3]));
154 s = s0[0]+z*(s0[1]+z*(s0[2]+z*s0[3]));
155 return (one + z*(r/s));
156 } else {
157 for (r = r1[8], s = s1[8], i = 7; i >= 0; i--) {
158 r = r*z + r1[i];
159 s = s*z + s1[i];
160 }
161 return (r/s);
162 }
163 }
164
165 static const GENERIC u0[13] = {
166 -7.380429510868722526754723020704317641941e-0002,
167 1.772607102684869924301459663049874294814e-0001,
168 -1.524370666542713828604078090970799356306e-0002,
169 4.650819100693891757143771557629924591915e-0004,
170 -7.125768872339528975036316108718239946022e-0006,
171 6.411017001656104598327565004771515257146e-0008,
172 -3.694275157433032553021246812379258781665e-0010,
173 1.434364544206266624252820889648445263842e-0012,
174 -3.852064731859936455895036286874139896861e-0015,
175 7.182052899726138381739945881914874579696e-0018,
176 -9.060556574619677567323741194079797987200e-0021,
177 7.124435467408860515265552217131230511455e-0024,
178 -2.709726774636397615328813121715432044771e-0027,
179 };
180 static const GENERIC v0[5] = {
181 1.0,
182 4.678678931512549002587702477349214886475e-0003,
183 9.486828955529948534822800829497565178985e-0006,
184 1.001495929158861646659010844136682454906e-0008,
185 4.725338116256021660204443235685358593611e-0012,
186 };
187
188 GENERIC
189 y0(GENERIC x) {
190 GENERIC z, /* d, */ s, c, ss, cc, u, v;
191 int i;
192
193 if (isnan(x))
194 return (x*x); /* + -> * for Cheetah */
195 if (x <= zero) {
196 if (x == zero)
197 /* d= -one/(x-x); */
198 return (_SVID_libm_err(x, x, 8));
199 else
200 /* d = zero/(x-x); */
201 return (_SVID_libm_err(x, x, 9));
202 }
203 if (x > 8.0) {
204 if (!finite(x))
205 return (zero);
206 s = sin(x);
207 c = cos(x);
208 /*
209 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
210 * where x0 = x-pi/4
211 * Better formula:
212 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
213 * = 1/sqrt(2) * (cos(x) + sin(x))
214 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
215 * = 1/sqrt(2) * (sin(x) - cos(x))
216 * To avoid cancellation, use
217 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
218 * to compute the worse one.
219 */
220 if (x > 8.9e307) { /* x+x may overflow */
221 ss = s-c;
222 cc = s+c;
223 } else if (signbit(s) != signbit(c)) {
224 ss = s - c;
225 cc = -cos(x+x)/ss;
226 } else {
227 cc = s + c;
228 ss = -cos(x+x)/cc;
229 }
230 /*
231 * j0(x) = 1/sqrt(pi*x) * (P(0,x)*cc - Q(0,x)*ss)
232 * y0(x) = 1/sqrt(pi*x) * (P(0,x)*ss + Q(0,x)*cc)
233 */
234 if (x > 1.0e40)
235 z = (invsqrtpi*ss)/sqrt(x);
236 else
237 z = invsqrtpi*(pzero(x)*ss+qzero(x)*cc)/sqrt(x);
238 if (x > X_TLOSS)
239 return (_SVID_libm_err(x, z, 35));
240 else
241 return (z);
242
243 }
244 if (x <= tiny) {
245 return (u0[0] + tpi*log(x));
246 }
247 z = x*x;
248 for (u = u0[12], i = 11; i >= 0; i--) u = u*z + u0[i];
249 v = v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4])));
250 return (u/v + tpi*(j0(x)*log(x)));
251 }
252
253 static const GENERIC pr[7] = { /* [8 -- inf] pzero 6550 */
254 .4861344183386052721391238447e5,
255 .1377662549407112278133438945e6,
256 .1222466364088289731869114004e6,
257 .4107070084315176135583353374e5,
258 .5026073801860637125889039915e4,
259 .1783193659125479654541542419e3,
260 .88010344055383421691677564e0,
261 };
262 static const GENERIC ps[7] = { /* [8 -- inf] pzero 6550 */
263 .4861344183386052721414037058e5,
264 .1378196632630384670477582699e6,
265 .1223967185341006542748936787e6,
266 .4120150243795353639995862617e5,
267 .5068271181053546392490184353e4,
268 .1829817905472769960535671664e3,
269 1.0,
270 };
271 static const GENERIC huge = 1.0e10;
272
273 static GENERIC
274 pzero(GENERIC x) {
275 GENERIC s, r, t, z;
276 int i;
277 if (x > huge)
278 return (one);
279 t = eight/x; z = t*t;
280 r = pr[5]+z*pr[6];
281 s = ps[5]+z;
282 for (i = 4; i >= 0; i--) {
283 r = r*z + pr[i];
284 s = s*z + ps[i];
285 }
286 return (r/s);
287 }
288
289 static const GENERIC qr[7] = { /* [8 -- inf] qzero 6950 */
290 -.1731210995701068539185611951e3,
291 -.5522559165936166961235240613e3,
292 -.5604935606637346590614529613e3,
293 -.2200430300226009379477365011e3,
294 -.323869355375648849771296746e2,
295 -.14294979207907956223499258e1,
296 -.834690374102384988158918e-2,
297 };
298 static const GENERIC qs[7] = { /* [8 -- inf] qzero 6950 */
299 .1107975037248683865326709645e5,
300 .3544581680627082674651471873e5,
301 .3619118937918394132179019059e5,
302 .1439895563565398007471485822e5,
303 .2190277023344363955930226234e4,
304 .106695157020407986137501682e3,
305 1.0,
306 };
307
308 static GENERIC
309 qzero(GENERIC x) {
310 GENERIC s, r, t, z;
311 int i;
312 if (x > huge)
313 return (-0.125/x);
314 t = eight/x; z = t*t;
315 r = qr[5]+z*qr[6];
316 s = qs[5]+z;
317 for (i = 4; i >= 0; i--) {
318 r = r*z + qr[i];
319 s = s*z + qs[i];
320 }
321 return (t*(r/s));
322 }