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