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 __j1l = j1l
40 #pragma weak __y1l = y1l
41
42 #include "libm.h"
43 #include "longdouble.h"
44 #include <math.h>
45 #if defined(__SUNPRO_C)
46 #include <sunmath.h>
47 #endif
48
49 #define GENERIC long double
50 static GENERIC
51 zero = 0.0L,
52 small = 1.0e-9L,
53 tiny = 1.0e-38L,
54 one = 1.0L,
55 five = 5.0L,
56 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
57 tpi = 0.636619772367581343075535053490057448L;
58
59 static GENERIC pone(), qone();
60 static GENERIC r0[7] = {
61 -6.249999999999999999999999999999999627320e-0002L,
62 1.940606727194041716205384618494641565464e-0003L,
63 -3.005630423155733701856481469986459043883e-0005L,
64 2.345586219403918667468341047369572169358e-0007L,
65 -9.976809285885253587529010109133336669724e-0010L,
66 2.218743258363623946078958783775107473381e-0012L,
67 -2.071079656218700604767650924103578046280e-0015L,
68 };
69 static GENERIC s0[7] = {
70 1.0e0L,
71 1.061695903156199920738051277075003059555e-0002L,
72 5.521860513111180371566951179398862692060e-0005L,
73 1.824214367413754193524107877084979441407e-0007L,
74 4.098957778439576834818838198039029353925e-0010L,
75 6.047735079699666389853240090925264056197e-0013L,
76 4.679044728878836197247923279512047035041e-0016L,
77 };
78
79 GENERIC
80 j1l(x) GENERIC x; {
81 GENERIC z, d, s, c, ss, cc, r;
82 int i, sgn;
83
84 if (!finitel(x))
85 return (one/x);
86 sgn = signbitl(x);
87 x = fabsl(x);
88 if (x > 1.28L) {
89 s = sinl(x);
90 c = cosl(x);
91 /*
92 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
93 * where x0 = x-3pi/4
94 * Better formula:
95 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
96 * = 1/sqrt(2) * (sin(x) - cos(x))
97 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
98 * = -1/sqrt(2) * (cos(x) + sin(x))
99 * To avoid cancellation, use
100 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
101 * to compute the worse one.
102 */
103 if (x > 1.0e2450L) { /* x+x may overflow */
104 ss = -s-c;
105 cc = s-c;
106 } else if (signbitl(s) != signbitl(c)) {
107 cc = s - c;
108 ss = cosl(x+x)/cc;
109 } else {
110 ss = -s-c;
111 cc = cosl(x+x)/ss;
112 }
113 /*
114 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
115 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
116 */
117 if (x > 1.0e120L)
118 return (invsqrtpi*cc)/sqrtl(x);
119 d = invsqrtpi*(pone(x)*cc-qone(x)*ss)/sqrtl(x);
120 if (sgn == 0)
121 return (d);
122 else
123 return (-d);
124 }
125 if (x <= small) {
126 if (x <= tiny) d = 0.5L*x;
127 else d = x*(0.5L-x*x*0.125L);
128 if (sgn == 0)
129 return (d);
130 else
131 return (-d);
132 }
133 z = x*x;
134 r = r0[6];
135 s = s0[6];
136 for (i = 5; i >= 0; i--) {
137 r = r*z + r0[i];
138 s = s*z + s0[i];
139 }
140 d = x*0.5L+x*(z*(r/s));
141 if (sgn == 0)
142 return (d);
143 else
144 return (-d);
145 }
146
147 static GENERIC u0[7] = {
148 -1.960570906462389484060557273467558703503e-0001L,
149 5.166389353148318460304315890665450006495e-0002L,
150 -2.229699464105910913337190798743451115604e-0003L,
151 3.625437034548863342715657067759078267158e-0005L,
152 -2.689902826993117212255524537353883987171e-0007L,
153 9.304570592456930912969387719010256018466e-0010L,
154 -1.234878126794286643318321347997500346131e-0012L,
155 };
156 static GENERIC v0[8] = {
157 1.0e0L,
158 1.369394302535807332517110204820556695644e-0002L,
159 9.508438148097659501433367062605935379588e-0005L,
160 4.399007309420092056052714797296467565655e-0007L,
161 1.488083087443756398305819693177715000787e-0009L,
162 3.751609832625793536245746965768587624922e-0012L,
163 6.680926434086257291872903276124244131448e-0015L,
164 6.676602383908906988160099057991121446058e-0018L,
165 };
166
167 GENERIC
168 y1l(x) GENERIC x; {
169 GENERIC z, s, c, ss, cc, u, v;
170 int i;
171
172 if (isnanl(x))
173 return (x+x);
174 if (x <= zero) {
175 if (x == zero)
176 return (-one/zero);
177 else
178 return (zero/zero);
179 }
180 if (x > 1.28L) {
181 if (!finitel(x))
182 return (zero);
183 s = sinl(x);
184 c = cosl(x);
185 /*
186 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
187 * where x0 = x-3pi/4
188 * Better formula:
189 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
190 * = 1/sqrt(2) * (sin(x) - cos(x))
191 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
192 * = -1/sqrt(2) * (cos(x) + sin(x))
193 * To avoid cancellation, use
194 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
195 * to compute the worse one.
196 */
197 if (x > 1.0e2450L) { /* x+x may overflow */
198 ss = -s-c;
199 cc = s-c;
200 } else if (signbitl(s) != signbitl(c)) {
201 cc = s - c;
202 ss = cosl(x+x)/cc;
203 } else {
204 ss = -s-c;
205 cc = cosl(x+x)/ss;
206 }
207 /*
208 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
209 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
210 */
211 if (x > 1.0e91L)
212 return (invsqrtpi*ss)/sqrtl(x);
213 return (invsqrtpi*(pone(x)*ss+qone(x)*cc)/sqrtl(x));
214 }
215 if (x <= tiny) {
216 return (-tpi/x);
217 }
218 z = x*x;
219 u = u0[6]; v = v0[6]+z*v0[7];
220 for (i = 5; i >= 0; i--) {
221 u = u*z + u0[i];
222 v = v*z + v0[i];
223 }
224 return (x*(u/v) + tpi*(j1l(x)*logl(x)-one/x));
225 }
226
227 static GENERIC pr0[12] = {
228 1.000000000000000000000000000000000000267e+0000L,
229 1.060717875045891455602180843276758003035e+0003L,
230 4.344347542892127024446687712181105852335e+0005L,
231 8.915680220724007016377924252717410457094e+0007L,
232 9.969502259938406062809873257569171272819e+0009L,
233 6.200290193138613035646510338707386316595e+0011L,
234 2.105978548788015119851815854422247330118e+0013L,
235 3.696635772784601239371730810311998368948e+0014L,
236 3.015913097920694682057958412534134515156e+0015L,
237 9.370298471339353098123277427328592725921e+0015L,
238 7.190349005196335967340799265074029443057e+0015L,
239 2.736097786240689996880391074927552517982e+0014L,
240 };
241 static GENERIC ps0[11] = {
242 1.0e0L,
243 1.060600687545891455602180843276758095107e+0003L,
244 4.343106093416975589147153906505338900961e+0005L,
245 8.910605869002176566582072242244353399059e+0007L,
246 9.959122058635087888690713917622056540190e+0009L,
247 6.188744967234948231792482949171041843894e+0011L,
248 2.098863976953783506401759873801990304907e+0013L,
249 3.672870357018063196746729751479938908450e+0014L,
250 2.975538419246824921049011529574385888420e+0015L,
251 9.063657659995043205018686029284479837091e+0015L,
252 6.401953344314747916729366441508892711691e+0015L,
253 };
254 static GENERIC pr1[12] = {
255 1.000000000000000000000023667524130660984e+0000L,
256 6.746154419979618754354803488126452971204e+0002L,
257 1.811210781083390154857018330296145970502e+0005L,
258 2.533098390379924268038005329095287842244e+0007L,
259 2.029683619805342145252338570875424600729e+0009L,
260 9.660859662192711465301069401598929980319e+0010L,
261 2.743396238644831519934098967716621316316e+0012L,
262 4.553097354140854377931023170263455246288e+0013L,
263 4.210245069852219757476169864974870720374e+0014L,
264 1.987334056229596485076645967176169801727e+0015L,
265 4.067120052787096893838970455751338930462e+0015L,
266 2.486539606380406398310845264910691398133e+0015L,
267 };
268 static GENERIC ps1[14] = {
269 1.0e0L,
270 6.744982544979618754355808680196859521782e+0002L,
271 1.810421795396966762032155290441364740350e+0005L,
272 2.530986460644310651529583759699988435573e+0007L,
273 2.026743276048023121360249288818290224145e+0009L,
274 9.637461924407405935245269407052641341836e+0010L,
275 2.732378628423766417402292797028314160831e+0012L,
276 4.522345274960527124354844364012184278488e+0013L,
277 4.160650668341743132685335758415469856545e+0014L,
278 1.943730242988858208243492424892435901211e+0015L,
279 3.880228532692127989901131618598067450001e+0015L,
280 2.178020816161154615841000173683302999728e+0015L,
281 -8.994062666842225551554346698171600634173e+0013L,
282 1.368520368508851253495764806934619574990e+0013L,
283 };
284 static GENERIC pr2[12] = {
285 1.000000000000000006938651621840396237282e+0000L,
286 3.658416291850404981407101077037948144698e+0002L,
287 5.267073772170356547709794670602812447537e+0004L,
288 3.912012101226837463014925210735894620442e+0006L,
289 1.651295648974103957193874928714180765625e+0008L,
290 4.114901144480797609972484998142146783499e+0009L,
291 6.092524309766036681542980572526335147672e+0010L,
292 5.263913178071282616719249969074134570577e+0011L,
293 2.538408581124324223367341020538081330994e+0012L,
294 6.288607929360291027895126983015365677648e+0012L,
295 6.848330048211148419047055075386525945280e+0012L,
296 2.290309646838867941423178163991423244690e+0012L,
297 };
298 static GENERIC ps2[14] = {
299 1.0e0L,
300 3.657244416850405086459410165762319861856e+0002L,
301 5.262802358425023243992387075861237306312e+0004L,
302 3.905896813959919648136295861661483848364e+0006L,
303 1.646791907791461220742694842108202772763e+0008L,
304 4.096132803064256022224954120208201437344e+0009L,
305 6.046665195915950447544429445730680236759e+0010L,
306 5.198061739781991313414052212328653295168e+0011L,
307 2.484233851814333966401527626421254279796e+0012L,
308 6.047868806925315879339651539434315255940e+0012L,
309 6.333103831254091652501642567294101813354e+0012L,
310 1.875143098754284994467609936924685024968e+0012L,
311 -5.238330920563392692965412762508813601534e+0010L,
312 4.656888609439364725427789198383779259957e+0009L,
313 };
314 static GENERIC pr3[13] = {
315 1.000000000000009336887318068056137842897e+0000L,
316 2.242719942728459588488051572002835729183e+0002L,
317 1.955450611382026550266257737331095691092e+0004L,
318 8.707143293993619899395400562409175590739e+0005L,
319 2.186267894487004565948324289010954505316e+0007L,
320 3.224328510541957792360691585667502864688e+0008L,
321 2.821057355151380597331792896882741364897e+0009L,
322 1.445371387295422404365584793796028979840e+0010L,
323 4.181743160669891357783011002656658107864e+0010L,
324 6.387371088767993119325536137794535513922e+0010L,
325 4.575619999412716078064070587767416436396e+0010L,
326 1.228415651211639160620284441690503550842e+0010L,
327 7.242170349875563053436050532153112882072e+0008L,
328 };
329 static GENERIC ps3[13] = {
330 1.0e0L,
331 2.241548067728529551049804610486061401070e+0002L,
332 1.952838216795552145132137932931237181307e+0004L,
333 8.684574926493185744628127341069974575526e+0005L,
334 2.176357771067037962940853412819852189164e+0007L,
335 3.199958682356132977319258783167122100567e+0008L,
336 2.786218931525334687844675219914201872570e+0009L,
337 1.416283776951741549631417572317916039767e+0010L,
338 4.042962659271567948735676834609348842922e+0010L,
339 6.028168462646694510083847222968444402161e+0010L,
340 4.118410226794641413833887606580085281111e+0010L,
341 9.918735736297038430744161253338202230263e+0009L,
342 4.092967198238098023219124487437130332038e+0008L,
343 };
344 static GENERIC pr4[13] = {
345 1.000000000001509220978157399042059553390e+0000L,
346 1.437551868378147851133499996323782607787e+0002L,
347 7.911335537418177296041518061404505428004e+0003L,
348 2.193710939115317214716518908935756104804e+0005L,
349 3.390662495136730962513489796538274984382e+0006L,
350 3.048655347929348891006070609293884274789e+0007L,
351 1.613781633489496606354045161527450975195e+0008L,
352 4.975089835037230277110156150038482159988e+0008L,
353 8.636047087015115403880904418339566323264e+0008L,
354 7.918202912328366140110671223076949101509e+0008L,
355 3.423294665798984733439650311722794853294e+0008L,
356 5.621904953441963961040503934782662613621e+0007L,
357 2.086303543310240260758670404509484499793e+0006L,
358 };
359 static GENERIC ps4[13] = {
360 1.0e0L,
361 1.436379993384532371670493319591847362304e+0002L,
362 7.894647154785430678061053848847436659499e+0003L,
363 2.184659753392097529008981741550878586174e+0005L,
364 3.366109083305465176803513738147049499361e+0006L,
365 3.011911545968996817697665866587226343186e+0007L,
366 1.582262913779689851316760148459414895301e+0008L,
367 4.819268809494937919217938589530138201770e+0008L,
368 8.201355762990450679702837123432527154830e+0008L,
369 7.268232093982510937417446421282341425212e+0008L,
370 2.950911909015572933262131323934036480462e+0008L,
371 4.242839924305934423010858966540621219396e+0007L,
372 1.064387620445090779182117666330405186866e+0006L,
373 };
374 static GENERIC pr5[13] = {
375 1.000000000102434805241171427253847353861e+0000L,
376 9.129332257083629259060502249025963234821e+0001L,
377 3.132238483586953037576119377504557191413e+0003L,
378 5.329782528269307971278943122454171107861e+0004L,
379 4.988460157184117790692873002103052944145e+0005L,
380 2.686602071615786816147010334256047469378e+0006L,
381 8.445418526028961197703799808701268301831e+0006L,
382 1.536575358646141157475725889907900827390e+0007L,
383 1.568405818236523821796862770586544811945e+0007L,
384 8.450876239888770102387618667362302173547e+0006L,
385 2.154414900139567328424026827163203446077e+0006L,
386 2.105656926565043898888460254808062352205e+0005L,
387 4.739165011023396507022134303736862812975e+0003L,
388 };
389 static GENERIC ps5[13] = {
390 1.0e0L,
391 9.117613509595327476509152673394703847793e+0001L,
392 3.121697972484015639301279229281770795147e+0003L,
393 5.294447222735893568040911873834576440255e+0004L,
394 4.930368882192772335798256684110887882807e+0005L,
395 2.634854685641165298302167435798357437768e+0006L,
396 8.185462775400326393555896157031818280918e+0006L,
397 1.462417423080215192609668642663030667086e+0007L,
398 1.450624993985851675982860844153954896015e+0007L,
399 7.460467647561995283219086567162006113864e+0006L,
400 1.754210981405612478869227142579056338965e+0006L,
401 1.463286721155271971526264914524746699596e+0005L,
402 2.155894725796702015341211116579827039459e+0003L,
403 };
404 static GENERIC pr6[13] = {
405 1.000000003564855546741735920315743157129e+0000L,
406 5.734003934862540458119423509909510288366e+0001L,
407 1.209572491935850486086559692291796887976e+0003L,
408 1.243398391422281247933674779163660286838e+0004L,
409 6.930996755181437937258220998601708278787e+0004L,
410 2.198067659532757598646722249966767620099e+0005L,
411 4.033659432712058633933179115820576858455e+0005L,
412 4.257759657219008027016047206574574358678e+0005L,
413 2.511917395876004349480721277445763916389e+0005L,
414 7.813756153070623654178731651381881953552e+0004L,
415 1.152069173381127881385588092905864352891e+0004L,
416 6.548580782804088553777816037551523398082e+0002L,
417 8.668725370116906132327542766127938496880e+0000L,
418 };
419 static GENERIC ps6[13] = {
420 1.0e0L,
421 5.722285236357114566499221525736286205184e+0001L,
422 1.203010842878317935444582950620339570506e+0003L,
423 1.230058335378583550155825502172435371208e+0004L,
424 6.800998550607861288865300438648089894412e+0004L,
425 2.130767829599304262987769347536850885921e+0005L,
426 3.840483466643916681759936972992155310026e+0005L,
427 3.947432373459225542861819148108081160393e+0005L,
428 2.237816239393081111481588434457838526738e+0005L,
429 6.545820495124419723398946273790921540774e+0004L,
430 8.729563630320892741500726213278834737196e+0003L,
431 4.130762660291894753450174794196998813709e+0002L,
432 3.480368898672684645130335786015075595598e+0000L,
433 };
434 static GENERIC sixteen = 16.0L;
435 static GENERIC eight = 8.0L;
436 static GENERIC huge = 1.0e30L;
437
438 static GENERIC pone(x)
439 GENERIC x;
440 {
441 GENERIC s, r, t, z;
442 int i;
443 if (x > huge)
444 return (one);
445 t = one/x; z = t*t;
446 if (x > sixteen) {
447 r = z*pr0[11]+pr0[10]; s = ps0[10];
448 for (i = 9; i >= 0; i--) {
449 r = z*r + pr0[i];
450 s = z*s + ps0[i];
451 }
452 } else if (x > eight) {
453 r = pr1[11]; s = ps1[11]+z*(ps1[12]+z*ps1[13]);
454 for (i = 10; i >= 0; i--) {
455 r = z*r + pr1[i];
456 s = z*s + ps1[i];
457 }
458 } else if (x > five) {
459 r = pr2[11]; s = ps2[11]+z*(ps2[12]+z*ps2[13]);
460 for (i = 10; i >= 0; i--) {
461 r = z*r + pr2[i];
462 s = z*s + ps2[i];
463 }
464 } else if (x > 3.5L) {
465 r = pr3[12]; s = ps3[12];
466 for (i = 11; i >= 0; i--) {
467 r = z*r + pr3[i];
468 s = z*s + ps3[i];
469 }
470 } else if (x > 2.5L) {
471 r = pr4[12]; s = ps4[12];
472 for (i = 11; i >= 0; i--) {
473 r = z*r + pr4[i];
474 s = z*s + ps4[i];
475 }
476 } else if (x > (1.0L/0.5625L)) {
477 r = pr5[12]; s = ps5[12];
478 for (i = 11; i >= 0; i--) {
479 r = z*r + pr5[i];
480 s = z*s + ps5[i];
481 }
482 } else { /* assume x > 1.28 */
483 r = pr6[12]; s = ps6[12];
484 for (i = 11; i >= 0; i--) {
485 r = z*r + pr6[i];
486 s = z*s + ps6[i];
487 }
488 }
489 return (r/s);
490 }
491
492
493 static GENERIC qr0[12] = {
494 3.749999999999999999999999999999999971033e-0001L,
495 4.256726035237050601607682277433094262226e+0002L,
496 1.875976490812878489192409978945401066066e+0005L,
497 4.170314268048041914273603680317745592790e+0007L,
498 5.092750132543855817293451118974555746551e+0009L,
499 3.494749676278488654103505795794139483404e+0011L,
500 1.327062148257437316997667817096694173709e+0013L,
501 2.648993138273427226907503742066551150490e+0014L,
502 2.511695665909547412222430494473998127684e+0015L,
503 9.274694506662289043224310499164702306096e+0015L,
504 8.150904170663663829331320302911792892002e+0015L,
505 -5.001918733707662355772037829620388765122e+0014L,
506 };
507 static GENERIC qs0[11] = {
508 1.0e0L,
509 1.135400380229880160428715273982155760093e+0003L,
510 5.005701183877126164326765545516590744360e+0005L,
511 1.113444200113712167984337603933040102987e+0008L,
512 1.361074819925223062778717565699039471124e+0010L,
513 9.355750985802849484438933905325982809653e+0011L,
514 3.563462786008988825003965543857998084828e+0013L,
515 7.155145113900094163648726863803802910454e+0014L,
516 6.871266835834472758055559013851843654113e+0015L,
517 2.622030899226736712644974988157345234092e+0016L,
518 2.602912729172876330650077021706139707746e+0016L,
519 };
520 static GENERIC qr1[12] = {
521 3.749999999999999999997762458207284405806e-0001L,
522 2.697883998881706839929255517498189980485e+0002L,
523 7.755195925781028489386938870473834411019e+0004L,
524 1.166777762104017777198211072895528968355e+0007L,
525 1.011504772984321168320010084520261069362e+0009L,
526 5.246007703574156853577754571720205550010e+0010L,
527 1.637692549885592683166116551691266537647e+0012L,
528 3.022303623698185669912990310925039382495e+0013L,
529 3.154769927290655684846107030265909987946e+0014L,
530 1.715819913441554770089730934808123360921e+0015L,
531 4.165044355759732622273534445131736188510e+0015L,
532 3.151381420874174705643100381708086287596e+0015L,
533 };
534 static GENERIC qs1[14] = {
535 1.0e0L,
536 7.197091705351218239785633172408276982828e+0002L,
537 2.070012799599548685544883041297609861055e+0005L,
538 3.117014815317656221871840152778458754516e+0007L,
539 2.705719678902554974863325877025902971727e+0009L,
540 1.406113614727345726925060648750867264098e+0011L,
541 4.403777536067131320363005978631674817359e+0012L,
542 8.170725690209322283061499386703167242894e+0013L,
543 8.609458844975495289227794126964431210566e+0014L,
544 4.766766367015473481257280600694952920204e+0015L,
545 1.202286587943342194863557940888115641650e+0016L,
546 1.012474328306200909525063936061756024120e+0016L,
547 6.183552022678917858273222879615824070703e+0014L,
548 -9.756731548558226997573737400988225722740e+0013L,
549 };
550 static GENERIC qr2[12] = {
551 3.749999999999999481245647262226994293189e-0001L,
552 1.471366807289771354491181140167359026735e+0002L,
553 2.279432486768448220142080962843526951250e+0004L,
554 1.828943048523771225163804043356958285893e+0006L,
555 8.379828388647823135832220596417725010837e+0007L,
556 2.279814029335044024585393671278378022053e+0009L,
557 3.711653952257118120832817785271466441420e+0010L,
558 3.557650914518554549916730572553105048068e+0011L,
559 1.924583483146095896259774329498934160650e+0012L,
560 5.424386256063736390759567088291887140278e+0012L,
561 6.839325621241776786206509704671746841737e+0012L,
562 2.702169563144001166291686452305436313971e+0012L,
563 };
564 static GENERIC qs2[14] = {
565 1.0e0L,
566 3.926379194439388135703211933895203191089e+0002L,
567 6.089148804106598297488336063007609312276e+0004L,
568 4.893546162973278583711376356041614150645e+0006L,
569 2.247571119114497845046388801813832219404e+0008L,
570 6.137635663350177751290469334200757872645e+0009L,
571 1.005115019784102856424493519524998953678e+0011L,
572 9.725664462014503832860151384604677240620e+0011L,
573 5.345525100485511116148634192844434636072e+0012L,
574 1.549944007398946691720862738173956994779e+0013L,
575 2.067148441178952625710302124163264760362e+0013L,
576 9.401565402641963611295119487242595462301e+0012L,
577 3.548217088622398274748837287769709374385e+0011L,
578 -2.934470341719047120076509938432417352365e+0010L,
579 };
580 static GENERIC qr3[13] = {
581 3.749999999999412724084579833297451472091e-0001L,
582 9.058478580291706212422978492938435582527e+0001L,
583 8.524056033161038750461083666711724381171e+0003L,
584 4.105967158629109427753434569223631014730e+0005L,
585 1.118326603378531348259783091972623333657e+0007L,
586 1.794636683403578918528064904714132329343e+0008L,
587 1.714314157463635959556133236004368896724e+0009L,
588 9.622092032236084846572067257267661456030e+0009L,
589 3.057759524485859159957762858780768355020e+0010L,
590 5.129306780754798531609621454415938890020e+0010L,
591 3.999122002794961070680636194346316041352e+0010L,
592 1.122298454643493485989721564358100345388e+0010L,
593 5.603981987645989709668830968522362582221e+0008L,
594 };
595 static GENERIC qs3[13] = {
596 1.0e0L,
597 2.418328663076578169836155170053634419922e+0002L,
598 2.279620205900121042587523541281272875520e+0004L,
599 1.100984222585729521470129014992217092794e+0006L,
600 3.010743223679247091004262516286654516282e+0007L,
601 4.860925542827367817289619265215599433996e+0008L,
602 4.686668111035348691982715864307839581243e+0009L,
603 2.668701788405102017427214705946730894074e+0010L,
604 8.677395746106802640390580944836650584903e+0010L,
605 1.511936455574951790658498795945106643036e+0011L,
606 1.260845604432623478002018696873608353093e+0011L,
607 4.052692278419853853911440231600864589805e+0010L,
608 2.965516519212226064983267822243329694729e+0009L,
609 };
610 static GENERIC qr4[13] = {
611 3.749999999919234164154669754440123072618e-0001L,
612 5.844218580776819864791168253485055101858e+0001L,
613 3.489273514092912982675669411371435670220e+0003L,
614 1.050523637774575684509663430018995479594e+0005L,
615 1.764549172059701565500717319792780115289e+0006L,
616 1.725532438844133795028063102681497371154e+0007L,
617 9.938114847359778539965140247590176334874e+0007L,
618 3.331710808184595545396883770200772842314e+0008L,
619 6.271970557641881511609560444872797282698e+0008L,
620 6.188529798677357075020774923903737913285e+0008L,
621 2.821905302742849974509982167877885011629e+0008L,
622 4.615467358646911976773290256984329814896e+0007L,
623 1.348140608731546467396685802693380693275e+0006L,
624 };
625 static GENERIC qs4[13] = {
626 1.0e0L,
627 1.561192663112345185261418296389902133372e+0002L,
628 9.346678031144098270547225423124213083072e+0003L,
629 2.825851246482293547838023847601704751590e+0005L,
630 4.776572711622156091710902891124911556293e+0006L,
631 4.715106953717135402977938048006267859302e+0007L,
632 2.753962350894311316439652227611209035193e+0008L,
633 9.428501434615463207768964787500411575223e+0008L,
634 1.832650858775206787088236896454141572617e+0009L,
635 1.901697378939743226948920874296595242257e+0009L,
636 9.433322226854293780627188599226380812725e+0008L,
637 1.808520540608671608680284520798858587370e+0008L,
638 7.983342331736662753157217446919462398008e+0006L,
639 };
640 static GENERIC qr5[13] = {
641 3.749999995331364437028988850515190446719e-0001L,
642 3.739356381766559882677514593041627547911e+0001L,
643 1.399562500629413529355265462912819802551e+0003L,
644 2.594154053098947925345332218062210111753e+0004L,
645 2.640149879297408640394163979394594318371e+0005L,
646 1.542471854873199142031889093591449397995e+0006L,
647 5.242272868972053374067572098992335425895e+0006L,
648 1.025834487769410221329633071426044839935e+0007L,
649 1.116553924239448940142230579060124209622e+0007L,
650 6.318076065595910176374916303525884653514e+0006L,
651 1.641218086168640408527639735915512881785e+0006L,
652 1.522369793529178644168813882912134706444e+0005L,
653 2.526530541062297200914180060208669584055e+0003L,
654 };
655 static GENERIC qs5[13] = {
656 1.0e0L,
657 9.998960735935075380397545659016287506660e+0001L,
658 3.758767417842043742686475060540416737562e+0003L,
659 7.013652806952306520121959742519780781653e+0004L,
660 7.208949808818615099246529616211730446850e+0005L,
661 4.272753927109614455417836186072202009252e+0006L,
662 1.482524411356470699336129814111025434703e+0007L,
663 2.988750366665678233425279237627700803473e+0007L,
664 3.396957890261080492694709150553619185065e+0007L,
665 2.050652487738593004111578091156304540386e+0007L,
666 5.900504120811732547616511555946279451316e+0006L,
667 6.563391409260160897024498082273183468347e+0005L,
668 1.692629845012790205348966731477187041419e+0004L,
669 };
670 static GENERIC qr6[13] = {
671 3.749999861516664133157566870858975421296e-0001L,
672 2.367863756747764863120797431599473468918e+0001L,
673 5.476715802114976248882067325630793143777e+0002L,
674 6.143190357869842894025012945444096170251e+0003L,
675 3.716250534677997850513733595140463851730e+0004L,
676 1.270883463823876752138326905022875657430e+0005L,
677 2.495301449636814481646371665429083801388e+0005L,
678 2.789578988212952248340486296254398601942e+0005L,
679 1.718247946911109055931819087137397324634e+0005L,
680 5.458973214011665714330326732204106364229e+0004L,
681 7.912102686687948786048943339759596652813e+0003L,
682 4.077961006160866935722030715149087938091e+0002L,
683 3.765206972770245085551057237882528510428e+0000L,
684 };
685 static GENERIC qs6[13] = {
686 1.0e0L,
687 6.341646532940517305641893852673926809601e+0001L,
688 1.477058277414040790932597537920671025359e+0003L,
689 1.674406564031044491436044253393536487604e+0004L,
690 1.028516501369755949895050806908994650768e+0005L,
691 3.593620042532885295087463507733285434207e+0005L,
692 7.267924991381020915185873399453724799625e+0005L,
693 8.462277510768818399961191426205006083088e+0005L,
694 5.514399892230892163373611895645500250514e+0005L,
695 1.898084241009259353540620272932188102299e+0005L,
696 3.102941242117739015721984123081026253068e+0004L,
697 1.958971184431466907681440650181421086143e+0003L,
698 2.878853357310495087181721613889455121867e+0001L,
699 };
700 static GENERIC qone(x)
701 GENERIC x;
702 {
703 GENERIC s, r, t, z;
704 int i;
705 if (x > huge)
706 return (0.375L/x);
707 t = one/x; z = t*t;
708 if (x > sixteen) {
709 r = z*qr0[11]+qr0[10]; s = qs0[10];
710 for (i = 9; i >= 0; i--) {
711 r = z*r + qr0[i];
712 s = z*s + qs0[i];
713 }
714 } else if (x > eight) {
715 r = qr1[11]; s = qs1[11]+z*(qs1[12]+z*qs1[13]);
716 for (i = 10; i >= 0; i--) {
717 r = z*r + qr1[i];
718 s = z*s + qs1[i];
719 }
720 } else if (x > five) { /* x > 5.0 */
721 r = qr2[11]; s = qs2[11]+z*(qs2[12]+z*qs2[13]);
722 for (i = 10; i >= 0; i--) {
723 r = z*r + qr2[i];
724 s = z*s + qs2[i];
725 }
726 } else if (x > 3.5L) {
727 r = qr3[12]; s = qs3[12];
728 for (i = 11; i >= 0; i--) {
729 r = z*r + qr3[i];
730 s = z*s + qs3[i];
731 }
732 } else if (x > 2.5L) {
733 r = qr4[12]; s = qs4[12];
734 for (i = 11; i >= 0; i--) {
735 r = z*r + qr4[i];
736 s = z*s + qs4[i];
737 }
738 } else if (x > (1.0L/0.5625L)) {
739 r = qr5[12]; s = qs5[12];
740 for (i = 11; i >= 0; i--) {
741 r = z*r + qr5[i];
742 s = z*s + qs5[i];
743 }
744 } else { /* assume x > 1.28 */
745 r = qr6[12]; s = qs6[12];
746 for (i = 11; i >= 0; i--) {
747 r = z*r + qr6[i];
748 s = z*s + qs6[i];
749 }
750 }
751 return (t*(r/s));
752 }
|
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 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 /*
32 * floating point Bessel's function of the first and second kinds
33 * of order zero: j1(x),y1(x);
34 *
35 * Special cases:
36 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
37 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
38 */
39
40 #pragma weak __j1l = j1l
41 #pragma weak __y1l = y1l
42
43 #include "libm.h"
44 #include "longdouble.h"
45 #include <math.h>
46 #if defined(__SUNPRO_C)
47 #include <sunmath.h>
48 #endif
49
50 #define GENERIC long double
51
52 static GENERIC zero = 0.0L,
53 small = 1.0e-9L,
54 tiny = 1.0e-38L,
55 one = 1.0L,
56 five = 5.0L,
57 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
58 tpi = 0.636619772367581343075535053490057448L;
59
60 static GENERIC pone(), qone();
61
62 static GENERIC r0[7] = {
63 -6.249999999999999999999999999999999627320e-0002L,
64 1.940606727194041716205384618494641565464e-0003L,
65 -3.005630423155733701856481469986459043883e-0005L,
66 2.345586219403918667468341047369572169358e-0007L,
67 -9.976809285885253587529010109133336669724e-0010L,
68 2.218743258363623946078958783775107473381e-0012L,
69 -2.071079656218700604767650924103578046280e-0015L,
70 };
71
72 static GENERIC s0[7] = {
73 1.0e0L,
74 1.061695903156199920738051277075003059555e-0002L,
75 5.521860513111180371566951179398862692060e-0005L,
76 1.824214367413754193524107877084979441407e-0007L,
77 4.098957778439576834818838198039029353925e-0010L,
78 6.047735079699666389853240090925264056197e-0013L,
79 4.679044728878836197247923279512047035041e-0016L,
80 };
81
82 GENERIC
83 j1l(GENERIC x)
84 {
85 GENERIC z, d, s, c, ss, cc, r;
86 int i, sgn;
87
88 if (!finitel(x))
89 return (one / x);
90
91 sgn = signbitl(x);
92 x = fabsl(x);
93
94 if (x > 1.28L) {
95 s = sinl(x);
96 c = cosl(x);
97
98 /* BEGIN CSTYLED */
99 /*
100 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
101 * where x0 = x-3pi/4
102 * Better formula:
103 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
104 * = 1/sqrt(2) * (sin(x) - cos(x))
105 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
106 * = -1/sqrt(2) * (cos(x) + sin(x))
107 * To avoid cancellation, use
108 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
109 * to compute the worse one.
110 */
111 /* END CSTYLED */
112 if (x > 1.0e2450L) { /* x+x may overflow */
113 ss = -s - c;
114 cc = s - c;
115 } else if (signbitl(s) != signbitl(c)) {
116 cc = s - c;
117 ss = cosl(x + x) / cc;
118 } else {
119 ss = -s - c;
120 cc = cosl(x + x) / ss;
121 }
122
123 /*
124 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
125 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
126 */
127 if (x > 1.0e120L)
128 return ((invsqrtpi * cc) / sqrtl(x));
129
130 d = invsqrtpi * (pone(x) * cc - qone(x) * ss) / sqrtl(x);
131
132 if (sgn == 0)
133 return (d);
134 else
135 return (-d);
136 }
137
138 if (x <= small) {
139 if (x <= tiny)
140 d = 0.5L * x;
141 else
142 d = x * (0.5L - x * x * 0.125L);
143
144 if (sgn == 0)
145 return (d);
146 else
147 return (-d);
148 }
149
150 z = x * x;
151 r = r0[6];
152 s = s0[6];
153
154 for (i = 5; i >= 0; i--) {
155 r = r * z + r0[i];
156 s = s * z + s0[i];
157 }
158
159 d = x * 0.5L + x * (z * (r / s));
160
161 if (sgn == 0)
162 return (d);
163 else
164 return (-d);
165 }
166
167 static GENERIC u0[7] = {
168 -1.960570906462389484060557273467558703503e-0001L,
169 5.166389353148318460304315890665450006495e-0002L,
170 -2.229699464105910913337190798743451115604e-0003L,
171 3.625437034548863342715657067759078267158e-0005L,
172 -2.689902826993117212255524537353883987171e-0007L,
173 9.304570592456930912969387719010256018466e-0010L,
174 -1.234878126794286643318321347997500346131e-0012L,
175 };
176
177 static GENERIC v0[8] = {
178 1.0e0L,
179 1.369394302535807332517110204820556695644e-0002L,
180 9.508438148097659501433367062605935379588e-0005L,
181 4.399007309420092056052714797296467565655e-0007L,
182 1.488083087443756398305819693177715000787e-0009L,
183 3.751609832625793536245746965768587624922e-0012L,
184 6.680926434086257291872903276124244131448e-0015L,
185 6.676602383908906988160099057991121446058e-0018L,
186 };
187
188 GENERIC
189 y1l(GENERIC x)
190 {
191 GENERIC z, s, c, ss, cc, u, v;
192 int i;
193
194 if (isnanl(x))
195 return (x + x);
196
197 if (x <= zero) {
198 if (x == zero)
199 return (-one / zero);
200 else
201 return (zero / zero);
202 }
203
204 if (x > 1.28L) {
205 if (!finitel(x))
206 return (zero);
207
208 s = sinl(x);
209 c = cosl(x);
210
211 /* BEGIN CSTYLED */
212 /*
213 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0))
214 * where x0 = x-3pi/4
215 * Better formula:
216 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
217 * = 1/sqrt(2) * (sin(x) - cos(x))
218 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
219 * = -1/sqrt(2) * (cos(x) + sin(x))
220 * To avoid cancellation, use
221 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
222 * to compute the worse one.
223 */
224 /* END CSTYLED */
225 if (x > 1.0e2450L) { /* x+x may overflow */
226 ss = -s - c;
227 cc = s - c;
228 } else if (signbitl(s) != signbitl(c)) {
229 cc = s - c;
230 ss = cosl(x + x) / cc;
231 } else {
232 ss = -s - c;
233 cc = cosl(x + x) / ss;
234 }
235
236 /*
237 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
238 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
239 */
240 if (x > 1.0e91L)
241 return ((invsqrtpi * ss) / sqrtl(x));
242
243 return (invsqrtpi * (pone(x) * ss + qone(x) * cc) / sqrtl(x));
244 }
245
246 if (x <= tiny)
247 return (-tpi / x);
248
249 z = x * x;
250 u = u0[6];
251 v = v0[6] + z * v0[7];
252
253 for (i = 5; i >= 0; i--) {
254 u = u * z + u0[i];
255 v = v * z + v0[i];
256 }
257
258 return (x * (u / v) + tpi * (j1l(x) * logl(x) - one / x));
259 }
260
261 static GENERIC pr0[12] = {
262 1.000000000000000000000000000000000000267e+0000L,
263 1.060717875045891455602180843276758003035e+0003L,
264 4.344347542892127024446687712181105852335e+0005L,
265 8.915680220724007016377924252717410457094e+0007L,
266 9.969502259938406062809873257569171272819e+0009L,
267 6.200290193138613035646510338707386316595e+0011L,
268 2.105978548788015119851815854422247330118e+0013L,
269 3.696635772784601239371730810311998368948e+0014L,
270 3.015913097920694682057958412534134515156e+0015L,
271 9.370298471339353098123277427328592725921e+0015L,
272 7.190349005196335967340799265074029443057e+0015L,
273 2.736097786240689996880391074927552517982e+0014L,
274 };
275
276 static GENERIC ps0[11] = {
277 1.0e0L,
278 1.060600687545891455602180843276758095107e+0003L,
279 4.343106093416975589147153906505338900961e+0005L,
280 8.910605869002176566582072242244353399059e+0007L,
281 9.959122058635087888690713917622056540190e+0009L,
282 6.188744967234948231792482949171041843894e+0011L,
283 2.098863976953783506401759873801990304907e+0013L,
284 3.672870357018063196746729751479938908450e+0014L,
285 2.975538419246824921049011529574385888420e+0015L,
286 9.063657659995043205018686029284479837091e+0015L,
287 6.401953344314747916729366441508892711691e+0015L,
288 };
289
290 static GENERIC pr1[12] = {
291 1.000000000000000000000023667524130660984e+0000L,
292 6.746154419979618754354803488126452971204e+0002L,
293 1.811210781083390154857018330296145970502e+0005L,
294 2.533098390379924268038005329095287842244e+0007L,
295 2.029683619805342145252338570875424600729e+0009L,
296 9.660859662192711465301069401598929980319e+0010L,
297 2.743396238644831519934098967716621316316e+0012L,
298 4.553097354140854377931023170263455246288e+0013L,
299 4.210245069852219757476169864974870720374e+0014L,
300 1.987334056229596485076645967176169801727e+0015L,
301 4.067120052787096893838970455751338930462e+0015L,
302 2.486539606380406398310845264910691398133e+0015L,
303 };
304
305 static GENERIC ps1[14] = {
306 1.0e0L,
307 6.744982544979618754355808680196859521782e+0002L,
308 1.810421795396966762032155290441364740350e+0005L,
309 2.530986460644310651529583759699988435573e+0007L,
310 2.026743276048023121360249288818290224145e+0009L,
311 9.637461924407405935245269407052641341836e+0010L,
312 2.732378628423766417402292797028314160831e+0012L,
313 4.522345274960527124354844364012184278488e+0013L,
314 4.160650668341743132685335758415469856545e+0014L,
315 1.943730242988858208243492424892435901211e+0015L,
316 3.880228532692127989901131618598067450001e+0015L,
317 2.178020816161154615841000173683302999728e+0015L,
318 -8.994062666842225551554346698171600634173e+0013L,
319 1.368520368508851253495764806934619574990e+0013L,
320 };
321
322 static GENERIC pr2[12] = {
323 1.000000000000000006938651621840396237282e+0000L,
324 3.658416291850404981407101077037948144698e+0002L,
325 5.267073772170356547709794670602812447537e+0004L,
326 3.912012101226837463014925210735894620442e+0006L,
327 1.651295648974103957193874928714180765625e+0008L,
328 4.114901144480797609972484998142146783499e+0009L,
329 6.092524309766036681542980572526335147672e+0010L,
330 5.263913178071282616719249969074134570577e+0011L,
331 2.538408581124324223367341020538081330994e+0012L,
332 6.288607929360291027895126983015365677648e+0012L,
333 6.848330048211148419047055075386525945280e+0012L,
334 2.290309646838867941423178163991423244690e+0012L,
335 };
336
337 static GENERIC ps2[14] = {
338 1.0e0L,
339 3.657244416850405086459410165762319861856e+0002L,
340 5.262802358425023243992387075861237306312e+0004L,
341 3.905896813959919648136295861661483848364e+0006L,
342 1.646791907791461220742694842108202772763e+0008L,
343 4.096132803064256022224954120208201437344e+0009L,
344 6.046665195915950447544429445730680236759e+0010L,
345 5.198061739781991313414052212328653295168e+0011L,
346 2.484233851814333966401527626421254279796e+0012L,
347 6.047868806925315879339651539434315255940e+0012L,
348 6.333103831254091652501642567294101813354e+0012L,
349 1.875143098754284994467609936924685024968e+0012L,
350 -5.238330920563392692965412762508813601534e+0010L,
351 4.656888609439364725427789198383779259957e+0009L,
352 };
353
354 static GENERIC pr3[13] = {
355 1.000000000000009336887318068056137842897e+0000L,
356 2.242719942728459588488051572002835729183e+0002L,
357 1.955450611382026550266257737331095691092e+0004L,
358 8.707143293993619899395400562409175590739e+0005L,
359 2.186267894487004565948324289010954505316e+0007L,
360 3.224328510541957792360691585667502864688e+0008L,
361 2.821057355151380597331792896882741364897e+0009L,
362 1.445371387295422404365584793796028979840e+0010L,
363 4.181743160669891357783011002656658107864e+0010L,
364 6.387371088767993119325536137794535513922e+0010L,
365 4.575619999412716078064070587767416436396e+0010L,
366 1.228415651211639160620284441690503550842e+0010L,
367 7.242170349875563053436050532153112882072e+0008L,
368 };
369
370 static GENERIC ps3[13] = {
371 1.0e0L,
372 2.241548067728529551049804610486061401070e+0002L,
373 1.952838216795552145132137932931237181307e+0004L,
374 8.684574926493185744628127341069974575526e+0005L,
375 2.176357771067037962940853412819852189164e+0007L,
376 3.199958682356132977319258783167122100567e+0008L,
377 2.786218931525334687844675219914201872570e+0009L,
378 1.416283776951741549631417572317916039767e+0010L,
379 4.042962659271567948735676834609348842922e+0010L,
380 6.028168462646694510083847222968444402161e+0010L,
381 4.118410226794641413833887606580085281111e+0010L,
382 9.918735736297038430744161253338202230263e+0009L,
383 4.092967198238098023219124487437130332038e+0008L,
384 };
385
386 static GENERIC pr4[13] = {
387 1.000000000001509220978157399042059553390e+0000L,
388 1.437551868378147851133499996323782607787e+0002L,
389 7.911335537418177296041518061404505428004e+0003L,
390 2.193710939115317214716518908935756104804e+0005L,
391 3.390662495136730962513489796538274984382e+0006L,
392 3.048655347929348891006070609293884274789e+0007L,
393 1.613781633489496606354045161527450975195e+0008L,
394 4.975089835037230277110156150038482159988e+0008L,
395 8.636047087015115403880904418339566323264e+0008L,
396 7.918202912328366140110671223076949101509e+0008L,
397 3.423294665798984733439650311722794853294e+0008L,
398 5.621904953441963961040503934782662613621e+0007L,
399 2.086303543310240260758670404509484499793e+0006L,
400 };
401
402 static GENERIC ps4[13] = {
403 1.0e0L,
404 1.436379993384532371670493319591847362304e+0002L,
405 7.894647154785430678061053848847436659499e+0003L,
406 2.184659753392097529008981741550878586174e+0005L,
407 3.366109083305465176803513738147049499361e+0006L,
408 3.011911545968996817697665866587226343186e+0007L,
409 1.582262913779689851316760148459414895301e+0008L,
410 4.819268809494937919217938589530138201770e+0008L,
411 8.201355762990450679702837123432527154830e+0008L,
412 7.268232093982510937417446421282341425212e+0008L,
413 2.950911909015572933262131323934036480462e+0008L,
414 4.242839924305934423010858966540621219396e+0007L,
415 1.064387620445090779182117666330405186866e+0006L,
416 };
417
418 static GENERIC pr5[13] = {
419 1.000000000102434805241171427253847353861e+0000L,
420 9.129332257083629259060502249025963234821e+0001L,
421 3.132238483586953037576119377504557191413e+0003L,
422 5.329782528269307971278943122454171107861e+0004L,
423 4.988460157184117790692873002103052944145e+0005L,
424 2.686602071615786816147010334256047469378e+0006L,
425 8.445418526028961197703799808701268301831e+0006L,
426 1.536575358646141157475725889907900827390e+0007L,
427 1.568405818236523821796862770586544811945e+0007L,
428 8.450876239888770102387618667362302173547e+0006L,
429 2.154414900139567328424026827163203446077e+0006L,
430 2.105656926565043898888460254808062352205e+0005L,
431 4.739165011023396507022134303736862812975e+0003L,
432 };
433
434 static GENERIC ps5[13] = {
435 1.0e0L,
436 9.117613509595327476509152673394703847793e+0001L,
437 3.121697972484015639301279229281770795147e+0003L,
438 5.294447222735893568040911873834576440255e+0004L,
439 4.930368882192772335798256684110887882807e+0005L,
440 2.634854685641165298302167435798357437768e+0006L,
441 8.185462775400326393555896157031818280918e+0006L,
442 1.462417423080215192609668642663030667086e+0007L,
443 1.450624993985851675982860844153954896015e+0007L,
444 7.460467647561995283219086567162006113864e+0006L,
445 1.754210981405612478869227142579056338965e+0006L,
446 1.463286721155271971526264914524746699596e+0005L,
447 2.155894725796702015341211116579827039459e+0003L,
448 };
449
450 static GENERIC pr6[13] = {
451 1.000000003564855546741735920315743157129e+0000L,
452 5.734003934862540458119423509909510288366e+0001L,
453 1.209572491935850486086559692291796887976e+0003L,
454 1.243398391422281247933674779163660286838e+0004L,
455 6.930996755181437937258220998601708278787e+0004L,
456 2.198067659532757598646722249966767620099e+0005L,
457 4.033659432712058633933179115820576858455e+0005L,
458 4.257759657219008027016047206574574358678e+0005L,
459 2.511917395876004349480721277445763916389e+0005L,
460 7.813756153070623654178731651381881953552e+0004L,
461 1.152069173381127881385588092905864352891e+0004L,
462 6.548580782804088553777816037551523398082e+0002L,
463 8.668725370116906132327542766127938496880e+0000L,
464 };
465
466 static GENERIC ps6[13] = {
467 1.0e0L,
468 5.722285236357114566499221525736286205184e+0001L,
469 1.203010842878317935444582950620339570506e+0003L,
470 1.230058335378583550155825502172435371208e+0004L,
471 6.800998550607861288865300438648089894412e+0004L,
472 2.130767829599304262987769347536850885921e+0005L,
473 3.840483466643916681759936972992155310026e+0005L,
474 3.947432373459225542861819148108081160393e+0005L,
475 2.237816239393081111481588434457838526738e+0005L,
476 6.545820495124419723398946273790921540774e+0004L,
477 8.729563630320892741500726213278834737196e+0003L,
478 4.130762660291894753450174794196998813709e+0002L,
479 3.480368898672684645130335786015075595598e+0000L,
480 };
481
482 static GENERIC sixteen = 16.0L;
483 static GENERIC eight = 8.0L;
484 static GENERIC huge = 1.0e30L;
485
486 static GENERIC
487 pone(x)
488 GENERIC x;
489 {
490 GENERIC s, r, t, z;
491 int i;
492
493 if (x > huge)
494 return (one);
495
496 t = one / x;
497 z = t * t;
498
499 if (x > sixteen) {
500 r = z * pr0[11] + pr0[10];
501 s = ps0[10];
502
503 for (i = 9; i >= 0; i--) {
504 r = z * r + pr0[i];
505 s = z * s + ps0[i];
506 }
507 } else if (x > eight) {
508 r = pr1[11];
509 s = ps1[11] + z * (ps1[12] + z * ps1[13]);
510
511 for (i = 10; i >= 0; i--) {
512 r = z * r + pr1[i];
513 s = z * s + ps1[i];
514 }
515 } else if (x > five) {
516 r = pr2[11];
517 s = ps2[11] + z * (ps2[12] + z * ps2[13]);
518
519 for (i = 10; i >= 0; i--) {
520 r = z * r + pr2[i];
521 s = z * s + ps2[i];
522 }
523 } else if (x > 3.5L) {
524 r = pr3[12];
525 s = ps3[12];
526
527 for (i = 11; i >= 0; i--) {
528 r = z * r + pr3[i];
529 s = z * s + ps3[i];
530 }
531 } else if (x > 2.5L) {
532 r = pr4[12];
533 s = ps4[12];
534
535 for (i = 11; i >= 0; i--) {
536 r = z * r + pr4[i];
537 s = z * s + ps4[i];
538 }
539 } else if (x > (1.0L / 0.5625L)) {
540 r = pr5[12];
541 s = ps5[12];
542
543 for (i = 11; i >= 0; i--) {
544 r = z * r + pr5[i];
545 s = z * s + ps5[i];
546 }
547 } else { /* assume x > 1.28 */
548 r = pr6[12];
549 s = ps6[12];
550
551 for (i = 11; i >= 0; i--) {
552 r = z * r + pr6[i];
553 s = z * s + ps6[i];
554 }
555 }
556
557 return (r / s);
558 }
559 static GENERIC qr0[12] = {
560 3.749999999999999999999999999999999971033e-0001L,
561 4.256726035237050601607682277433094262226e+0002L,
562 1.875976490812878489192409978945401066066e+0005L,
563 4.170314268048041914273603680317745592790e+0007L,
564 5.092750132543855817293451118974555746551e+0009L,
565 3.494749676278488654103505795794139483404e+0011L,
566 1.327062148257437316997667817096694173709e+0013L,
567 2.648993138273427226907503742066551150490e+0014L,
568 2.511695665909547412222430494473998127684e+0015L,
569 9.274694506662289043224310499164702306096e+0015L,
570 8.150904170663663829331320302911792892002e+0015L,
571 -5.001918733707662355772037829620388765122e+0014L,
572 };
573
574 static GENERIC qs0[11] = {
575 1.0e0L,
576 1.135400380229880160428715273982155760093e+0003L,
577 5.005701183877126164326765545516590744360e+0005L,
578 1.113444200113712167984337603933040102987e+0008L,
579 1.361074819925223062778717565699039471124e+0010L,
580 9.355750985802849484438933905325982809653e+0011L,
581 3.563462786008988825003965543857998084828e+0013L,
582 7.155145113900094163648726863803802910454e+0014L,
583 6.871266835834472758055559013851843654113e+0015L,
584 2.622030899226736712644974988157345234092e+0016L,
585 2.602912729172876330650077021706139707746e+0016L,
586 };
587
588 static GENERIC qr1[12] = {
589 3.749999999999999999997762458207284405806e-0001L,
590 2.697883998881706839929255517498189980485e+0002L,
591 7.755195925781028489386938870473834411019e+0004L,
592 1.166777762104017777198211072895528968355e+0007L,
593 1.011504772984321168320010084520261069362e+0009L,
594 5.246007703574156853577754571720205550010e+0010L,
595 1.637692549885592683166116551691266537647e+0012L,
596 3.022303623698185669912990310925039382495e+0013L,
597 3.154769927290655684846107030265909987946e+0014L,
598 1.715819913441554770089730934808123360921e+0015L,
599 4.165044355759732622273534445131736188510e+0015L,
600 3.151381420874174705643100381708086287596e+0015L,
601 };
602
603 static GENERIC qs1[14] = {
604 1.0e0L,
605 7.197091705351218239785633172408276982828e+0002L,
606 2.070012799599548685544883041297609861055e+0005L,
607 3.117014815317656221871840152778458754516e+0007L,
608 2.705719678902554974863325877025902971727e+0009L,
609 1.406113614727345726925060648750867264098e+0011L,
610 4.403777536067131320363005978631674817359e+0012L,
611 8.170725690209322283061499386703167242894e+0013L,
612 8.609458844975495289227794126964431210566e+0014L,
613 4.766766367015473481257280600694952920204e+0015L,
614 1.202286587943342194863557940888115641650e+0016L,
615 1.012474328306200909525063936061756024120e+0016L,
616 6.183552022678917858273222879615824070703e+0014L,
617 -9.756731548558226997573737400988225722740e+0013L,
618 };
619
620 static GENERIC qr2[12] = {
621 3.749999999999999481245647262226994293189e-0001L,
622 1.471366807289771354491181140167359026735e+0002L,
623 2.279432486768448220142080962843526951250e+0004L,
624 1.828943048523771225163804043356958285893e+0006L,
625 8.379828388647823135832220596417725010837e+0007L,
626 2.279814029335044024585393671278378022053e+0009L,
627 3.711653952257118120832817785271466441420e+0010L,
628 3.557650914518554549916730572553105048068e+0011L,
629 1.924583483146095896259774329498934160650e+0012L,
630 5.424386256063736390759567088291887140278e+0012L,
631 6.839325621241776786206509704671746841737e+0012L,
632 2.702169563144001166291686452305436313971e+0012L,
633 };
634
635 static GENERIC qs2[14] = {
636 1.0e0L,
637 3.926379194439388135703211933895203191089e+0002L,
638 6.089148804106598297488336063007609312276e+0004L,
639 4.893546162973278583711376356041614150645e+0006L,
640 2.247571119114497845046388801813832219404e+0008L,
641 6.137635663350177751290469334200757872645e+0009L,
642 1.005115019784102856424493519524998953678e+0011L,
643 9.725664462014503832860151384604677240620e+0011L,
644 5.345525100485511116148634192844434636072e+0012L,
645 1.549944007398946691720862738173956994779e+0013L,
646 2.067148441178952625710302124163264760362e+0013L,
647 9.401565402641963611295119487242595462301e+0012L,
648 3.548217088622398274748837287769709374385e+0011L,
649 -2.934470341719047120076509938432417352365e+0010L,
650 };
651
652 static GENERIC qr3[13] = {
653 3.749999999999412724084579833297451472091e-0001L,
654 9.058478580291706212422978492938435582527e+0001L,
655 8.524056033161038750461083666711724381171e+0003L,
656 4.105967158629109427753434569223631014730e+0005L,
657 1.118326603378531348259783091972623333657e+0007L,
658 1.794636683403578918528064904714132329343e+0008L,
659 1.714314157463635959556133236004368896724e+0009L,
660 9.622092032236084846572067257267661456030e+0009L,
661 3.057759524485859159957762858780768355020e+0010L,
662 5.129306780754798531609621454415938890020e+0010L,
663 3.999122002794961070680636194346316041352e+0010L,
664 1.122298454643493485989721564358100345388e+0010L,
665 5.603981987645989709668830968522362582221e+0008L,
666 };
667
668 static GENERIC qs3[13] = {
669 1.0e0L,
670 2.418328663076578169836155170053634419922e+0002L,
671 2.279620205900121042587523541281272875520e+0004L,
672 1.100984222585729521470129014992217092794e+0006L,
673 3.010743223679247091004262516286654516282e+0007L,
674 4.860925542827367817289619265215599433996e+0008L,
675 4.686668111035348691982715864307839581243e+0009L,
676 2.668701788405102017427214705946730894074e+0010L,
677 8.677395746106802640390580944836650584903e+0010L,
678 1.511936455574951790658498795945106643036e+0011L,
679 1.260845604432623478002018696873608353093e+0011L,
680 4.052692278419853853911440231600864589805e+0010L,
681 2.965516519212226064983267822243329694729e+0009L,
682 };
683
684 static GENERIC qr4[13] = {
685 3.749999999919234164154669754440123072618e-0001L,
686 5.844218580776819864791168253485055101858e+0001L,
687 3.489273514092912982675669411371435670220e+0003L,
688 1.050523637774575684509663430018995479594e+0005L,
689 1.764549172059701565500717319792780115289e+0006L,
690 1.725532438844133795028063102681497371154e+0007L,
691 9.938114847359778539965140247590176334874e+0007L,
692 3.331710808184595545396883770200772842314e+0008L,
693 6.271970557641881511609560444872797282698e+0008L,
694 6.188529798677357075020774923903737913285e+0008L,
695 2.821905302742849974509982167877885011629e+0008L,
696 4.615467358646911976773290256984329814896e+0007L,
697 1.348140608731546467396685802693380693275e+0006L,
698 };
699
700 static GENERIC qs4[13] = {
701 1.0e0L,
702 1.561192663112345185261418296389902133372e+0002L,
703 9.346678031144098270547225423124213083072e+0003L,
704 2.825851246482293547838023847601704751590e+0005L,
705 4.776572711622156091710902891124911556293e+0006L,
706 4.715106953717135402977938048006267859302e+0007L,
707 2.753962350894311316439652227611209035193e+0008L,
708 9.428501434615463207768964787500411575223e+0008L,
709 1.832650858775206787088236896454141572617e+0009L,
710 1.901697378939743226948920874296595242257e+0009L,
711 9.433322226854293780627188599226380812725e+0008L,
712 1.808520540608671608680284520798858587370e+0008L,
713 7.983342331736662753157217446919462398008e+0006L,
714 };
715
716 static GENERIC qr5[13] = {
717 3.749999995331364437028988850515190446719e-0001L,
718 3.739356381766559882677514593041627547911e+0001L,
719 1.399562500629413529355265462912819802551e+0003L,
720 2.594154053098947925345332218062210111753e+0004L,
721 2.640149879297408640394163979394594318371e+0005L,
722 1.542471854873199142031889093591449397995e+0006L,
723 5.242272868972053374067572098992335425895e+0006L,
724 1.025834487769410221329633071426044839935e+0007L,
725 1.116553924239448940142230579060124209622e+0007L,
726 6.318076065595910176374916303525884653514e+0006L,
727 1.641218086168640408527639735915512881785e+0006L,
728 1.522369793529178644168813882912134706444e+0005L,
729 2.526530541062297200914180060208669584055e+0003L,
730 };
731
732 static GENERIC qs5[13] = {
733 1.0e0L,
734 9.998960735935075380397545659016287506660e+0001L,
735 3.758767417842043742686475060540416737562e+0003L,
736 7.013652806952306520121959742519780781653e+0004L,
737 7.208949808818615099246529616211730446850e+0005L,
738 4.272753927109614455417836186072202009252e+0006L,
739 1.482524411356470699336129814111025434703e+0007L,
740 2.988750366665678233425279237627700803473e+0007L,
741 3.396957890261080492694709150553619185065e+0007L,
742 2.050652487738593004111578091156304540386e+0007L,
743 5.900504120811732547616511555946279451316e+0006L,
744 6.563391409260160897024498082273183468347e+0005L,
745 1.692629845012790205348966731477187041419e+0004L,
746 };
747
748 static GENERIC qr6[13] = {
749 3.749999861516664133157566870858975421296e-0001L,
750 2.367863756747764863120797431599473468918e+0001L,
751 5.476715802114976248882067325630793143777e+0002L,
752 6.143190357869842894025012945444096170251e+0003L,
753 3.716250534677997850513733595140463851730e+0004L,
754 1.270883463823876752138326905022875657430e+0005L,
755 2.495301449636814481646371665429083801388e+0005L,
756 2.789578988212952248340486296254398601942e+0005L,
757 1.718247946911109055931819087137397324634e+0005L,
758 5.458973214011665714330326732204106364229e+0004L,
759 7.912102686687948786048943339759596652813e+0003L,
760 4.077961006160866935722030715149087938091e+0002L,
761 3.765206972770245085551057237882528510428e+0000L,
762 };
763
764 static GENERIC qs6[13] = {
765 1.0e0L,
766 6.341646532940517305641893852673926809601e+0001L,
767 1.477058277414040790932597537920671025359e+0003L,
768 1.674406564031044491436044253393536487604e+0004L,
769 1.028516501369755949895050806908994650768e+0005L,
770 3.593620042532885295087463507733285434207e+0005L,
771 7.267924991381020915185873399453724799625e+0005L,
772 8.462277510768818399961191426205006083088e+0005L,
773 5.514399892230892163373611895645500250514e+0005L,
774 1.898084241009259353540620272932188102299e+0005L,
775 3.102941242117739015721984123081026253068e+0004L,
776 1.958971184431466907681440650181421086143e+0003L,
777 2.878853357310495087181721613889455121867e+0001L,
778 };
779
780 static GENERIC
781 qone(x)
782 GENERIC x;
783 {
784 GENERIC s, r, t, z;
785 int i;
786
787 if (x > huge)
788 return (0.375L / x);
789
790 t = one / x;
791 z = t * t;
792
793 if (x > sixteen) {
794 r = z * qr0[11] + qr0[10];
795 s = qs0[10];
796
797 for (i = 9; i >= 0; i--) {
798 r = z * r + qr0[i];
799 s = z * s + qs0[i];
800 }
801 } else if (x > eight) {
802 r = qr1[11];
803 s = qs1[11] + z * (qs1[12] + z * qs1[13]);
804
805 for (i = 10; i >= 0; i--) {
806 r = z * r + qr1[i];
807 s = z * s + qs1[i];
808 }
809 } else if (x > five) { /* x > 5.0 */
810 r = qr2[11];
811 s = qs2[11] + z * (qs2[12] + z * qs2[13]);
812
813 for (i = 10; i >= 0; i--) {
814 r = z * r + qr2[i];
815 s = z * s + qs2[i];
816 }
817 } else if (x > 3.5L) {
818 r = qr3[12];
819 s = qs3[12];
820
821 for (i = 11; i >= 0; i--) {
822 r = z * r + qr3[i];
823 s = z * s + qs3[i];
824 }
825 } else if (x > 2.5L) {
826 r = qr4[12];
827 s = qs4[12];
828
829 for (i = 11; i >= 0; i--) {
830 r = z * r + qr4[i];
831 s = z * s + qs4[i];
832 }
833 } else if (x > (1.0L / 0.5625L)) {
834 r = qr5[12];
835 s = qs5[12];
836
837 for (i = 11; i >= 0; i--) {
838 r = z * r + qr5[i];
839 s = z * s + qs5[i];
840 }
841 } else { /* assume x > 1.28 */
842 r = qr6[12];
843 s = qs6[12];
844
845 for (i = 11; i >= 0; i--) {
846 r = z * r + qr6[i];
847 s = z * s + qs6[i];
848 }
849 }
850
851 return (t * (r / s));
852 }
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