1/* j0l.c
2 *
3 * Bessel function of order zero
4 *
5 *
6 *
7 * SYNOPSIS:
8 *
9 * __float128 x, y, j0l();
10 *
11 * y = j0l( x );
12 *
13 *
14 *
15 * DESCRIPTION:
16 *
17 * Returns Bessel function of first kind, order zero of the argument.
18 *
19 * The domain is divided into two major intervals [0, 2] and
20 * (2, infinity). In the first interval the rational approximation
21 * is J0(x) = 1 - x^2 / 4 + x^4 R(x^2)
22 * The second interval is further partitioned into eight equal segments
23 * of 1/x.
24 *
25 * J0(x) = sqrt(2/(pi x)) (P0(x) cos(X) - Q0(x) sin(X)),
26 * X = x - pi/4,
27 *
28 * and the auxiliary functions are given by
29 *
30 * J0(x)cos(X) + Y0(x)sin(X) = sqrt( 2/(pi x)) P0(x),
31 * P0(x) = 1 + 1/x^2 R(1/x^2)
32 *
33 * Y0(x)cos(X) - J0(x)sin(X) = sqrt( 2/(pi x)) Q0(x),
34 * Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
35 *
36 *
37 *
38 * ACCURACY:
39 *
40 * Absolute error:
41 * arithmetic domain # trials peak rms
42 * IEEE 0, 30 100000 1.7e-34 2.4e-35
43 *
44 *
45 */
46
47/* y0l.c
48 *
49 * Bessel function of the second kind, order zero
50 *
51 *
52 *
53 * SYNOPSIS:
54 *
55 * __float128 x, y, y0l();
56 *
57 * y = y0l( x );
58 *
59 *
60 *
61 * DESCRIPTION:
62 *
63 * Returns Bessel function of the second kind, of order
64 * zero, of the argument.
65 *
66 * The approximation is the same as for J0(x), and
67 * Y0(x) = sqrt(2/(pi x)) (P0(x) sin(X) + Q0(x) cos(X)).
68 *
69 * ACCURACY:
70 *
71 * Absolute error, when y0(x) < 1; else relative error:
72 *
73 * arithmetic domain # trials peak rms
74 * IEEE 0, 30 100000 3.0e-34 2.7e-35
75 *
76 */
77
78/* Copyright 2001 by Stephen L. Moshier (moshier@na-net.ornl.gov).
79
80 This library is free software; you can redistribute it and/or
81 modify it under the terms of the GNU Lesser General Public
82 License as published by the Free Software Foundation; either
83 version 2.1 of the License, or (at your option) any later version.
84
85 This library is distributed in the hope that it will be useful,
86 but WITHOUT ANY WARRANTY; without even the implied warranty of
87 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
88 Lesser General Public License for more details.
89
90 You should have received a copy of the GNU Lesser General Public
91 License along with this library; if not, write to the Free Software
92 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
93
94#include "quadmath-imp.h"
95
96/* 1 / sqrt(pi) */
97static const __float128 ONEOSQPI = 5.6418958354775628694807945156077258584405E-1Q;
98/* 2 / pi */
99static const __float128 TWOOPI = 6.3661977236758134307553505349005744813784E-1Q;
100static const __float128 zero = 0.0Q;
101
102/* J0(x) = 1 - x^2/4 + x^2 x^2 R(x^2)
103 Peak relative error 3.4e-37
104 0 <= x <= 2 */
105#define NJ0_2N 6
106static const __float128 J0_2N[NJ0_2N + 1] = {
107 3.133239376997663645548490085151484674892E16Q,
108 -5.479944965767990821079467311839107722107E14Q,
109 6.290828903904724265980249871997551894090E12Q,
110 -3.633750176832769659849028554429106299915E10Q,
111 1.207743757532429576399485415069244807022E8Q,
112 -2.107485999925074577174305650549367415465E5Q,
113 1.562826808020631846245296572935547005859E2Q,
114};
115#define NJ0_2D 6
116static const __float128 J0_2D[NJ0_2D + 1] = {
117 2.005273201278504733151033654496928968261E18Q,
118 2.063038558793221244373123294054149790864E16Q,
119 1.053350447931127971406896594022010524994E14Q,
120 3.496556557558702583143527876385508882310E11Q,
121 8.249114511878616075860654484367133976306E8Q,
122 1.402965782449571800199759247964242790589E6Q,
123 1.619910762853439600957801751815074787351E3Q,
124 /* 1.000000000000000000000000000000000000000E0 */
125};
126
127/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2),
128 0 <= 1/x <= .0625
129 Peak relative error 3.3e-36 */
130#define NP16_IN 9
131static const __float128 P16_IN[NP16_IN + 1] = {
132 -1.901689868258117463979611259731176301065E-16Q,
133 -1.798743043824071514483008340803573980931E-13Q,
134 -6.481746687115262291873324132944647438959E-11Q,
135 -1.150651553745409037257197798528294248012E-8Q,
136 -1.088408467297401082271185599507222695995E-6Q,
137 -5.551996725183495852661022587879817546508E-5Q,
138 -1.477286941214245433866838787454880214736E-3Q,
139 -1.882877976157714592017345347609200402472E-2Q,
140 -9.620983176855405325086530374317855880515E-2Q,
141 -1.271468546258855781530458854476627766233E-1Q,
142};
143#define NP16_ID 9
144static const __float128 P16_ID[NP16_ID + 1] = {
145 2.704625590411544837659891569420764475007E-15Q,
146 2.562526347676857624104306349421985403573E-12Q,
147 9.259137589952741054108665570122085036246E-10Q,
148 1.651044705794378365237454962653430805272E-7Q,
149 1.573561544138733044977714063100859136660E-5Q,
150 8.134482112334882274688298469629884804056E-4Q,
151 2.219259239404080863919375103673593571689E-2Q,
152 2.976990606226596289580242451096393862792E-1Q,
153 1.713895630454693931742734911930937246254E0Q,
154 3.231552290717904041465898249160757368855E0Q,
155 /* 1.000000000000000000000000000000000000000E0 */
156};
157
158/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
159 0.0625 <= 1/x <= 0.125
160 Peak relative error 2.4e-35 */
161#define NP8_16N 10
162static const __float128 P8_16N[NP8_16N + 1] = {
163 -2.335166846111159458466553806683579003632E-15Q,
164 -1.382763674252402720401020004169367089975E-12Q,
165 -3.192160804534716696058987967592784857907E-10Q,
166 -3.744199606283752333686144670572632116899E-8Q,
167 -2.439161236879511162078619292571922772224E-6Q,
168 -9.068436986859420951664151060267045346549E-5Q,
169 -1.905407090637058116299757292660002697359E-3Q,
170 -2.164456143936718388053842376884252978872E-2Q,
171 -1.212178415116411222341491717748696499966E-1Q,
172 -2.782433626588541494473277445959593334494E-1Q,
173 -1.670703190068873186016102289227646035035E-1Q,
174};
175#define NP8_16D 10
176static const __float128 P8_16D[NP8_16D + 1] = {
177 3.321126181135871232648331450082662856743E-14Q,
178 1.971894594837650840586859228510007703641E-11Q,
179 4.571144364787008285981633719513897281690E-9Q,
180 5.396419143536287457142904742849052402103E-7Q,
181 3.551548222385845912370226756036899901549E-5Q,
182 1.342353874566932014705609788054598013516E-3Q,
183 2.899133293006771317589357444614157734385E-2Q,
184 3.455374978185770197704507681491574261545E-1Q,
185 2.116616964297512311314454834712634820514E0Q,
186 5.850768316827915470087758636881584174432E0Q,
187 5.655273858938766830855753983631132928968E0Q,
188 /* 1.000000000000000000000000000000000000000E0 */
189};
190
191/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
192 0.125 <= 1/x <= 0.1875
193 Peak relative error 2.7e-35 */
194#define NP5_8N 10
195static const __float128 P5_8N[NP5_8N + 1] = {
196 -1.270478335089770355749591358934012019596E-12Q,
197 -4.007588712145412921057254992155810347245E-10Q,
198 -4.815187822989597568124520080486652009281E-8Q,
199 -2.867070063972764880024598300408284868021E-6Q,
200 -9.218742195161302204046454768106063638006E-5Q,
201 -1.635746821447052827526320629828043529997E-3Q,
202 -1.570376886640308408247709616497261011707E-2Q,
203 -7.656484795303305596941813361786219477807E-2Q,
204 -1.659371030767513274944805479908858628053E-1Q,
205 -1.185340550030955660015841796219919804915E-1Q,
206 -8.920026499909994671248893388013790366712E-3Q,
207};
208#define NP5_8D 9
209static const __float128 P5_8D[NP5_8D + 1] = {
210 1.806902521016705225778045904631543990314E-11Q,
211 5.728502760243502431663549179135868966031E-9Q,
212 6.938168504826004255287618819550667978450E-7Q,
213 4.183769964807453250763325026573037785902E-5Q,
214 1.372660678476925468014882230851637878587E-3Q,
215 2.516452105242920335873286419212708961771E-2Q,
216 2.550502712902647803796267951846557316182E-1Q,
217 1.365861559418983216913629123778747617072E0Q,
218 3.523825618308783966723472468855042541407E0Q,
219 3.656365803506136165615111349150536282434E0Q,
220 /* 1.000000000000000000000000000000000000000E0 */
221};
222
223/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
224 Peak relative error 3.5e-35
225 0.1875 <= 1/x <= 0.25 */
226#define NP4_5N 9
227static const __float128 P4_5N[NP4_5N + 1] = {
228 -9.791405771694098960254468859195175708252E-10Q,
229 -1.917193059944531970421626610188102836352E-7Q,
230 -1.393597539508855262243816152893982002084E-5Q,
231 -4.881863490846771259880606911667479860077E-4Q,
232 -8.946571245022470127331892085881699269853E-3Q,
233 -8.707474232568097513415336886103899434251E-2Q,
234 -4.362042697474650737898551272505525973766E-1Q,
235 -1.032712171267523975431451359962375617386E0Q,
236 -9.630502683169895107062182070514713702346E-1Q,
237 -2.251804386252969656586810309252357233320E-1Q,
238};
239#define NP4_5D 9
240static const __float128 P4_5D[NP4_5D + 1] = {
241 1.392555487577717669739688337895791213139E-8Q,
242 2.748886559120659027172816051276451376854E-6Q,
243 2.024717710644378047477189849678576659290E-4Q,
244 7.244868609350416002930624752604670292469E-3Q,
245 1.373631762292244371102989739300382152416E-1Q,
246 1.412298581400224267910294815260613240668E0Q,
247 7.742495637843445079276397723849017617210E0Q,
248 2.138429269198406512028307045259503811861E1Q,
249 2.651547684548423476506826951831712762610E1Q,
250 1.167499382465291931571685222882909166935E1Q,
251 /* 1.000000000000000000000000000000000000000E0 */
252};
253
254/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
255 Peak relative error 2.3e-36
256 0.25 <= 1/x <= 0.3125 */
257#define NP3r2_4N 9
258static const __float128 P3r2_4N[NP3r2_4N + 1] = {
259 -2.589155123706348361249809342508270121788E-8Q,
260 -3.746254369796115441118148490849195516593E-6Q,
261 -1.985595497390808544622893738135529701062E-4Q,
262 -5.008253705202932091290132760394976551426E-3Q,
263 -6.529469780539591572179155511840853077232E-2Q,
264 -4.468736064761814602927408833818990271514E-1Q,
265 -1.556391252586395038089729428444444823380E0Q,
266 -2.533135309840530224072920725976994981638E0Q,
267 -1.605509621731068453869408718565392869560E0Q,
268 -2.518966692256192789269859830255724429375E-1Q,
269};
270#define NP3r2_4D 9
271static const __float128 P3r2_4D[NP3r2_4D + 1] = {
272 3.682353957237979993646169732962573930237E-7Q,
273 5.386741661883067824698973455566332102029E-5Q,
274 2.906881154171822780345134853794241037053E-3Q,
275 7.545832595801289519475806339863492074126E-2Q,
276 1.029405357245594877344360389469584526654E0Q,
277 7.565706120589873131187989560509757626725E0Q,
278 2.951172890699569545357692207898667665796E1Q,
279 5.785723537170311456298467310529815457536E1Q,
280 5.095621464598267889126015412522773474467E1Q,
281 1.602958484169953109437547474953308401442E1Q,
282 /* 1.000000000000000000000000000000000000000E0 */
283};
284
285/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
286 Peak relative error 1.0e-35
287 0.3125 <= 1/x <= 0.375 */
288#define NP2r7_3r2N 9
289static const __float128 P2r7_3r2N[NP2r7_3r2N + 1] = {
290 -1.917322340814391131073820537027234322550E-7Q,
291 -1.966595744473227183846019639723259011906E-5Q,
292 -7.177081163619679403212623526632690465290E-4Q,
293 -1.206467373860974695661544653741899755695E-2Q,
294 -1.008656452188539812154551482286328107316E-1Q,
295 -4.216016116408810856620947307438823892707E-1Q,
296 -8.378631013025721741744285026537009814161E-1Q,
297 -6.973895635309960850033762745957946272579E-1Q,
298 -1.797864718878320770670740413285763554812E-1Q,
299 -4.098025357743657347681137871388402849581E-3Q,
300};
301#define NP2r7_3r2D 8
302static const __float128 P2r7_3r2D[NP2r7_3r2D + 1] = {
303 2.726858489303036441686496086962545034018E-6Q,
304 2.840430827557109238386808968234848081424E-4Q,
305 1.063826772041781947891481054529454088832E-2Q,
306 1.864775537138364773178044431045514405468E-1Q,
307 1.665660052857205170440952607701728254211E0Q,
308 7.723745889544331153080842168958348568395E0Q,
309 1.810726427571829798856428548102077799835E1Q,
310 1.986460672157794440666187503833545388527E1Q,
311 8.645503204552282306364296517220055815488E0Q,
312 /* 1.000000000000000000000000000000000000000E0 */
313};
314
315/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
316 Peak relative error 1.3e-36
317 0.3125 <= 1/x <= 0.4375 */
318#define NP2r3_2r7N 9
319static const __float128 P2r3_2r7N[NP2r3_2r7N + 1] = {
320 -1.594642785584856746358609622003310312622E-6Q,
321 -1.323238196302221554194031733595194539794E-4Q,
322 -3.856087818696874802689922536987100372345E-3Q,
323 -5.113241710697777193011470733601522047399E-2Q,
324 -3.334229537209911914449990372942022350558E-1Q,
325 -1.075703518198127096179198549659283422832E0Q,
326 -1.634174803414062725476343124267110981807E0Q,
327 -1.030133247434119595616826842367268304880E0Q,
328 -1.989811539080358501229347481000707289391E-1Q,
329 -3.246859189246653459359775001466924610236E-3Q,
330};
331#define NP2r3_2r7D 8
332static const __float128 P2r3_2r7D[NP2r3_2r7D + 1] = {
333 2.267936634217251403663034189684284173018E-5Q,
334 1.918112982168673386858072491437971732237E-3Q,
335 5.771704085468423159125856786653868219522E-2Q,
336 8.056124451167969333717642810661498890507E-1Q,
337 5.687897967531010276788680634413789328776E0Q,
338 2.072596760717695491085444438270778394421E1Q,
339 3.801722099819929988585197088613160496684E1Q,
340 3.254620235902912339534998592085115836829E1Q,
341 1.104847772130720331801884344645060675036E1Q,
342 /* 1.000000000000000000000000000000000000000E0 */
343};
344
345/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
346 Peak relative error 1.2e-35
347 0.4375 <= 1/x <= 0.5 */
348#define NP2_2r3N 8
349static const __float128 P2_2r3N[NP2_2r3N + 1] = {
350 -1.001042324337684297465071506097365389123E-4Q,
351 -6.289034524673365824853547252689991418981E-3Q,
352 -1.346527918018624234373664526930736205806E-1Q,
353 -1.268808313614288355444506172560463315102E0Q,
354 -5.654126123607146048354132115649177406163E0Q,
355 -1.186649511267312652171775803270911971693E1Q,
356 -1.094032424931998612551588246779200724257E1Q,
357 -3.728792136814520055025256353193674625267E0Q,
358 -3.000348318524471807839934764596331810608E-1Q,
359};
360#define NP2_2r3D 8
361static const __float128 P2_2r3D[NP2_2r3D + 1] = {
362 1.423705538269770974803901422532055612980E-3Q,
363 9.171476630091439978533535167485230575894E-2Q,
364 2.049776318166637248868444600215942828537E0Q,
365 2.068970329743769804547326701946144899583E1Q,
366 1.025103500560831035592731539565060347709E2Q,
367 2.528088049697570728252145557167066708284E2Q,
368 2.992160327587558573740271294804830114205E2Q,
369 1.540193761146551025832707739468679973036E2Q,
370 2.779516701986912132637672140709452502650E1Q,
371 /* 1.000000000000000000000000000000000000000E0 */
372};
373
374/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
375 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
376 Peak relative error 2.2e-35
377 0 <= 1/x <= .0625 */
378#define NQ16_IN 10
379static const __float128 Q16_IN[NQ16_IN + 1] = {
380 2.343640834407975740545326632205999437469E-18Q,
381 2.667978112927811452221176781536278257448E-15Q,
382 1.178415018484555397390098879501969116536E-12Q,
383 2.622049767502719728905924701288614016597E-10Q,
384 3.196908059607618864801313380896308968673E-8Q,
385 2.179466154171673958770030655199434798494E-6Q,
386 8.139959091628545225221976413795645177291E-5Q,
387 1.563900725721039825236927137885747138654E-3Q,
388 1.355172364265825167113562519307194840307E-2Q,
389 3.928058355906967977269780046844768588532E-2Q,
390 1.107891967702173292405380993183694932208E-2Q,
391};
392#define NQ16_ID 9
393static const __float128 Q16_ID[NQ16_ID + 1] = {
394 3.199850952578356211091219295199301766718E-17Q,
395 3.652601488020654842194486058637953363918E-14Q,
396 1.620179741394865258354608590461839031281E-11Q,
397 3.629359209474609630056463248923684371426E-9Q,
398 4.473680923894354600193264347733477363305E-7Q,
399 3.106368086644715743265603656011050476736E-5Q,
400 1.198239259946770604954664925153424252622E-3Q,
401 2.446041004004283102372887804475767568272E-2Q,
402 2.403235525011860603014707768815113698768E-1Q,
403 9.491006790682158612266270665136910927149E-1Q,
404 /* 1.000000000000000000000000000000000000000E0 */
405 };
406
407/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
408 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
409 Peak relative error 5.1e-36
410 0.0625 <= 1/x <= 0.125 */
411#define NQ8_16N 11
412static const __float128 Q8_16N[NQ8_16N + 1] = {
413 1.001954266485599464105669390693597125904E-17Q,
414 7.545499865295034556206475956620160007849E-15Q,
415 2.267838684785673931024792538193202559922E-12Q,
416 3.561909705814420373609574999542459912419E-10Q,
417 3.216201422768092505214730633842924944671E-8Q,
418 1.731194793857907454569364622452058554314E-6Q,
419 5.576944613034537050396518509871004586039E-5Q,
420 1.051787760316848982655967052985391418146E-3Q,
421 1.102852974036687441600678598019883746959E-2Q,
422 5.834647019292460494254225988766702933571E-2Q,
423 1.290281921604364618912425380717127576529E-1Q,
424 7.598886310387075708640370806458926458301E-2Q,
425};
426#define NQ8_16D 11
427static const __float128 Q8_16D[NQ8_16D + 1] = {
428 1.368001558508338469503329967729951830843E-16Q,
429 1.034454121857542147020549303317348297289E-13Q,
430 3.128109209247090744354764050629381674436E-11Q,
431 4.957795214328501986562102573522064468671E-9Q,
432 4.537872468606711261992676606899273588899E-7Q,
433 2.493639207101727713192687060517509774182E-5Q,
434 8.294957278145328349785532236663051405805E-4Q,
435 1.646471258966713577374948205279380115839E-2Q,
436 1.878910092770966718491814497982191447073E-1Q,
437 1.152641605706170353727903052525652504075E0Q,
438 3.383550240669773485412333679367792932235E0Q,
439 3.823875252882035706910024716609908473970E0Q,
440 /* 1.000000000000000000000000000000000000000E0 */
441};
442
443/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
444 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
445 Peak relative error 3.9e-35
446 0.125 <= 1/x <= 0.1875 */
447#define NQ5_8N 10
448static const __float128 Q5_8N[NQ5_8N + 1] = {
449 1.750399094021293722243426623211733898747E-13Q,
450 6.483426211748008735242909236490115050294E-11Q,
451 9.279430665656575457141747875716899958373E-9Q,
452 6.696634968526907231258534757736576340266E-7Q,
453 2.666560823798895649685231292142838188061E-5Q,
454 6.025087697259436271271562769707550594540E-4Q,
455 7.652807734168613251901945778921336353485E-3Q,
456 5.226269002589406461622551452343519078905E-2Q,
457 1.748390159751117658969324896330142895079E-1Q,
458 2.378188719097006494782174902213083589660E-1Q,
459 8.383984859679804095463699702165659216831E-2Q,
460};
461#define NQ5_8D 10
462static const __float128 Q5_8D[NQ5_8D + 1] = {
463 2.389878229704327939008104855942987615715E-12Q,
464 8.926142817142546018703814194987786425099E-10Q,
465 1.294065862406745901206588525833274399038E-7Q,
466 9.524139899457666250828752185212769682191E-6Q,
467 3.908332488377770886091936221573123353489E-4Q,
468 9.250427033957236609624199884089916836748E-3Q,
469 1.263420066165922645975830877751588421451E-1Q,
470 9.692527053860420229711317379861733180654E-1Q,
471 3.937813834630430172221329298841520707954E0Q,
472 7.603126427436356534498908111445191312181E0Q,
473 5.670677653334105479259958485084550934305E0Q,
474 /* 1.000000000000000000000000000000000000000E0 */
475};
476
477/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
478 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
479 Peak relative error 3.2e-35
480 0.1875 <= 1/x <= 0.25 */
481#define NQ4_5N 10
482static const __float128 Q4_5N[NQ4_5N + 1] = {
483 2.233870042925895644234072357400122854086E-11Q,
484 5.146223225761993222808463878999151699792E-9Q,
485 4.459114531468296461688753521109797474523E-7Q,
486 1.891397692931537975547242165291668056276E-5Q,
487 4.279519145911541776938964806470674565504E-4Q,
488 5.275239415656560634702073291768904783989E-3Q,
489 3.468698403240744801278238473898432608887E-2Q,
490 1.138773146337708415188856882915457888274E-1Q,
491 1.622717518946443013587108598334636458955E-1Q,
492 7.249040006390586123760992346453034628227E-2Q,
493 1.941595365256460232175236758506411486667E-3Q,
494};
495#define NQ4_5D 9
496static const __float128 Q4_5D[NQ4_5D + 1] = {
497 3.049977232266999249626430127217988047453E-10Q,
498 7.120883230531035857746096928889676144099E-8Q,
499 6.301786064753734446784637919554359588859E-6Q,
500 2.762010530095069598480766869426308077192E-4Q,
501 6.572163250572867859316828886203406361251E-3Q,
502 8.752566114841221958200215255461843397776E-2Q,
503 6.487654992874805093499285311075289932664E-1Q,
504 2.576550017826654579451615283022812801435E0Q,
505 5.056392229924022835364779562707348096036E0Q,
506 4.179770081068251464907531367859072157773E0Q,
507 /* 1.000000000000000000000000000000000000000E0 */
508};
509
510/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
511 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
512 Peak relative error 1.4e-36
513 0.25 <= 1/x <= 0.3125 */
514#define NQ3r2_4N 10
515static const __float128 Q3r2_4N[NQ3r2_4N + 1] = {
516 6.126167301024815034423262653066023684411E-10Q,
517 1.043969327113173261820028225053598975128E-7Q,
518 6.592927270288697027757438170153763220190E-6Q,
519 2.009103660938497963095652951912071336730E-4Q,
520 3.220543385492643525985862356352195896964E-3Q,
521 2.774405975730545157543417650436941650990E-2Q,
522 1.258114008023826384487378016636555041129E-1Q,
523 2.811724258266902502344701449984698323860E-1Q,
524 2.691837665193548059322831687432415014067E-1Q,
525 7.949087384900985370683770525312735605034E-2Q,
526 1.229509543620976530030153018986910810747E-3Q,
527};
528#define NQ3r2_4D 9
529static const __float128 Q3r2_4D[NQ3r2_4D + 1] = {
530 8.364260446128475461539941389210166156568E-9Q,
531 1.451301850638956578622154585560759862764E-6Q,
532 9.431830010924603664244578867057141839463E-5Q,
533 3.004105101667433434196388593004526182741E-3Q,
534 5.148157397848271739710011717102773780221E-2Q,
535 4.901089301726939576055285374953887874895E-1Q,
536 2.581760991981709901216967665934142240346E0Q,
537 7.257105880775059281391729708630912791847E0Q,
538 1.006014717326362868007913423810737369312E1Q,
539 5.879416600465399514404064187445293212470E0Q,
540 /* 1.000000000000000000000000000000000000000E0*/
541};
542
543/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
544 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
545 Peak relative error 3.8e-36
546 0.3125 <= 1/x <= 0.375 */
547#define NQ2r7_3r2N 9
548static const __float128 Q2r7_3r2N[NQ2r7_3r2N + 1] = {
549 7.584861620402450302063691901886141875454E-8Q,
550 9.300939338814216296064659459966041794591E-6Q,
551 4.112108906197521696032158235392604947895E-4Q,
552 8.515168851578898791897038357239630654431E-3Q,
553 8.971286321017307400142720556749573229058E-2Q,
554 4.885856732902956303343015636331874194498E-1Q,
555 1.334506268733103291656253500506406045846E0Q,
556 1.681207956863028164179042145803851824654E0Q,
557 8.165042692571721959157677701625853772271E-1Q,
558 9.805848115375053300608712721986235900715E-2Q,
559};
560#define NQ2r7_3r2D 9
561static const __float128 Q2r7_3r2D[NQ2r7_3r2D + 1] = {
562 1.035586492113036586458163971239438078160E-6Q,
563 1.301999337731768381683593636500979713689E-4Q,
564 5.993695702564527062553071126719088859654E-3Q,
565 1.321184892887881883489141186815457808785E-1Q,
566 1.528766555485015021144963194165165083312E0Q,
567 9.561463309176490874525827051566494939295E0Q,
568 3.203719484883967351729513662089163356911E1Q,
569 5.497294687660930446641539152123568668447E1Q,
570 4.391158169390578768508675452986948391118E1Q,
571 1.347836630730048077907818943625789418378E1Q,
572 /* 1.000000000000000000000000000000000000000E0 */
573};
574
575/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
576 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
577 Peak relative error 2.2e-35
578 0.375 <= 1/x <= 0.4375 */
579#define NQ2r3_2r7N 9
580static const __float128 Q2r3_2r7N[NQ2r3_2r7N + 1] = {
581 4.455027774980750211349941766420190722088E-7Q,
582 4.031998274578520170631601850866780366466E-5Q,
583 1.273987274325947007856695677491340636339E-3Q,
584 1.818754543377448509897226554179659122873E-2Q,
585 1.266748858326568264126353051352269875352E-1Q,
586 4.327578594728723821137731555139472880414E-1Q,
587 6.892532471436503074928194969154192615359E-1Q,
588 4.490775818438716873422163588640262036506E-1Q,
589 8.649615949297322440032000346117031581572E-2Q,
590 7.261345286655345047417257611469066147561E-4Q,
591};
592#define NQ2r3_2r7D 8
593static const __float128 Q2r3_2r7D[NQ2r3_2r7D + 1] = {
594 6.082600739680555266312417978064954793142E-6Q,
595 5.693622538165494742945717226571441747567E-4Q,
596 1.901625907009092204458328768129666975975E-2Q,
597 2.958689532697857335456896889409923371570E-1Q,
598 2.343124711045660081603809437993368799568E0Q,
599 9.665894032187458293568704885528192804376E0Q,
600 2.035273104990617136065743426322454881353E1Q,
601 2.044102010478792896815088858740075165531E1Q,
602 8.445937177863155827844146643468706599304E0Q,
603 /* 1.000000000000000000000000000000000000000E0 */
604};
605
606/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
607 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
608 Peak relative error 3.1e-36
609 0.4375 <= 1/x <= 0.5 */
610#define NQ2_2r3N 9
611static const __float128 Q2_2r3N[NQ2_2r3N + 1] = {
612 2.817566786579768804844367382809101929314E-6Q,
613 2.122772176396691634147024348373539744935E-4Q,
614 5.501378031780457828919593905395747517585E-3Q,
615 6.355374424341762686099147452020466524659E-2Q,
616 3.539652320122661637429658698954748337223E-1Q,
617 9.571721066119617436343740541777014319695E-1Q,
618 1.196258777828426399432550698612171955305E0Q,
619 6.069388659458926158392384709893753793967E-1Q,
620 9.026746127269713176512359976978248763621E-2Q,
621 5.317668723070450235320878117210807236375E-4Q,
622};
623#define NQ2_2r3D 8
624static const __float128 Q2_2r3D[NQ2_2r3D + 1] = {
625 3.846924354014260866793741072933159380158E-5Q,
626 3.017562820057704325510067178327449946763E-3Q,
627 8.356305620686867949798885808540444210935E-2Q,
628 1.068314930499906838814019619594424586273E0Q,
629 6.900279623894821067017966573640732685233E0Q,
630 2.307667390886377924509090271780839563141E1Q,
631 3.921043465412723970791036825401273528513E1Q,
632 3.167569478939719383241775717095729233436E1Q,
633 1.051023841699200920276198346301543665909E1Q,
634 /* 1.000000000000000000000000000000000000000E0*/
635};
636
637
638/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
639
640static __float128
641neval (__float128 x, const __float128 *p, int n)
642{
643 __float128 y;
644
645 p += n;
646 y = *p--;
647 do
648 {
649 y = y * x + *p--;
650 }
651 while (--n > 0);
652 return y;
653}
654
655
656/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
657
658static __float128
659deval (__float128 x, const __float128 *p, int n)
660{
661 __float128 y;
662
663 p += n;
664 y = x + *p--;
665 do
666 {
667 y = y * x + *p--;
668 }
669 while (--n > 0);
670 return y;
671}
672
673
674/* Bessel function of the first kind, order zero. */
675
676__float128
677j0q (__float128 x)
678{
679 __float128 xx, xinv, z, p, q, c, s, cc, ss;
680
681 if (! finiteq (x))
682 {
683 if (x != x)
684 return x + x;
685 else
686 return 0.0Q;
687 }
688 if (x == 0.0Q)
689 return 1.0Q;
690
691 xx = fabsq (x);
692 if (xx <= 2.0Q)
693 {
694 if (xx < 0x1p-57Q)
695 return 1.0Q;
696 /* 0 <= x <= 2 */
697 z = xx * xx;
698 p = z * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D);
699 p -= 0.25Q * z;
700 p += 1.0Q;
701 return p;
702 }
703
704 /* X = x - pi/4
705 cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4)
706 = 1/sqrt(2) * (cos(x) + sin(x))
707 sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4)
708 = 1/sqrt(2) * (sin(x) - cos(x))
709 sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
710 cf. Fdlibm. */
711 sincosq (xx, &s, &c);
712 ss = s - c;
713 cc = s + c;
714 if (xx <= FLT128_MAX / 2.0Q)
715 {
716 z = -cosq (xx + xx);
717 if ((s * c) < 0)
718 cc = z / ss;
719 else
720 ss = z / cc;
721 }
722
723 if (xx > 0x1p256Q)
724 return ONEOSQPI * cc / sqrtq (xx);
725
726 xinv = 1.0Q / xx;
727 z = xinv * xinv;
728 if (xinv <= 0.25)
729 {
730 if (xinv <= 0.125)
731 {
732 if (xinv <= 0.0625)
733 {
734 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
735 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
736 }
737 else
738 {
739 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
740 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
741 }
742 }
743 else if (xinv <= 0.1875)
744 {
745 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
746 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
747 }
748 else
749 {
750 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
751 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
752 }
753 } /* .25 */
754 else /* if (xinv <= 0.5) */
755 {
756 if (xinv <= 0.375)
757 {
758 if (xinv <= 0.3125)
759 {
760 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
761 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
762 }
763 else
764 {
765 p = neval (z, P2r7_3r2N, NP2r7_3r2N)
766 / deval (z, P2r7_3r2D, NP2r7_3r2D);
767 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
768 / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
769 }
770 }
771 else if (xinv <= 0.4375)
772 {
773 p = neval (z, P2r3_2r7N, NP2r3_2r7N)
774 / deval (z, P2r3_2r7D, NP2r3_2r7D);
775 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
776 / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
777 }
778 else
779 {
780 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
781 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
782 }
783 }
784 p = 1.0Q + z * p;
785 q = z * xinv * q;
786 q = q - 0.125Q * xinv;
787 z = ONEOSQPI * (p * cc - q * ss) / sqrtq (xx);
788 return z;
789}
790
791
792/* Y0(x) = 2/pi * log(x) * J0(x) + R(x^2)
793 Peak absolute error 1.7e-36 (relative where Y0 > 1)
794 0 <= x <= 2 */
795#define NY0_2N 7
796static __float128 Y0_2N[NY0_2N + 1] = {
797 -1.062023609591350692692296993537002558155E19Q,
798 2.542000883190248639104127452714966858866E19Q,
799 -1.984190771278515324281415820316054696545E18Q,
800 4.982586044371592942465373274440222033891E16Q,
801 -5.529326354780295177243773419090123407550E14Q,
802 3.013431465522152289279088265336861140391E12Q,
803 -7.959436160727126750732203098982718347785E9Q,
804 8.230845651379566339707130644134372793322E6Q,
805};
806#define NY0_2D 7
807static __float128 Y0_2D[NY0_2D + 1] = {
808 1.438972634353286978700329883122253752192E20Q,
809 1.856409101981569254247700169486907405500E18Q,
810 1.219693352678218589553725579802986255614E16Q,
811 5.389428943282838648918475915779958097958E13Q,
812 1.774125762108874864433872173544743051653E11Q,
813 4.522104832545149534808218252434693007036E8Q,
814 8.872187401232943927082914504125234454930E5Q,
815 1.251945613186787532055610876304669413955E3Q,
816 /* 1.000000000000000000000000000000000000000E0 */
817};
818
819static const __float128 U0 = -7.3804295108687225274343927948483016310862e-02Q;
820
821/* Bessel function of the second kind, order zero. */
822
823__float128
824y0q (__float128 x)
825{
826 __float128 xx, xinv, z, p, q, c, s, cc, ss;
827
828 if (! finiteq (x))
829 return 1 / (x + x * x);
830 if (x <= 0.0Q)
831 {
832 if (x < 0.0Q)
833 return (zero / (zero * x));
834 return -1 / zero; /* -inf and divide by zero exception. */
835 }
836 xx = fabsq (x);
837 if (xx <= 0x1p-57)
838 return U0 + TWOOPI * logq (x);
839 if (xx <= 2.0Q)
840 {
841 /* 0 <= x <= 2 */
842 z = xx * xx;
843 p = neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D);
844 p = TWOOPI * logq (x) * j0q (x) + p;
845 return p;
846 }
847
848 /* X = x - pi/4
849 cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4)
850 = 1/sqrt(2) * (cos(x) + sin(x))
851 sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4)
852 = 1/sqrt(2) * (sin(x) - cos(x))
853 sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
854 cf. Fdlibm. */
855 sincosq (x, &s, &c);
856 ss = s - c;
857 cc = s + c;
858 if (xx <= FLT128_MAX / 2.0Q)
859 {
860 z = -cosq (x + x);
861 if ((s * c) < 0)
862 cc = z / ss;
863 else
864 ss = z / cc;
865 }
866
867 if (xx > 0x1p256Q)
868 return ONEOSQPI * ss / sqrtq (x);
869
870 xinv = 1.0Q / xx;
871 z = xinv * xinv;
872 if (xinv <= 0.25)
873 {
874 if (xinv <= 0.125)
875 {
876 if (xinv <= 0.0625)
877 {
878 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
879 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
880 }
881 else
882 {
883 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
884 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
885 }
886 }
887 else if (xinv <= 0.1875)
888 {
889 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
890 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
891 }
892 else
893 {
894 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
895 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
896 }
897 } /* .25 */
898 else /* if (xinv <= 0.5) */
899 {
900 if (xinv <= 0.375)
901 {
902 if (xinv <= 0.3125)
903 {
904 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
905 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
906 }
907 else
908 {
909 p = neval (z, P2r7_3r2N, NP2r7_3r2N)
910 / deval (z, P2r7_3r2D, NP2r7_3r2D);
911 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
912 / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
913 }
914 }
915 else if (xinv <= 0.4375)
916 {
917 p = neval (z, P2r3_2r7N, NP2r3_2r7N)
918 / deval (z, P2r3_2r7D, NP2r3_2r7D);
919 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
920 / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
921 }
922 else
923 {
924 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
925 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
926 }
927 }
928 p = 1.0Q + z * p;
929 q = z * xinv * q;
930 q = q - 0.125Q * xinv;
931 z = ONEOSQPI * (p * ss + q * cc) / sqrtq (x);
932 return z;
933}
934