1 /* Bessel function of order zero. IBM Extended Precision version.
2 Copyright 2001 by Stephen L. Moshier (moshier@na-net.ornl.gov).
3
4 This library is free software; you can redistribute it and/or
5 modify it under the terms of the GNU Lesser General Public
6 License as published by the Free Software Foundation; either
7 version 2.1 of the License, or (at your option) any later version.
8
9 This library is distributed in the hope that it will be useful,
10 but WITHOUT ANY WARRANTY; without even the implied warranty of
11 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
12 Lesser General Public License for more details.
13
14 You should have received a copy of the GNU Lesser General Public
15 License along with this library; if not, see
16 <https://www.gnu.org/licenses/>. */
17
18 /* This file was copied from sysdeps/ieee754/ldbl-128/e_j0l.c. */
19
20
21 #include <math.h>
22 #include <math_private.h>
23 #include <float.h>
24 #include <libm-alias-finite.h>
25
26 /* 1 / sqrt(pi) */
27 static const long double ONEOSQPI = 5.6418958354775628694807945156077258584405E-1L;
28 /* 2 / pi */
29 static const long double TWOOPI = 6.3661977236758134307553505349005744813784E-1L;
30 static const long double zero = 0;
31
32 /* J0(x) = 1 - x^2/4 + x^2 x^2 R(x^2)
33 Peak relative error 3.4e-37
34 0 <= x <= 2 */
35 #define NJ0_2N 6
36 static const long double J0_2N[NJ0_2N + 1] = {
37 3.133239376997663645548490085151484674892E16L,
38 -5.479944965767990821079467311839107722107E14L,
39 6.290828903904724265980249871997551894090E12L,
40 -3.633750176832769659849028554429106299915E10L,
41 1.207743757532429576399485415069244807022E8L,
42 -2.107485999925074577174305650549367415465E5L,
43 1.562826808020631846245296572935547005859E2L,
44 };
45 #define NJ0_2D 6
46 static const long double J0_2D[NJ0_2D + 1] = {
47 2.005273201278504733151033654496928968261E18L,
48 2.063038558793221244373123294054149790864E16L,
49 1.053350447931127971406896594022010524994E14L,
50 3.496556557558702583143527876385508882310E11L,
51 8.249114511878616075860654484367133976306E8L,
52 1.402965782449571800199759247964242790589E6L,
53 1.619910762853439600957801751815074787351E3L,
54 /* 1.000000000000000000000000000000000000000E0 */
55 };
56
57 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2),
58 0 <= 1/x <= .0625
59 Peak relative error 3.3e-36 */
60 #define NP16_IN 9
61 static const long double P16_IN[NP16_IN + 1] = {
62 -1.901689868258117463979611259731176301065E-16L,
63 -1.798743043824071514483008340803573980931E-13L,
64 -6.481746687115262291873324132944647438959E-11L,
65 -1.150651553745409037257197798528294248012E-8L,
66 -1.088408467297401082271185599507222695995E-6L,
67 -5.551996725183495852661022587879817546508E-5L,
68 -1.477286941214245433866838787454880214736E-3L,
69 -1.882877976157714592017345347609200402472E-2L,
70 -9.620983176855405325086530374317855880515E-2L,
71 -1.271468546258855781530458854476627766233E-1L,
72 };
73 #define NP16_ID 9
74 static const long double P16_ID[NP16_ID + 1] = {
75 2.704625590411544837659891569420764475007E-15L,
76 2.562526347676857624104306349421985403573E-12L,
77 9.259137589952741054108665570122085036246E-10L,
78 1.651044705794378365237454962653430805272E-7L,
79 1.573561544138733044977714063100859136660E-5L,
80 8.134482112334882274688298469629884804056E-4L,
81 2.219259239404080863919375103673593571689E-2L,
82 2.976990606226596289580242451096393862792E-1L,
83 1.713895630454693931742734911930937246254E0L,
84 3.231552290717904041465898249160757368855E0L,
85 /* 1.000000000000000000000000000000000000000E0 */
86 };
87
88 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
89 0.0625 <= 1/x <= 0.125
90 Peak relative error 2.4e-35 */
91 #define NP8_16N 10
92 static const long double P8_16N[NP8_16N + 1] = {
93 -2.335166846111159458466553806683579003632E-15L,
94 -1.382763674252402720401020004169367089975E-12L,
95 -3.192160804534716696058987967592784857907E-10L,
96 -3.744199606283752333686144670572632116899E-8L,
97 -2.439161236879511162078619292571922772224E-6L,
98 -9.068436986859420951664151060267045346549E-5L,
99 -1.905407090637058116299757292660002697359E-3L,
100 -2.164456143936718388053842376884252978872E-2L,
101 -1.212178415116411222341491717748696499966E-1L,
102 -2.782433626588541494473277445959593334494E-1L,
103 -1.670703190068873186016102289227646035035E-1L,
104 };
105 #define NP8_16D 10
106 static const long double P8_16D[NP8_16D + 1] = {
107 3.321126181135871232648331450082662856743E-14L,
108 1.971894594837650840586859228510007703641E-11L,
109 4.571144364787008285981633719513897281690E-9L,
110 5.396419143536287457142904742849052402103E-7L,
111 3.551548222385845912370226756036899901549E-5L,
112 1.342353874566932014705609788054598013516E-3L,
113 2.899133293006771317589357444614157734385E-2L,
114 3.455374978185770197704507681491574261545E-1L,
115 2.116616964297512311314454834712634820514E0L,
116 5.850768316827915470087758636881584174432E0L,
117 5.655273858938766830855753983631132928968E0L,
118 /* 1.000000000000000000000000000000000000000E0 */
119 };
120
121 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
122 0.125 <= 1/x <= 0.1875
123 Peak relative error 2.7e-35 */
124 #define NP5_8N 10
125 static const long double P5_8N[NP5_8N + 1] = {
126 -1.270478335089770355749591358934012019596E-12L,
127 -4.007588712145412921057254992155810347245E-10L,
128 -4.815187822989597568124520080486652009281E-8L,
129 -2.867070063972764880024598300408284868021E-6L,
130 -9.218742195161302204046454768106063638006E-5L,
131 -1.635746821447052827526320629828043529997E-3L,
132 -1.570376886640308408247709616497261011707E-2L,
133 -7.656484795303305596941813361786219477807E-2L,
134 -1.659371030767513274944805479908858628053E-1L,
135 -1.185340550030955660015841796219919804915E-1L,
136 -8.920026499909994671248893388013790366712E-3L,
137 };
138 #define NP5_8D 9
139 static const long double P5_8D[NP5_8D + 1] = {
140 1.806902521016705225778045904631543990314E-11L,
141 5.728502760243502431663549179135868966031E-9L,
142 6.938168504826004255287618819550667978450E-7L,
143 4.183769964807453250763325026573037785902E-5L,
144 1.372660678476925468014882230851637878587E-3L,
145 2.516452105242920335873286419212708961771E-2L,
146 2.550502712902647803796267951846557316182E-1L,
147 1.365861559418983216913629123778747617072E0L,
148 3.523825618308783966723472468855042541407E0L,
149 3.656365803506136165615111349150536282434E0L,
150 /* 1.000000000000000000000000000000000000000E0 */
151 };
152
153 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
154 Peak relative error 3.5e-35
155 0.1875 <= 1/x <= 0.25 */
156 #define NP4_5N 9
157 static const long double P4_5N[NP4_5N + 1] = {
158 -9.791405771694098960254468859195175708252E-10L,
159 -1.917193059944531970421626610188102836352E-7L,
160 -1.393597539508855262243816152893982002084E-5L,
161 -4.881863490846771259880606911667479860077E-4L,
162 -8.946571245022470127331892085881699269853E-3L,
163 -8.707474232568097513415336886103899434251E-2L,
164 -4.362042697474650737898551272505525973766E-1L,
165 -1.032712171267523975431451359962375617386E0L,
166 -9.630502683169895107062182070514713702346E-1L,
167 -2.251804386252969656586810309252357233320E-1L,
168 };
169 #define NP4_5D 9
170 static const long double P4_5D[NP4_5D + 1] = {
171 1.392555487577717669739688337895791213139E-8L,
172 2.748886559120659027172816051276451376854E-6L,
173 2.024717710644378047477189849678576659290E-4L,
174 7.244868609350416002930624752604670292469E-3L,
175 1.373631762292244371102989739300382152416E-1L,
176 1.412298581400224267910294815260613240668E0L,
177 7.742495637843445079276397723849017617210E0L,
178 2.138429269198406512028307045259503811861E1L,
179 2.651547684548423476506826951831712762610E1L,
180 1.167499382465291931571685222882909166935E1L,
181 /* 1.000000000000000000000000000000000000000E0 */
182 };
183
184 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
185 Peak relative error 2.3e-36
186 0.25 <= 1/x <= 0.3125 */
187 #define NP3r2_4N 9
188 static const long double P3r2_4N[NP3r2_4N + 1] = {
189 -2.589155123706348361249809342508270121788E-8L,
190 -3.746254369796115441118148490849195516593E-6L,
191 -1.985595497390808544622893738135529701062E-4L,
192 -5.008253705202932091290132760394976551426E-3L,
193 -6.529469780539591572179155511840853077232E-2L,
194 -4.468736064761814602927408833818990271514E-1L,
195 -1.556391252586395038089729428444444823380E0L,
196 -2.533135309840530224072920725976994981638E0L,
197 -1.605509621731068453869408718565392869560E0L,
198 -2.518966692256192789269859830255724429375E-1L,
199 };
200 #define NP3r2_4D 9
201 static const long double P3r2_4D[NP3r2_4D + 1] = {
202 3.682353957237979993646169732962573930237E-7L,
203 5.386741661883067824698973455566332102029E-5L,
204 2.906881154171822780345134853794241037053E-3L,
205 7.545832595801289519475806339863492074126E-2L,
206 1.029405357245594877344360389469584526654E0L,
207 7.565706120589873131187989560509757626725E0L,
208 2.951172890699569545357692207898667665796E1L,
209 5.785723537170311456298467310529815457536E1L,
210 5.095621464598267889126015412522773474467E1L,
211 1.602958484169953109437547474953308401442E1L,
212 /* 1.000000000000000000000000000000000000000E0 */
213 };
214
215 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
216 Peak relative error 1.0e-35
217 0.3125 <= 1/x <= 0.375 */
218 #define NP2r7_3r2N 9
219 static const long double P2r7_3r2N[NP2r7_3r2N + 1] = {
220 -1.917322340814391131073820537027234322550E-7L,
221 -1.966595744473227183846019639723259011906E-5L,
222 -7.177081163619679403212623526632690465290E-4L,
223 -1.206467373860974695661544653741899755695E-2L,
224 -1.008656452188539812154551482286328107316E-1L,
225 -4.216016116408810856620947307438823892707E-1L,
226 -8.378631013025721741744285026537009814161E-1L,
227 -6.973895635309960850033762745957946272579E-1L,
228 -1.797864718878320770670740413285763554812E-1L,
229 -4.098025357743657347681137871388402849581E-3L,
230 };
231 #define NP2r7_3r2D 8
232 static const long double P2r7_3r2D[NP2r7_3r2D + 1] = {
233 2.726858489303036441686496086962545034018E-6L,
234 2.840430827557109238386808968234848081424E-4L,
235 1.063826772041781947891481054529454088832E-2L,
236 1.864775537138364773178044431045514405468E-1L,
237 1.665660052857205170440952607701728254211E0L,
238 7.723745889544331153080842168958348568395E0L,
239 1.810726427571829798856428548102077799835E1L,
240 1.986460672157794440666187503833545388527E1L,
241 8.645503204552282306364296517220055815488E0L,
242 /* 1.000000000000000000000000000000000000000E0 */
243 };
244
245 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
246 Peak relative error 1.3e-36
247 0.3125 <= 1/x <= 0.4375 */
248 #define NP2r3_2r7N 9
249 static const long double P2r3_2r7N[NP2r3_2r7N + 1] = {
250 -1.594642785584856746358609622003310312622E-6L,
251 -1.323238196302221554194031733595194539794E-4L,
252 -3.856087818696874802689922536987100372345E-3L,
253 -5.113241710697777193011470733601522047399E-2L,
254 -3.334229537209911914449990372942022350558E-1L,
255 -1.075703518198127096179198549659283422832E0L,
256 -1.634174803414062725476343124267110981807E0L,
257 -1.030133247434119595616826842367268304880E0L,
258 -1.989811539080358501229347481000707289391E-1L,
259 -3.246859189246653459359775001466924610236E-3L,
260 };
261 #define NP2r3_2r7D 8
262 static const long double P2r3_2r7D[NP2r3_2r7D + 1] = {
263 2.267936634217251403663034189684284173018E-5L,
264 1.918112982168673386858072491437971732237E-3L,
265 5.771704085468423159125856786653868219522E-2L,
266 8.056124451167969333717642810661498890507E-1L,
267 5.687897967531010276788680634413789328776E0L,
268 2.072596760717695491085444438270778394421E1L,
269 3.801722099819929988585197088613160496684E1L,
270 3.254620235902912339534998592085115836829E1L,
271 1.104847772130720331801884344645060675036E1L,
272 /* 1.000000000000000000000000000000000000000E0 */
273 };
274
275 /* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
276 Peak relative error 1.2e-35
277 0.4375 <= 1/x <= 0.5 */
278 #define NP2_2r3N 8
279 static const long double P2_2r3N[NP2_2r3N + 1] = {
280 -1.001042324337684297465071506097365389123E-4L,
281 -6.289034524673365824853547252689991418981E-3L,
282 -1.346527918018624234373664526930736205806E-1L,
283 -1.268808313614288355444506172560463315102E0L,
284 -5.654126123607146048354132115649177406163E0L,
285 -1.186649511267312652171775803270911971693E1L,
286 -1.094032424931998612551588246779200724257E1L,
287 -3.728792136814520055025256353193674625267E0L,
288 -3.000348318524471807839934764596331810608E-1L,
289 };
290 #define NP2_2r3D 8
291 static const long double P2_2r3D[NP2_2r3D + 1] = {
292 1.423705538269770974803901422532055612980E-3L,
293 9.171476630091439978533535167485230575894E-2L,
294 2.049776318166637248868444600215942828537E0L,
295 2.068970329743769804547326701946144899583E1L,
296 1.025103500560831035592731539565060347709E2L,
297 2.528088049697570728252145557167066708284E2L,
298 2.992160327587558573740271294804830114205E2L,
299 1.540193761146551025832707739468679973036E2L,
300 2.779516701986912132637672140709452502650E1L,
301 /* 1.000000000000000000000000000000000000000E0 */
302 };
303
304 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
305 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
306 Peak relative error 2.2e-35
307 0 <= 1/x <= .0625 */
308 #define NQ16_IN 10
309 static const long double Q16_IN[NQ16_IN + 1] = {
310 2.343640834407975740545326632205999437469E-18L,
311 2.667978112927811452221176781536278257448E-15L,
312 1.178415018484555397390098879501969116536E-12L,
313 2.622049767502719728905924701288614016597E-10L,
314 3.196908059607618864801313380896308968673E-8L,
315 2.179466154171673958770030655199434798494E-6L,
316 8.139959091628545225221976413795645177291E-5L,
317 1.563900725721039825236927137885747138654E-3L,
318 1.355172364265825167113562519307194840307E-2L,
319 3.928058355906967977269780046844768588532E-2L,
320 1.107891967702173292405380993183694932208E-2L,
321 };
322 #define NQ16_ID 9
323 static const long double Q16_ID[NQ16_ID + 1] = {
324 3.199850952578356211091219295199301766718E-17L,
325 3.652601488020654842194486058637953363918E-14L,
326 1.620179741394865258354608590461839031281E-11L,
327 3.629359209474609630056463248923684371426E-9L,
328 4.473680923894354600193264347733477363305E-7L,
329 3.106368086644715743265603656011050476736E-5L,
330 1.198239259946770604954664925153424252622E-3L,
331 2.446041004004283102372887804475767568272E-2L,
332 2.403235525011860603014707768815113698768E-1L,
333 9.491006790682158612266270665136910927149E-1L,
334 /* 1.000000000000000000000000000000000000000E0 */
335 };
336
337 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
338 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
339 Peak relative error 5.1e-36
340 0.0625 <= 1/x <= 0.125 */
341 #define NQ8_16N 11
342 static const long double Q8_16N[NQ8_16N + 1] = {
343 1.001954266485599464105669390693597125904E-17L,
344 7.545499865295034556206475956620160007849E-15L,
345 2.267838684785673931024792538193202559922E-12L,
346 3.561909705814420373609574999542459912419E-10L,
347 3.216201422768092505214730633842924944671E-8L,
348 1.731194793857907454569364622452058554314E-6L,
349 5.576944613034537050396518509871004586039E-5L,
350 1.051787760316848982655967052985391418146E-3L,
351 1.102852974036687441600678598019883746959E-2L,
352 5.834647019292460494254225988766702933571E-2L,
353 1.290281921604364618912425380717127576529E-1L,
354 7.598886310387075708640370806458926458301E-2L,
355 };
356 #define NQ8_16D 11
357 static const long double Q8_16D[NQ8_16D + 1] = {
358 1.368001558508338469503329967729951830843E-16L,
359 1.034454121857542147020549303317348297289E-13L,
360 3.128109209247090744354764050629381674436E-11L,
361 4.957795214328501986562102573522064468671E-9L,
362 4.537872468606711261992676606899273588899E-7L,
363 2.493639207101727713192687060517509774182E-5L,
364 8.294957278145328349785532236663051405805E-4L,
365 1.646471258966713577374948205279380115839E-2L,
366 1.878910092770966718491814497982191447073E-1L,
367 1.152641605706170353727903052525652504075E0L,
368 3.383550240669773485412333679367792932235E0L,
369 3.823875252882035706910024716609908473970E0L,
370 /* 1.000000000000000000000000000000000000000E0 */
371 };
372
373 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
374 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
375 Peak relative error 3.9e-35
376 0.125 <= 1/x <= 0.1875 */
377 #define NQ5_8N 10
378 static const long double Q5_8N[NQ5_8N + 1] = {
379 1.750399094021293722243426623211733898747E-13L,
380 6.483426211748008735242909236490115050294E-11L,
381 9.279430665656575457141747875716899958373E-9L,
382 6.696634968526907231258534757736576340266E-7L,
383 2.666560823798895649685231292142838188061E-5L,
384 6.025087697259436271271562769707550594540E-4L,
385 7.652807734168613251901945778921336353485E-3L,
386 5.226269002589406461622551452343519078905E-2L,
387 1.748390159751117658969324896330142895079E-1L,
388 2.378188719097006494782174902213083589660E-1L,
389 8.383984859679804095463699702165659216831E-2L,
390 };
391 #define NQ5_8D 10
392 static const long double Q5_8D[NQ5_8D + 1] = {
393 2.389878229704327939008104855942987615715E-12L,
394 8.926142817142546018703814194987786425099E-10L,
395 1.294065862406745901206588525833274399038E-7L,
396 9.524139899457666250828752185212769682191E-6L,
397 3.908332488377770886091936221573123353489E-4L,
398 9.250427033957236609624199884089916836748E-3L,
399 1.263420066165922645975830877751588421451E-1L,
400 9.692527053860420229711317379861733180654E-1L,
401 3.937813834630430172221329298841520707954E0L,
402 7.603126427436356534498908111445191312181E0L,
403 5.670677653334105479259958485084550934305E0L,
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 3.2e-35
410 0.1875 <= 1/x <= 0.25 */
411 #define NQ4_5N 10
412 static const long double Q4_5N[NQ4_5N + 1] = {
413 2.233870042925895644234072357400122854086E-11L,
414 5.146223225761993222808463878999151699792E-9L,
415 4.459114531468296461688753521109797474523E-7L,
416 1.891397692931537975547242165291668056276E-5L,
417 4.279519145911541776938964806470674565504E-4L,
418 5.275239415656560634702073291768904783989E-3L,
419 3.468698403240744801278238473898432608887E-2L,
420 1.138773146337708415188856882915457888274E-1L,
421 1.622717518946443013587108598334636458955E-1L,
422 7.249040006390586123760992346453034628227E-2L,
423 1.941595365256460232175236758506411486667E-3L,
424 };
425 #define NQ4_5D 9
426 static const long double Q4_5D[NQ4_5D + 1] = {
427 3.049977232266999249626430127217988047453E-10L,
428 7.120883230531035857746096928889676144099E-8L,
429 6.301786064753734446784637919554359588859E-6L,
430 2.762010530095069598480766869426308077192E-4L,
431 6.572163250572867859316828886203406361251E-3L,
432 8.752566114841221958200215255461843397776E-2L,
433 6.487654992874805093499285311075289932664E-1L,
434 2.576550017826654579451615283022812801435E0L,
435 5.056392229924022835364779562707348096036E0L,
436 4.179770081068251464907531367859072157773E0L,
437 /* 1.000000000000000000000000000000000000000E0 */
438 };
439
440 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
441 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
442 Peak relative error 1.4e-36
443 0.25 <= 1/x <= 0.3125 */
444 #define NQ3r2_4N 10
445 static const long double Q3r2_4N[NQ3r2_4N + 1] = {
446 6.126167301024815034423262653066023684411E-10L,
447 1.043969327113173261820028225053598975128E-7L,
448 6.592927270288697027757438170153763220190E-6L,
449 2.009103660938497963095652951912071336730E-4L,
450 3.220543385492643525985862356352195896964E-3L,
451 2.774405975730545157543417650436941650990E-2L,
452 1.258114008023826384487378016636555041129E-1L,
453 2.811724258266902502344701449984698323860E-1L,
454 2.691837665193548059322831687432415014067E-1L,
455 7.949087384900985370683770525312735605034E-2L,
456 1.229509543620976530030153018986910810747E-3L,
457 };
458 #define NQ3r2_4D 9
459 static const long double Q3r2_4D[NQ3r2_4D + 1] = {
460 8.364260446128475461539941389210166156568E-9L,
461 1.451301850638956578622154585560759862764E-6L,
462 9.431830010924603664244578867057141839463E-5L,
463 3.004105101667433434196388593004526182741E-3L,
464 5.148157397848271739710011717102773780221E-2L,
465 4.901089301726939576055285374953887874895E-1L,
466 2.581760991981709901216967665934142240346E0L,
467 7.257105880775059281391729708630912791847E0L,
468 1.006014717326362868007913423810737369312E1L,
469 5.879416600465399514404064187445293212470E0L,
470 /* 1.000000000000000000000000000000000000000E0*/
471 };
472
473 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
474 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
475 Peak relative error 3.8e-36
476 0.3125 <= 1/x <= 0.375 */
477 #define NQ2r7_3r2N 9
478 static const long double Q2r7_3r2N[NQ2r7_3r2N + 1] = {
479 7.584861620402450302063691901886141875454E-8L,
480 9.300939338814216296064659459966041794591E-6L,
481 4.112108906197521696032158235392604947895E-4L,
482 8.515168851578898791897038357239630654431E-3L,
483 8.971286321017307400142720556749573229058E-2L,
484 4.885856732902956303343015636331874194498E-1L,
485 1.334506268733103291656253500506406045846E0L,
486 1.681207956863028164179042145803851824654E0L,
487 8.165042692571721959157677701625853772271E-1L,
488 9.805848115375053300608712721986235900715E-2L,
489 };
490 #define NQ2r7_3r2D 9
491 static const long double Q2r7_3r2D[NQ2r7_3r2D + 1] = {
492 1.035586492113036586458163971239438078160E-6L,
493 1.301999337731768381683593636500979713689E-4L,
494 5.993695702564527062553071126719088859654E-3L,
495 1.321184892887881883489141186815457808785E-1L,
496 1.528766555485015021144963194165165083312E0L,
497 9.561463309176490874525827051566494939295E0L,
498 3.203719484883967351729513662089163356911E1L,
499 5.497294687660930446641539152123568668447E1L,
500 4.391158169390578768508675452986948391118E1L,
501 1.347836630730048077907818943625789418378E1L,
502 /* 1.000000000000000000000000000000000000000E0 */
503 };
504
505 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
506 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
507 Peak relative error 2.2e-35
508 0.375 <= 1/x <= 0.4375 */
509 #define NQ2r3_2r7N 9
510 static const long double Q2r3_2r7N[NQ2r3_2r7N + 1] = {
511 4.455027774980750211349941766420190722088E-7L,
512 4.031998274578520170631601850866780366466E-5L,
513 1.273987274325947007856695677491340636339E-3L,
514 1.818754543377448509897226554179659122873E-2L,
515 1.266748858326568264126353051352269875352E-1L,
516 4.327578594728723821137731555139472880414E-1L,
517 6.892532471436503074928194969154192615359E-1L,
518 4.490775818438716873422163588640262036506E-1L,
519 8.649615949297322440032000346117031581572E-2L,
520 7.261345286655345047417257611469066147561E-4L,
521 };
522 #define NQ2r3_2r7D 8
523 static const long double Q2r3_2r7D[NQ2r3_2r7D + 1] = {
524 6.082600739680555266312417978064954793142E-6L,
525 5.693622538165494742945717226571441747567E-4L,
526 1.901625907009092204458328768129666975975E-2L,
527 2.958689532697857335456896889409923371570E-1L,
528 2.343124711045660081603809437993368799568E0L,
529 9.665894032187458293568704885528192804376E0L,
530 2.035273104990617136065743426322454881353E1L,
531 2.044102010478792896815088858740075165531E1L,
532 8.445937177863155827844146643468706599304E0L,
533 /* 1.000000000000000000000000000000000000000E0 */
534 };
535
536 /* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
537 Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
538 Peak relative error 3.1e-36
539 0.4375 <= 1/x <= 0.5 */
540 #define NQ2_2r3N 9
541 static const long double Q2_2r3N[NQ2_2r3N + 1] = {
542 2.817566786579768804844367382809101929314E-6L,
543 2.122772176396691634147024348373539744935E-4L,
544 5.501378031780457828919593905395747517585E-3L,
545 6.355374424341762686099147452020466524659E-2L,
546 3.539652320122661637429658698954748337223E-1L,
547 9.571721066119617436343740541777014319695E-1L,
548 1.196258777828426399432550698612171955305E0L,
549 6.069388659458926158392384709893753793967E-1L,
550 9.026746127269713176512359976978248763621E-2L,
551 5.317668723070450235320878117210807236375E-4L,
552 };
553 #define NQ2_2r3D 8
554 static const long double Q2_2r3D[NQ2_2r3D + 1] = {
555 3.846924354014260866793741072933159380158E-5L,
556 3.017562820057704325510067178327449946763E-3L,
557 8.356305620686867949798885808540444210935E-2L,
558 1.068314930499906838814019619594424586273E0L,
559 6.900279623894821067017966573640732685233E0L,
560 2.307667390886377924509090271780839563141E1L,
561 3.921043465412723970791036825401273528513E1L,
562 3.167569478939719383241775717095729233436E1L,
563 1.051023841699200920276198346301543665909E1L,
564 /* 1.000000000000000000000000000000000000000E0*/
565 };
566
567
568 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
569
570 static long double
571 neval (long double x, const long double *p, int n)
572 {
573 long double y;
574
575 p += n;
576 y = *p--;
577 do
578 {
579 y = y * x + *p--;
580 }
581 while (--n > 0);
582 return y;
583 }
584
585
586 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
587
588 static long double
589 deval (long double x, const long double *p, int n)
590 {
591 long double y;
592
593 p += n;
594 y = x + *p--;
595 do
596 {
597 y = y * x + *p--;
598 }
599 while (--n > 0);
600 return y;
601 }
602
603
604 /* Bessel function of the first kind, order zero. */
605
606 long double
607 __ieee754_j0l (long double x)
608 {
609 long double xx, xinv, z, p, q, c, s, cc, ss;
610
611 if (! isfinite (x))
612 {
613 if (x != x)
614 return x + x;
615 else
616 return 0;
617 }
618 if (x == 0)
619 return 1;
620
621 xx = fabsl (x);
622 if (xx <= 2)
623 {
624 if (xx < 0x1p-57L)
625 return 1;
626 /* 0 <= x <= 2 */
627 z = xx * xx;
628 p = z * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D);
629 p -= 0.25L * z;
630 p += 1;
631 return p;
632 }
633
634 /* X = x - pi/4
635 cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4)
636 = 1/sqrt(2) * (cos(x) + sin(x))
637 sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4)
638 = 1/sqrt(2) * (sin(x) - cos(x))
639 sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
640 cf. Fdlibm. */
641 __sincosl (xx, &s, &c);
642 ss = s - c;
643 cc = s + c;
644 if (xx <= LDBL_MAX / 2)
645 {
646 z = -__cosl (xx + xx);
647 if ((s * c) < 0)
648 cc = z / ss;
649 else
650 ss = z / cc;
651 }
652
653 if (xx > 0x1p256L)
654 return ONEOSQPI * cc / sqrtl (xx);
655
656 xinv = 1 / xx;
657 z = xinv * xinv;
658 if (xinv <= 0.25)
659 {
660 if (xinv <= 0.125)
661 {
662 if (xinv <= 0.0625)
663 {
664 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
665 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
666 }
667 else
668 {
669 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
670 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
671 }
672 }
673 else if (xinv <= 0.1875)
674 {
675 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
676 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
677 }
678 else
679 {
680 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
681 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
682 }
683 } /* .25 */
684 else /* if (xinv <= 0.5) */
685 {
686 if (xinv <= 0.375)
687 {
688 if (xinv <= 0.3125)
689 {
690 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
691 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
692 }
693 else
694 {
695 p = neval (z, P2r7_3r2N, NP2r7_3r2N)
696 / deval (z, P2r7_3r2D, NP2r7_3r2D);
697 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
698 / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
699 }
700 }
701 else if (xinv <= 0.4375)
702 {
703 p = neval (z, P2r3_2r7N, NP2r3_2r7N)
704 / deval (z, P2r3_2r7D, NP2r3_2r7D);
705 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
706 / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
707 }
708 else
709 {
710 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
711 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
712 }
713 }
714 p = 1 + z * p;
715 q = z * xinv * q;
716 q = q - 0.125L * xinv;
717 z = ONEOSQPI * (p * cc - q * ss) / sqrtl (xx);
718 return z;
719 }
720 libm_alias_finite (__ieee754_j0l, __j0l)
721
722
723 /* Y0(x) = 2/pi * log(x) * J0(x) + R(x^2)
724 Peak absolute error 1.7e-36 (relative where Y0 > 1)
725 0 <= x <= 2 */
726 #define NY0_2N 7
727 static const long double Y0_2N[NY0_2N + 1] = {
728 -1.062023609591350692692296993537002558155E19L,
729 2.542000883190248639104127452714966858866E19L,
730 -1.984190771278515324281415820316054696545E18L,
731 4.982586044371592942465373274440222033891E16L,
732 -5.529326354780295177243773419090123407550E14L,
733 3.013431465522152289279088265336861140391E12L,
734 -7.959436160727126750732203098982718347785E9L,
735 8.230845651379566339707130644134372793322E6L,
736 };
737 #define NY0_2D 7
738 static const long double Y0_2D[NY0_2D + 1] = {
739 1.438972634353286978700329883122253752192E20L,
740 1.856409101981569254247700169486907405500E18L,
741 1.219693352678218589553725579802986255614E16L,
742 5.389428943282838648918475915779958097958E13L,
743 1.774125762108874864433872173544743051653E11L,
744 4.522104832545149534808218252434693007036E8L,
745 8.872187401232943927082914504125234454930E5L,
746 1.251945613186787532055610876304669413955E3L,
747 /* 1.000000000000000000000000000000000000000E0 */
748 };
749
750 static const long double U0 = -7.3804295108687225274343927948483016310862e-02L;
751
752 /* Bessel function of the second kind, order zero. */
753
754 long double
755 __ieee754_y0l(long double x)
756 {
757 long double xx, xinv, z, p, q, c, s, cc, ss;
758
759 if (! isfinite (x))
760 return 1 / (x + x * x);
761 if (x <= 0)
762 {
763 if (x < 0)
764 return (zero / (zero * x));
765 return -1 / zero; /* -inf and divide by zero exception. */
766 }
767 xx = fabsl (x);
768 if (xx <= 0x1p-57)
769 return U0 + TWOOPI * __ieee754_logl (x);
770 if (xx <= 2)
771 {
772 /* 0 <= x <= 2 */
773 z = xx * xx;
774 p = neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D);
775 p = TWOOPI * __ieee754_logl (x) * __ieee754_j0l (x) + p;
776 return p;
777 }
778
779 /* X = x - pi/4
780 cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4)
781 = 1/sqrt(2) * (cos(x) + sin(x))
782 sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4)
783 = 1/sqrt(2) * (sin(x) - cos(x))
784 sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
785 cf. Fdlibm. */
786 __sincosl (x, &s, &c);
787 ss = s - c;
788 cc = s + c;
789 if (xx <= LDBL_MAX / 2)
790 {
791 z = -__cosl (x + x);
792 if ((s * c) < 0)
793 cc = z / ss;
794 else
795 ss = z / cc;
796 }
797
798 if (xx > 0x1p256L)
799 return ONEOSQPI * ss / sqrtl (x);
800
801 xinv = 1 / xx;
802 z = xinv * xinv;
803 if (xinv <= 0.25)
804 {
805 if (xinv <= 0.125)
806 {
807 if (xinv <= 0.0625)
808 {
809 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
810 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
811 }
812 else
813 {
814 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
815 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
816 }
817 }
818 else if (xinv <= 0.1875)
819 {
820 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
821 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
822 }
823 else
824 {
825 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
826 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
827 }
828 } /* .25 */
829 else /* if (xinv <= 0.5) */
830 {
831 if (xinv <= 0.375)
832 {
833 if (xinv <= 0.3125)
834 {
835 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
836 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
837 }
838 else
839 {
840 p = neval (z, P2r7_3r2N, NP2r7_3r2N)
841 / deval (z, P2r7_3r2D, NP2r7_3r2D);
842 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
843 / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
844 }
845 }
846 else if (xinv <= 0.4375)
847 {
848 p = neval (z, P2r3_2r7N, NP2r3_2r7N)
849 / deval (z, P2r3_2r7D, NP2r3_2r7D);
850 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
851 / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
852 }
853 else
854 {
855 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
856 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
857 }
858 }
859 p = 1 + z * p;
860 q = z * xinv * q;
861 q = q - 0.125L * xinv;
862 z = ONEOSQPI * (p * ss + q * cc) / sqrtl (x);
863 return z;
864 }
865 libm_alias_finite (__ieee754_y0l, __y0l)