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: 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 (x: z, p: J0_2N, NJ0_2N) / deval (x: z, p: 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 (x: xx, sinx: &s, cosx: &c); |
642 | ss = s - c; |
643 | cc = s + c; |
644 | if (xx <= LDBL_MAX / 2) |
645 | { |
646 | z = -__cosl (x: 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 (x: z, p: P16_IN, NP16_IN) / deval (x: z, p: P16_ID, NP16_ID); |
665 | q = neval (x: z, p: Q16_IN, NQ16_IN) / deval (x: z, p: Q16_ID, NQ16_ID); |
666 | } |
667 | else |
668 | { |
669 | p = neval (x: z, p: P8_16N, NP8_16N) / deval (x: z, p: P8_16D, NP8_16D); |
670 | q = neval (x: z, p: Q8_16N, NQ8_16N) / deval (x: z, p: Q8_16D, NQ8_16D); |
671 | } |
672 | } |
673 | else if (xinv <= 0.1875) |
674 | { |
675 | p = neval (x: z, p: P5_8N, NP5_8N) / deval (x: z, p: P5_8D, NP5_8D); |
676 | q = neval (x: z, p: Q5_8N, NQ5_8N) / deval (x: z, p: Q5_8D, NQ5_8D); |
677 | } |
678 | else |
679 | { |
680 | p = neval (x: z, p: P4_5N, NP4_5N) / deval (x: z, p: P4_5D, NP4_5D); |
681 | q = neval (x: z, p: Q4_5N, NQ4_5N) / deval (x: z, p: 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 (x: z, p: P3r2_4N, NP3r2_4N) / deval (x: z, p: P3r2_4D, NP3r2_4D); |
691 | q = neval (x: z, p: Q3r2_4N, NQ3r2_4N) / deval (x: z, p: Q3r2_4D, NQ3r2_4D); |
692 | } |
693 | else |
694 | { |
695 | p = neval (x: z, p: P2r7_3r2N, NP2r7_3r2N) |
696 | / deval (x: z, p: P2r7_3r2D, NP2r7_3r2D); |
697 | q = neval (x: z, p: Q2r7_3r2N, NQ2r7_3r2N) |
698 | / deval (x: z, p: Q2r7_3r2D, NQ2r7_3r2D); |
699 | } |
700 | } |
701 | else if (xinv <= 0.4375) |
702 | { |
703 | p = neval (x: z, p: P2r3_2r7N, NP2r3_2r7N) |
704 | / deval (x: z, p: P2r3_2r7D, NP2r3_2r7D); |
705 | q = neval (x: z, p: Q2r3_2r7N, NQ2r3_2r7N) |
706 | / deval (x: z, p: Q2r3_2r7D, NQ2r3_2r7D); |
707 | } |
708 | else |
709 | { |
710 | p = neval (x: z, p: P2_2r3N, NP2_2r3N) / deval (x: z, p: P2_2r3D, NP2_2r3D); |
711 | q = neval (x: z, p: Q2_2r3N, NQ2_2r3N) / deval (x: z, p: 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: 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 (x: z, p: Y0_2N, NY0_2N) / deval (x: z, p: 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: x, sinx: &s, cosx: &c); |
787 | ss = s - c; |
788 | cc = s + c; |
789 | if (xx <= LDBL_MAX / 2) |
790 | { |
791 | z = -__cosl (x: 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 (x: z, p: P16_IN, NP16_IN) / deval (x: z, p: P16_ID, NP16_ID); |
810 | q = neval (x: z, p: Q16_IN, NQ16_IN) / deval (x: z, p: Q16_ID, NQ16_ID); |
811 | } |
812 | else |
813 | { |
814 | p = neval (x: z, p: P8_16N, NP8_16N) / deval (x: z, p: P8_16D, NP8_16D); |
815 | q = neval (x: z, p: Q8_16N, NQ8_16N) / deval (x: z, p: Q8_16D, NQ8_16D); |
816 | } |
817 | } |
818 | else if (xinv <= 0.1875) |
819 | { |
820 | p = neval (x: z, p: P5_8N, NP5_8N) / deval (x: z, p: P5_8D, NP5_8D); |
821 | q = neval (x: z, p: Q5_8N, NQ5_8N) / deval (x: z, p: Q5_8D, NQ5_8D); |
822 | } |
823 | else |
824 | { |
825 | p = neval (x: z, p: P4_5N, NP4_5N) / deval (x: z, p: P4_5D, NP4_5D); |
826 | q = neval (x: z, p: Q4_5N, NQ4_5N) / deval (x: z, p: 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 (x: z, p: P3r2_4N, NP3r2_4N) / deval (x: z, p: P3r2_4D, NP3r2_4D); |
836 | q = neval (x: z, p: Q3r2_4N, NQ3r2_4N) / deval (x: z, p: Q3r2_4D, NQ3r2_4D); |
837 | } |
838 | else |
839 | { |
840 | p = neval (x: z, p: P2r7_3r2N, NP2r7_3r2N) |
841 | / deval (x: z, p: P2r7_3r2D, NP2r7_3r2D); |
842 | q = neval (x: z, p: Q2r7_3r2N, NQ2r7_3r2N) |
843 | / deval (x: z, p: Q2r7_3r2D, NQ2r7_3r2D); |
844 | } |
845 | } |
846 | else if (xinv <= 0.4375) |
847 | { |
848 | p = neval (x: z, p: P2r3_2r7N, NP2r3_2r7N) |
849 | / deval (x: z, p: P2r3_2r7D, NP2r3_2r7D); |
850 | q = neval (x: z, p: Q2r3_2r7N, NQ2r3_2r7N) |
851 | / deval (x: z, p: Q2r3_2r7D, NQ2r3_2r7D); |
852 | } |
853 | else |
854 | { |
855 | p = neval (x: z, p: P2_2r3N, NP2_2r3N) / deval (x: z, p: P2_2r3D, NP2_2r3D); |
856 | q = neval (x: z, p: Q2_2r3N, NQ2_2r3N) / deval (x: z, p: 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) |
866 | |