1 /* Bessel function of order one. IBM Extended Precision version.
2 Copyright 2001 by Stephen L. Moshier (moshier@na-net.onrl.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 <errno.h>
22 #include <math.h>
23 #include <math_private.h>
24 #include <fenv_private.h>
25 #include <math-underflow.h>
26 #include <float.h>
27 #include <libm-alias-finite.h>
28
29 /* 1 / sqrt(pi) */
30 static const long double ONEOSQPI = 5.6418958354775628694807945156077258584405E-1L;
31 /* 2 / pi */
32 static const long double TWOOPI = 6.3661977236758134307553505349005744813784E-1L;
33 static const long double zero = 0;
34
35 /* J1(x) = .5x + x x^2 R(x^2)
36 Peak relative error 1.9e-35
37 0 <= x <= 2 */
38 #define NJ0_2N 6
39 static const long double J0_2N[NJ0_2N + 1] = {
40 -5.943799577386942855938508697619735179660E16L,
41 1.812087021305009192259946997014044074711E15L,
42 -2.761698314264509665075127515729146460895E13L,
43 2.091089497823600978949389109350658815972E11L,
44 -8.546413231387036372945453565654130054307E8L,
45 1.797229225249742247475464052741320612261E6L,
46 -1.559552840946694171346552770008812083969E3L
47 };
48 #define NJ0_2D 6
49 static const long double J0_2D[NJ0_2D + 1] = {
50 9.510079323819108569501613916191477479397E17L,
51 1.063193817503280529676423936545854693915E16L,
52 5.934143516050192600795972192791775226920E13L,
53 2.168000911950620999091479265214368352883E11L,
54 5.673775894803172808323058205986256928794E8L,
55 1.080329960080981204840966206372671147224E6L,
56 1.411951256636576283942477881535283304912E3L,
57 /* 1.000000000000000000000000000000000000000E0L */
58 };
59
60 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
61 0 <= 1/x <= .0625
62 Peak relative error 3.6e-36 */
63 #define NP16_IN 9
64 static const long double P16_IN[NP16_IN + 1] = {
65 5.143674369359646114999545149085139822905E-16L,
66 4.836645664124562546056389268546233577376E-13L,
67 1.730945562285804805325011561498453013673E-10L,
68 3.047976856147077889834905908605310585810E-8L,
69 2.855227609107969710407464739188141162386E-6L,
70 1.439362407936705484122143713643023998457E-4L,
71 3.774489768532936551500999699815873422073E-3L,
72 4.723962172984642566142399678920790598426E-2L,
73 2.359289678988743939925017240478818248735E-1L,
74 3.032580002220628812728954785118117124520E-1L,
75 };
76 #define NP16_ID 9
77 static const long double P16_ID[NP16_ID + 1] = {
78 4.389268795186898018132945193912677177553E-15L,
79 4.132671824807454334388868363256830961655E-12L,
80 1.482133328179508835835963635130894413136E-9L,
81 2.618941412861122118906353737117067376236E-7L,
82 2.467854246740858470815714426201888034270E-5L,
83 1.257192927368839847825938545925340230490E-3L,
84 3.362739031941574274949719324644120720341E-2L,
85 4.384458231338934105875343439265370178858E-1L,
86 2.412830809841095249170909628197264854651E0L,
87 4.176078204111348059102962617368214856874E0L,
88 /* 1.000000000000000000000000000000000000000E0 */
89 };
90
91 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
92 0.0625 <= 1/x <= 0.125
93 Peak relative error 1.9e-36 */
94 #define NP8_16N 11
95 static const long double P8_16N[NP8_16N + 1] = {
96 2.984612480763362345647303274082071598135E-16L,
97 1.923651877544126103941232173085475682334E-13L,
98 4.881258879388869396043760693256024307743E-11L,
99 6.368866572475045408480898921866869811889E-9L,
100 4.684818344104910450523906967821090796737E-7L,
101 2.005177298271593587095982211091300382796E-5L,
102 4.979808067163957634120681477207147536182E-4L,
103 6.946005761642579085284689047091173581127E-3L,
104 5.074601112955765012750207555985299026204E-2L,
105 1.698599455896180893191766195194231825379E-1L,
106 1.957536905259237627737222775573623779638E-1L,
107 2.991314703282528370270179989044994319374E-2L,
108 };
109 #define NP8_16D 10
110 static const long double P8_16D[NP8_16D + 1] = {
111 2.546869316918069202079580939942463010937E-15L,
112 1.644650111942455804019788382157745229955E-12L,
113 4.185430770291694079925607420808011147173E-10L,
114 5.485331966975218025368698195861074143153E-8L,
115 4.062884421686912042335466327098932678905E-6L,
116 1.758139661060905948870523641319556816772E-4L,
117 4.445143889306356207566032244985607493096E-3L,
118 6.391901016293512632765621532571159071158E-2L,
119 4.933040207519900471177016015718145795434E-1L,
120 1.839144086168947712971630337250761842976E0L,
121 2.715120873995490920415616716916149586579E0L,
122 /* 1.000000000000000000000000000000000000000E0 */
123 };
124
125 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
126 0.125 <= 1/x <= 0.1875
127 Peak relative error 1.3e-36 */
128 #define NP5_8N 10
129 static const long double P5_8N[NP5_8N + 1] = {
130 2.837678373978003452653763806968237227234E-12L,
131 9.726641165590364928442128579282742354806E-10L,
132 1.284408003604131382028112171490633956539E-7L,
133 8.524624695868291291250573339272194285008E-6L,
134 3.111516908953172249853673787748841282846E-4L,
135 6.423175156126364104172801983096596409176E-3L,
136 7.430220589989104581004416356260692450652E-2L,
137 4.608315409833682489016656279567605536619E-1L,
138 1.396870223510964882676225042258855977512E0L,
139 1.718500293904122365894630460672081526236E0L,
140 5.465927698800862172307352821870223855365E-1L
141 };
142 #define NP5_8D 10
143 static const long double P5_8D[NP5_8D + 1] = {
144 2.421485545794616609951168511612060482715E-11L,
145 8.329862750896452929030058039752327232310E-9L,
146 1.106137992233383429630592081375289010720E-6L,
147 7.405786153760681090127497796448503306939E-5L,
148 2.740364785433195322492093333127633465227E-3L,
149 5.781246470403095224872243564165254652198E-2L,
150 6.927711353039742469918754111511109983546E-1L,
151 4.558679283460430281188304515922826156690E0L,
152 1.534468499844879487013168065728837900009E1L,
153 2.313927430889218597919624843161569422745E1L,
154 1.194506341319498844336768473218382828637E1L,
155 /* 1.000000000000000000000000000000000000000E0 */
156 };
157
158 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
159 Peak relative error 1.4e-36
160 0.1875 <= 1/x <= 0.25 */
161 #define NP4_5N 10
162 static const long double P4_5N[NP4_5N + 1] = {
163 1.846029078268368685834261260420933914621E-10L,
164 3.916295939611376119377869680335444207768E-8L,
165 3.122158792018920627984597530935323997312E-6L,
166 1.218073444893078303994045653603392272450E-4L,
167 2.536420827983485448140477159977981844883E-3L,
168 2.883011322006690823959367922241169171315E-2L,
169 1.755255190734902907438042414495469810830E-1L,
170 5.379317079922628599870898285488723736599E-1L,
171 7.284904050194300773890303361501726561938E-1L,
172 3.270110346613085348094396323925000362813E-1L,
173 1.804473805689725610052078464951722064757E-2L,
174 };
175 #define NP4_5D 9
176 static const long double P4_5D[NP4_5D + 1] = {
177 1.575278146806816970152174364308980863569E-9L,
178 3.361289173657099516191331123405675054321E-7L,
179 2.704692281550877810424745289838790693708E-5L,
180 1.070854930483999749316546199273521063543E-3L,
181 2.282373093495295842598097265627962125411E-2L,
182 2.692025460665354148328762368240343249830E-1L,
183 1.739892942593664447220951225734811133759E0L,
184 5.890727576752230385342377570386657229324E0L,
185 9.517442287057841500750256954117735128153E0L,
186 6.100616353935338240775363403030137736013E0L,
187 /* 1.000000000000000000000000000000000000000E0 */
188 };
189
190 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
191 Peak relative error 3.0e-36
192 0.25 <= 1/x <= 0.3125 */
193 #define NP3r2_4N 9
194 static const long double P3r2_4N[NP3r2_4N + 1] = {
195 8.240803130988044478595580300846665863782E-8L,
196 1.179418958381961224222969866406483744580E-5L,
197 6.179787320956386624336959112503824397755E-4L,
198 1.540270833608687596420595830747166658383E-2L,
199 1.983904219491512618376375619598837355076E-1L,
200 1.341465722692038870390470651608301155565E0L,
201 4.617865326696612898792238245990854646057E0L,
202 7.435574801812346424460233180412308000587E0L,
203 4.671327027414635292514599201278557680420E0L,
204 7.299530852495776936690976966995187714739E-1L,
205 };
206 #define NP3r2_4D 9
207 static const long double P3r2_4D[NP3r2_4D + 1] = {
208 7.032152009675729604487575753279187576521E-7L,
209 1.015090352324577615777511269928856742848E-4L,
210 5.394262184808448484302067955186308730620E-3L,
211 1.375291438480256110455809354836988584325E-1L,
212 1.836247144461106304788160919310404376670E0L,
213 1.314378564254376655001094503090935880349E1L,
214 4.957184590465712006934452500894672343488E1L,
215 9.287394244300647738855415178790263465398E1L,
216 7.652563275535900609085229286020552768399E1L,
217 2.147042473003074533150718117770093209096E1L,
218 /* 1.000000000000000000000000000000000000000E0 */
219 };
220
221 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
222 Peak relative error 1.0e-35
223 0.3125 <= 1/x <= 0.375 */
224 #define NP2r7_3r2N 9
225 static const long double P2r7_3r2N[NP2r7_3r2N + 1] = {
226 4.599033469240421554219816935160627085991E-7L,
227 4.665724440345003914596647144630893997284E-5L,
228 1.684348845667764271596142716944374892756E-3L,
229 2.802446446884455707845985913454440176223E-2L,
230 2.321937586453963310008279956042545173930E-1L,
231 9.640277413988055668692438709376437553804E-1L,
232 1.911021064710270904508663334033003246028E0L,
233 1.600811610164341450262992138893970224971E0L,
234 4.266299218652587901171386591543457861138E-1L,
235 1.316470424456061252962568223251247207325E-2L,
236 };
237 #define NP2r7_3r2D 8
238 static const long double P2r7_3r2D[NP2r7_3r2D + 1] = {
239 3.924508608545520758883457108453520099610E-6L,
240 4.029707889408829273226495756222078039823E-4L,
241 1.484629715787703260797886463307469600219E-2L,
242 2.553136379967180865331706538897231588685E-1L,
243 2.229457223891676394409880026887106228740E0L,
244 1.005708903856384091956550845198392117318E1L,
245 2.277082659664386953166629360352385889558E1L,
246 2.384726835193630788249826630376533988245E1L,
247 9.700989749041320895890113781610939632410E0L,
248 /* 1.000000000000000000000000000000000000000E0 */
249 };
250
251 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
252 Peak relative error 1.7e-36
253 0.3125 <= 1/x <= 0.4375 */
254 #define NP2r3_2r7N 9
255 static const long double P2r3_2r7N[NP2r3_2r7N + 1] = {
256 3.916766777108274628543759603786857387402E-6L,
257 3.212176636756546217390661984304645137013E-4L,
258 9.255768488524816445220126081207248947118E-3L,
259 1.214853146369078277453080641911700735354E-1L,
260 7.855163309847214136198449861311404633665E-1L,
261 2.520058073282978403655488662066019816540E0L,
262 3.825136484837545257209234285382183711466E0L,
263 2.432569427554248006229715163865569506873E0L,
264 4.877934835018231178495030117729800489743E-1L,
265 1.109902737860249670981355149101343427885E-2L,
266 };
267 #define NP2r3_2r7D 8
268 static const long double P2r3_2r7D[NP2r3_2r7D + 1] = {
269 3.342307880794065640312646341190547184461E-5L,
270 2.782182891138893201544978009012096558265E-3L,
271 8.221304931614200702142049236141249929207E-2L,
272 1.123728246291165812392918571987858010949E0L,
273 7.740482453652715577233858317133423434590E0L,
274 2.737624677567945952953322566311201919139E1L,
275 4.837181477096062403118304137851260715475E1L,
276 3.941098643468580791437772701093795299274E1L,
277 1.245821247166544627558323920382547533630E1L,
278 /* 1.000000000000000000000000000000000000000E0 */
279 };
280
281 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
282 Peak relative error 1.7e-35
283 0.4375 <= 1/x <= 0.5 */
284 #define NP2_2r3N 8
285 static const long double P2_2r3N[NP2_2r3N + 1] = {
286 3.397930802851248553545191160608731940751E-4L,
287 2.104020902735482418784312825637833698217E-2L,
288 4.442291771608095963935342749477836181939E-1L,
289 4.131797328716583282869183304291833754967E0L,
290 1.819920169779026500146134832455189917589E1L,
291 3.781779616522937565300309684282401791291E1L,
292 3.459605449728864218972931220783543410347E1L,
293 1.173594248397603882049066603238568316561E1L,
294 9.455702270242780642835086549285560316461E-1L,
295 };
296 #define NP2_2r3D 8
297 static const long double P2_2r3D[NP2_2r3D + 1] = {
298 2.899568897241432883079888249845707400614E-3L,
299 1.831107138190848460767699919531132426356E-1L,
300 3.999350044057883839080258832758908825165E0L,
301 3.929041535867957938340569419874195303712E1L,
302 1.884245613422523323068802689915538908291E2L,
303 4.461469948819229734353852978424629815929E2L,
304 5.004998753999796821224085972610636347903E2L,
305 2.386342520092608513170837883757163414100E2L,
306 3.791322528149347975999851588922424189957E1L,
307 /* 1.000000000000000000000000000000000000000E0 */
308 };
309
310 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
311 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
312 Peak relative error 8.0e-36
313 0 <= 1/x <= .0625 */
314 #define NQ16_IN 10
315 static const long double Q16_IN[NQ16_IN + 1] = {
316 -3.917420835712508001321875734030357393421E-18L,
317 -4.440311387483014485304387406538069930457E-15L,
318 -1.951635424076926487780929645954007139616E-12L,
319 -4.318256438421012555040546775651612810513E-10L,
320 -5.231244131926180765270446557146989238020E-8L,
321 -3.540072702902043752460711989234732357653E-6L,
322 -1.311017536555269966928228052917534882984E-4L,
323 -2.495184669674631806622008769674827575088E-3L,
324 -2.141868222987209028118086708697998506716E-2L,
325 -6.184031415202148901863605871197272650090E-2L,
326 -1.922298704033332356899546792898156493887E-2L,
327 };
328 #define NQ16_ID 9
329 static const long double Q16_ID[NQ16_ID + 1] = {
330 3.820418034066293517479619763498400162314E-17L,
331 4.340702810799239909648911373329149354911E-14L,
332 1.914985356383416140706179933075303538524E-11L,
333 4.262333682610888819476498617261895474330E-9L,
334 5.213481314722233980346462747902942182792E-7L,
335 3.585741697694069399299005316809954590558E-5L,
336 1.366513429642842006385029778105539457546E-3L,
337 2.745282599850704662726337474371355160594E-2L,
338 2.637644521611867647651200098449903330074E-1L,
339 1.006953426110765984590782655598680488746E0L,
340 /* 1.000000000000000000000000000000000000000E0 */
341 };
342
343 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
344 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
345 Peak relative error 1.9e-36
346 0.0625 <= 1/x <= 0.125 */
347 #define NQ8_16N 11
348 static const long double Q8_16N[NQ8_16N + 1] = {
349 -2.028630366670228670781362543615221542291E-17L,
350 -1.519634620380959966438130374006858864624E-14L,
351 -4.540596528116104986388796594639405114524E-12L,
352 -7.085151756671466559280490913558388648274E-10L,
353 -6.351062671323970823761883833531546885452E-8L,
354 -3.390817171111032905297982523519503522491E-6L,
355 -1.082340897018886970282138836861233213972E-4L,
356 -2.020120801187226444822977006648252379508E-3L,
357 -2.093169910981725694937457070649605557555E-2L,
358 -1.092176538874275712359269481414448063393E-1L,
359 -2.374790947854765809203590474789108718733E-1L,
360 -1.365364204556573800719985118029601401323E-1L,
361 };
362 #define NQ8_16D 11
363 static const long double Q8_16D[NQ8_16D + 1] = {
364 1.978397614733632533581207058069628242280E-16L,
365 1.487361156806202736877009608336766720560E-13L,
366 4.468041406888412086042576067133365913456E-11L,
367 7.027822074821007443672290507210594648877E-9L,
368 6.375740580686101224127290062867976007374E-7L,
369 3.466887658320002225888644977076410421940E-5L,
370 1.138625640905289601186353909213719596986E-3L,
371 2.224470799470414663443449818235008486439E-2L,
372 2.487052928527244907490589787691478482358E-1L,
373 1.483927406564349124649083853892380899217E0L,
374 4.182773513276056975777258788903489507705E0L,
375 4.419665392573449746043880892524360870944E0L,
376 /* 1.000000000000000000000000000000000000000E0 */
377 };
378
379 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
380 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
381 Peak relative error 1.5e-35
382 0.125 <= 1/x <= 0.1875 */
383 #define NQ5_8N 10
384 static const long double Q5_8N[NQ5_8N + 1] = {
385 -3.656082407740970534915918390488336879763E-13L,
386 -1.344660308497244804752334556734121771023E-10L,
387 -1.909765035234071738548629788698150760791E-8L,
388 -1.366668038160120210269389551283666716453E-6L,
389 -5.392327355984269366895210704976314135683E-5L,
390 -1.206268245713024564674432357634540343884E-3L,
391 -1.515456784370354374066417703736088291287E-2L,
392 -1.022454301137286306933217746545237098518E-1L,
393 -3.373438906472495080504907858424251082240E-1L,
394 -4.510782522110845697262323973549178453405E-1L,
395 -1.549000892545288676809660828213589804884E-1L,
396 };
397 #define NQ5_8D 10
398 static const long double Q5_8D[NQ5_8D + 1] = {
399 3.565550843359501079050699598913828460036E-12L,
400 1.321016015556560621591847454285330528045E-9L,
401 1.897542728662346479999969679234270605975E-7L,
402 1.381720283068706710298734234287456219474E-5L,
403 5.599248147286524662305325795203422873725E-4L,
404 1.305442352653121436697064782499122164843E-2L,
405 1.750234079626943298160445750078631894985E-1L,
406 1.311420542073436520965439883806946678491E0L,
407 5.162757689856842406744504211089724926650E0L,
408 9.527760296384704425618556332087850581308E0L,
409 6.604648207463236667912921642545100248584E0L,
410 /* 1.000000000000000000000000000000000000000E0 */
411 };
412
413 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
414 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
415 Peak relative error 1.3e-35
416 0.1875 <= 1/x <= 0.25 */
417 #define NQ4_5N 10
418 static const long double Q4_5N[NQ4_5N + 1] = {
419 -4.079513568708891749424783046520200903755E-11L,
420 -9.326548104106791766891812583019664893311E-9L,
421 -8.016795121318423066292906123815687003356E-7L,
422 -3.372350544043594415609295225664186750995E-5L,
423 -7.566238665947967882207277686375417983917E-4L,
424 -9.248861580055565402130441618521591282617E-3L,
425 -6.033106131055851432267702948850231270338E-2L,
426 -1.966908754799996793730369265431584303447E-1L,
427 -2.791062741179964150755788226623462207560E-1L,
428 -1.255478605849190549914610121863534191666E-1L,
429 -4.320429862021265463213168186061696944062E-3L,
430 };
431 #define NQ4_5D 9
432 static const long double Q4_5D[NQ4_5D + 1] = {
433 3.978497042580921479003851216297330701056E-10L,
434 9.203304163828145809278568906420772246666E-8L,
435 8.059685467088175644915010485174545743798E-6L,
436 3.490187375993956409171098277561669167446E-4L,
437 8.189109654456872150100501732073810028829E-3L,
438 1.072572867311023640958725265762483033769E-1L,
439 7.790606862409960053675717185714576937994E-1L,
440 3.016049768232011196434185423512777656328E0L,
441 5.722963851442769787733717162314477949360E0L,
442 4.510527838428473279647251350931380867663E0L,
443 /* 1.000000000000000000000000000000000000000E0 */
444 };
445
446 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
447 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
448 Peak relative error 2.1e-35
449 0.25 <= 1/x <= 0.3125 */
450 #define NQ3r2_4N 9
451 static const long double Q3r2_4N[NQ3r2_4N + 1] = {
452 -1.087480809271383885936921889040388133627E-8L,
453 -1.690067828697463740906962973479310170932E-6L,
454 -9.608064416995105532790745641974762550982E-5L,
455 -2.594198839156517191858208513873961837410E-3L,
456 -3.610954144421543968160459863048062977822E-2L,
457 -2.629866798251843212210482269563961685666E-1L,
458 -9.709186825881775885917984975685752956660E-1L,
459 -1.667521829918185121727268867619982417317E0L,
460 -1.109255082925540057138766105229900943501E0L,
461 -1.812932453006641348145049323713469043328E-1L,
462 };
463 #define NQ3r2_4D 9
464 static const long double Q3r2_4D[NQ3r2_4D + 1] = {
465 1.060552717496912381388763753841473407026E-7L,
466 1.676928002024920520786883649102388708024E-5L,
467 9.803481712245420839301400601140812255737E-4L,
468 2.765559874262309494758505158089249012930E-2L,
469 4.117921827792571791298862613287549140706E-1L,
470 3.323769515244751267093378361930279161413E0L,
471 1.436602494405814164724810151689705353670E1L,
472 3.163087869617098638064881410646782408297E1L,
473 3.198181264977021649489103980298349589419E1L,
474 1.203649258862068431199471076202897823272E1L,
475 /* 1.000000000000000000000000000000000000000E0 */
476 };
477
478 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
479 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
480 Peak relative error 1.6e-36
481 0.3125 <= 1/x <= 0.375 */
482 #define NQ2r7_3r2N 9
483 static const long double Q2r7_3r2N[NQ2r7_3r2N + 1] = {
484 -1.723405393982209853244278760171643219530E-7L,
485 -2.090508758514655456365709712333460087442E-5L,
486 -9.140104013370974823232873472192719263019E-4L,
487 -1.871349499990714843332742160292474780128E-2L,
488 -1.948930738119938669637865956162512983416E-1L,
489 -1.048764684978978127908439526343174139788E0L,
490 -2.827714929925679500237476105843643064698E0L,
491 -3.508761569156476114276988181329773987314E0L,
492 -1.669332202790211090973255098624488308989E0L,
493 -1.930796319299022954013840684651016077770E-1L,
494 };
495 #define NQ2r7_3r2D 9
496 static const long double Q2r7_3r2D[NQ2r7_3r2D + 1] = {
497 1.680730662300831976234547482334347983474E-6L,
498 2.084241442440551016475972218719621841120E-4L,
499 9.445316642108367479043541702688736295579E-3L,
500 2.044637889456631896650179477133252184672E-1L,
501 2.316091982244297350829522534435350078205E0L,
502 1.412031891783015085196708811890448488865E1L,
503 4.583830154673223384837091077279595496149E1L,
504 7.549520609270909439885998474045974122261E1L,
505 5.697605832808113367197494052388203310638E1L,
506 1.601496240876192444526383314589371686234E1L,
507 /* 1.000000000000000000000000000000000000000E0 */
508 };
509
510 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
511 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
512 Peak relative error 9.5e-36
513 0.375 <= 1/x <= 0.4375 */
514 #define NQ2r3_2r7N 9
515 static const long double Q2r3_2r7N[NQ2r3_2r7N + 1] = {
516 -8.603042076329122085722385914954878953775E-7L,
517 -7.701746260451647874214968882605186675720E-5L,
518 -2.407932004380727587382493696877569654271E-3L,
519 -3.403434217607634279028110636919987224188E-2L,
520 -2.348707332185238159192422084985713102877E-1L,
521 -7.957498841538254916147095255700637463207E-1L,
522 -1.258469078442635106431098063707934348577E0L,
523 -8.162415474676345812459353639449971369890E-1L,
524 -1.581783890269379690141513949609572806898E-1L,
525 -1.890595651683552228232308756569450822905E-3L,
526 };
527 #define NQ2r3_2r7D 8
528 static const long double Q2r3_2r7D[NQ2r3_2r7D + 1] = {
529 8.390017524798316921170710533381568175665E-6L,
530 7.738148683730826286477254659973968763659E-4L,
531 2.541480810958665794368759558791634341779E-2L,
532 3.878879789711276799058486068562386244873E-1L,
533 3.003783779325811292142957336802456109333E0L,
534 1.206480374773322029883039064575464497400E1L,
535 2.458414064785315978408974662900438351782E1L,
536 2.367237826273668567199042088835448715228E1L,
537 9.231451197519171090875569102116321676763E0L,
538 /* 1.000000000000000000000000000000000000000E0 */
539 };
540
541 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
542 Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
543 Peak relative error 1.4e-36
544 0.4375 <= 1/x <= 0.5 */
545 #define NQ2_2r3N 9
546 static const long double Q2_2r3N[NQ2_2r3N + 1] = {
547 -5.552507516089087822166822364590806076174E-6L,
548 -4.135067659799500521040944087433752970297E-4L,
549 -1.059928728869218962607068840646564457980E-2L,
550 -1.212070036005832342565792241385459023801E-1L,
551 -6.688350110633603958684302153362735625156E-1L,
552 -1.793587878197360221340277951304429821582E0L,
553 -2.225407682237197485644647380483725045326E0L,
554 -1.123402135458940189438898496348239744403E0L,
555 -1.679187241566347077204805190763597299805E-1L,
556 -1.458550613639093752909985189067233504148E-3L,
557 };
558 #define NQ2_2r3D 8
559 static const long double Q2_2r3D[NQ2_2r3D + 1] = {
560 5.415024336507980465169023996403597916115E-5L,
561 4.179246497380453022046357404266022870788E-3L,
562 1.136306384261959483095442402929502368598E-1L,
563 1.422640343719842213484515445393284072830E0L,
564 8.968786703393158374728850922289204805764E0L,
565 2.914542473339246127533384118781216495934E1L,
566 4.781605421020380669870197378210457054685E1L,
567 3.693865837171883152382820584714795072937E1L,
568 1.153220502744204904763115556224395893076E1L,
569 /* 1.000000000000000000000000000000000000000E0 */
570 };
571
572
573 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
574
575 static long double
576 neval (long double x, const long double *p, int n)
577 {
578 long double y;
579
580 p += n;
581 y = *p--;
582 do
583 {
584 y = y * x + *p--;
585 }
586 while (--n > 0);
587 return y;
588 }
589
590
591 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
592
593 static long double
594 deval (long double x, const long double *p, int n)
595 {
596 long double y;
597
598 p += n;
599 y = x + *p--;
600 do
601 {
602 y = y * x + *p--;
603 }
604 while (--n > 0);
605 return y;
606 }
607
608
609 /* Bessel function of the first kind, order one. */
610
611 long double
612 __ieee754_j1l (long double x)
613 {
614 long double xx, xinv, z, p, q, c, s, cc, ss;
615
616 if (! isfinite (x))
617 {
618 if (x != x)
619 return x + x;
620 else
621 return 0;
622 }
623 if (x == 0)
624 return x;
625 xx = fabsl (x);
626 if (xx <= 0x1p-58L)
627 {
628 long double ret = x * 0.5L;
629 math_check_force_underflow (ret);
630 if (ret == 0)
631 __set_errno (ERANGE);
632 return ret;
633 }
634 if (xx <= 2)
635 {
636 /* 0 <= x <= 2 */
637 z = xx * xx;
638 p = xx * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D);
639 p += 0.5L * xx;
640 if (x < 0)
641 p = -p;
642 return p;
643 }
644
645 /* X = x - 3 pi/4
646 cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4)
647 = 1/sqrt(2) * (-cos(x) + sin(x))
648 sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4)
649 = -1/sqrt(2) * (sin(x) + cos(x))
650 cf. Fdlibm. */
651 __sincosl (xx, &s, &c);
652 ss = -s - c;
653 cc = s - c;
654 if (xx <= LDBL_MAX / 2)
655 {
656 z = __cosl (xx + xx);
657 if ((s * c) > 0)
658 cc = z / ss;
659 else
660 ss = z / cc;
661 }
662
663 if (xx > 0x1p256L)
664 {
665 z = ONEOSQPI * cc / sqrtl (xx);
666 if (x < 0)
667 z = -z;
668 return z;
669 }
670
671 xinv = 1 / xx;
672 z = xinv * xinv;
673 if (xinv <= 0.25)
674 {
675 if (xinv <= 0.125)
676 {
677 if (xinv <= 0.0625)
678 {
679 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
680 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
681 }
682 else
683 {
684 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
685 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
686 }
687 }
688 else if (xinv <= 0.1875)
689 {
690 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
691 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
692 }
693 else
694 {
695 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
696 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
697 }
698 } /* .25 */
699 else /* if (xinv <= 0.5) */
700 {
701 if (xinv <= 0.375)
702 {
703 if (xinv <= 0.3125)
704 {
705 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
706 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
707 }
708 else
709 {
710 p = neval (z, P2r7_3r2N, NP2r7_3r2N)
711 / deval (z, P2r7_3r2D, NP2r7_3r2D);
712 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
713 / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
714 }
715 }
716 else if (xinv <= 0.4375)
717 {
718 p = neval (z, P2r3_2r7N, NP2r3_2r7N)
719 / deval (z, P2r3_2r7D, NP2r3_2r7D);
720 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
721 / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
722 }
723 else
724 {
725 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
726 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
727 }
728 }
729 p = 1 + z * p;
730 q = z * q;
731 q = q * xinv + 0.375L * xinv;
732 z = ONEOSQPI * (p * cc - q * ss) / sqrtl (xx);
733 if (x < 0)
734 z = -z;
735 return z;
736 }
737 libm_alias_finite (__ieee754_j1l, __j1l)
738
739
740 /* Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2)
741 Peak relative error 6.2e-38
742 0 <= x <= 2 */
743 #define NY0_2N 7
744 static const long double Y0_2N[NY0_2N + 1] = {
745 -6.804415404830253804408698161694720833249E19L,
746 1.805450517967019908027153056150465849237E19L,
747 -8.065747497063694098810419456383006737312E17L,
748 1.401336667383028259295830955439028236299E16L,
749 -1.171654432898137585000399489686629680230E14L,
750 5.061267920943853732895341125243428129150E11L,
751 -1.096677850566094204586208610960870217970E9L,
752 9.541172044989995856117187515882879304461E5L,
753 };
754 #define NY0_2D 7
755 static const long double Y0_2D[NY0_2D + 1] = {
756 3.470629591820267059538637461549677594549E20L,
757 4.120796439009916326855848107545425217219E18L,
758 2.477653371652018249749350657387030814542E16L,
759 9.954678543353888958177169349272167762797E13L,
760 2.957927997613630118216218290262851197754E11L,
761 6.748421382188864486018861197614025972118E8L,
762 1.173453425218010888004562071020305709319E6L,
763 1.450335662961034949894009554536003377187E3L,
764 /* 1.000000000000000000000000000000000000000E0 */
765 };
766
767
768 /* Bessel function of the second kind, order one. */
769
770 long double
771 __ieee754_y1l (long double x)
772 {
773 long double xx, xinv, z, p, q, c, s, cc, ss;
774
775 if (! isfinite (x))
776 return 1 / (x + x * x);
777 if (x <= 0)
778 {
779 if (x < 0)
780 return (zero / (zero * x));
781 return -1 / zero; /* -inf and divide by zero exception. */
782 }
783 xx = fabsl (x);
784 if (xx <= 0x1p-114)
785 {
786 z = -TWOOPI / x;
787 if (isinf (z))
788 __set_errno (ERANGE);
789 return z;
790 }
791 if (xx <= 2)
792 {
793 /* 0 <= x <= 2 */
794 SET_RESTORE_ROUNDL (FE_TONEAREST);
795 xx = math_opt_barrier (xx);
796 x = math_opt_barrier (x);
797 z = xx * xx;
798 p = xx * neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D);
799 p = -TWOOPI / xx + p;
800 p = TWOOPI * __ieee754_logl (x) * __ieee754_j1l (x) + p;
801 math_force_eval (p);
802 return p;
803 }
804
805 /* X = x - 3 pi/4
806 cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4)
807 = 1/sqrt(2) * (-cos(x) + sin(x))
808 sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4)
809 = -1/sqrt(2) * (sin(x) + cos(x))
810 cf. Fdlibm. */
811 __sincosl (xx, &s, &c);
812 ss = -s - c;
813 cc = s - c;
814 if (xx <= LDBL_MAX / 2)
815 {
816 z = __cosl (xx + xx);
817 if ((s * c) > 0)
818 cc = z / ss;
819 else
820 ss = z / cc;
821 }
822
823 if (xx > 0x1p256L)
824 return ONEOSQPI * ss / sqrtl (xx);
825
826 xinv = 1 / xx;
827 z = xinv * xinv;
828 if (xinv <= 0.25)
829 {
830 if (xinv <= 0.125)
831 {
832 if (xinv <= 0.0625)
833 {
834 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
835 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
836 }
837 else
838 {
839 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
840 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
841 }
842 }
843 else if (xinv <= 0.1875)
844 {
845 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
846 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
847 }
848 else
849 {
850 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
851 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
852 }
853 } /* .25 */
854 else /* if (xinv <= 0.5) */
855 {
856 if (xinv <= 0.375)
857 {
858 if (xinv <= 0.3125)
859 {
860 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
861 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
862 }
863 else
864 {
865 p = neval (z, P2r7_3r2N, NP2r7_3r2N)
866 / deval (z, P2r7_3r2D, NP2r7_3r2D);
867 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
868 / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
869 }
870 }
871 else if (xinv <= 0.4375)
872 {
873 p = neval (z, P2r3_2r7N, NP2r3_2r7N)
874 / deval (z, P2r3_2r7D, NP2r3_2r7D);
875 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
876 / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
877 }
878 else
879 {
880 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
881 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
882 }
883 }
884 p = 1 + z * p;
885 q = z * q;
886 q = q * xinv + 0.375L * xinv;
887 z = ONEOSQPI * (p * ss + q * cc) / sqrtl (xx);
888 return z;
889 }
890 libm_alias_finite (__ieee754_y1l, __y1l)