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87969c8c | 1 | /* j1l.c |
2 | * | |
3 | * Bessel function of order one | |
4 | * | |
5 | * | |
6 | * | |
7 | * SYNOPSIS: | |
8 | * | |
e409716d | 9 | * long double x, y, j1l(); |
87969c8c | 10 | * |
e409716d | 11 | * y = j1l( x ); |
87969c8c | 12 | * |
13 | * | |
14 | * | |
15 | * DESCRIPTION: | |
16 | * | |
17 | * Returns Bessel function of first kind, order one of the argument. | |
18 | * | |
19 | * The domain is divided into two major intervals [0, 2] and | |
20 | * (2, infinity). In the first interval the rational approximation is | |
21 | * J1(x) = .5x + x x^2 R(x^2) | |
22 | * | |
23 | * The second interval is further partitioned into eight equal segments | |
24 | * of 1/x. | |
25 | * J1(x) = sqrt(2/(pi x)) (P1(x) cos(X) - Q1(x) sin(X)), | |
26 | * X = x - 3 pi / 4, | |
27 | * | |
28 | * and the auxiliary functions are given by | |
29 | * | |
30 | * J1(x)cos(X) + Y1(x)sin(X) = sqrt( 2/(pi x)) P1(x), | |
31 | * P1(x) = 1 + 1/x^2 R(1/x^2) | |
32 | * | |
33 | * Y1(x)cos(X) - J1(x)sin(X) = sqrt( 2/(pi x)) Q1(x), | |
34 | * Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)). | |
35 | * | |
36 | * | |
37 | * | |
38 | * ACCURACY: | |
39 | * | |
40 | * Absolute error: | |
41 | * arithmetic domain # trials peak rms | |
42 | * IEEE 0, 30 100000 2.8e-34 2.7e-35 | |
43 | * | |
44 | * | |
45 | */ | |
46 | ||
47 | /* y1l.c | |
48 | * | |
49 | * Bessel function of the second kind, order one | |
50 | * | |
51 | * | |
52 | * | |
53 | * SYNOPSIS: | |
54 | * | |
e409716d | 55 | * double x, y, y1l(); |
87969c8c | 56 | * |
e409716d | 57 | * y = y1l( x ); |
87969c8c | 58 | * |
59 | * | |
60 | * | |
61 | * DESCRIPTION: | |
62 | * | |
63 | * Returns Bessel function of the second kind, of order | |
64 | * one, of the argument. | |
65 | * | |
66 | * The domain is divided into two major intervals [0, 2] and | |
67 | * (2, infinity). In the first interval the rational approximation is | |
68 | * Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2) . | |
69 | * In the second interval the approximation is the same as for J1(x), and | |
70 | * Y1(x) = sqrt(2/(pi x)) (P1(x) sin(X) + Q1(x) cos(X)), | |
71 | * X = x - 3 pi / 4. | |
72 | * | |
73 | * ACCURACY: | |
74 | * | |
75 | * Absolute error, when y0(x) < 1; else relative error: | |
76 | * | |
77 | * arithmetic domain # trials peak rms | |
78 | * IEEE 0, 30 100000 2.7e-34 2.9e-35 | |
79 | * | |
80 | */ | |
81 | ||
82 | /* Copyright 2001 by Stephen L. Moshier (moshier@na-net.onrl.gov). | |
83 | ||
84 | This library is free software; you can redistribute it and/or | |
85 | modify it under the terms of the GNU Lesser General Public | |
86 | License as published by the Free Software Foundation; either | |
87 | version 2.1 of the License, or (at your option) any later version. | |
88 | ||
89 | This library is distributed in the hope that it will be useful, | |
90 | but WITHOUT ANY WARRANTY; without even the implied warranty of | |
91 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
92 | Lesser General Public License for more details. | |
93 | ||
94 | You should have received a copy of the GNU Lesser General Public | |
e409716d | 95 | License along with this library; if not, see |
96 | <http://www.gnu.org/licenses/>. */ | |
87969c8c | 97 | |
98 | #include "quadmath-imp.h" | |
99 | ||
100 | /* 1 / sqrt(pi) */ | |
101 | static const __float128 ONEOSQPI = 5.6418958354775628694807945156077258584405E-1Q; | |
102 | /* 2 / pi */ | |
103 | static const __float128 TWOOPI = 6.3661977236758134307553505349005744813784E-1Q; | |
e409716d | 104 | static const __float128 zero = 0; |
87969c8c | 105 | |
106 | /* J1(x) = .5x + x x^2 R(x^2) | |
107 | Peak relative error 1.9e-35 | |
108 | 0 <= x <= 2 */ | |
109 | #define NJ0_2N 6 | |
110 | static const __float128 J0_2N[NJ0_2N + 1] = { | |
111 | -5.943799577386942855938508697619735179660E16Q, | |
112 | 1.812087021305009192259946997014044074711E15Q, | |
113 | -2.761698314264509665075127515729146460895E13Q, | |
114 | 2.091089497823600978949389109350658815972E11Q, | |
115 | -8.546413231387036372945453565654130054307E8Q, | |
116 | 1.797229225249742247475464052741320612261E6Q, | |
117 | -1.559552840946694171346552770008812083969E3Q | |
118 | }; | |
119 | #define NJ0_2D 6 | |
120 | static const __float128 J0_2D[NJ0_2D + 1] = { | |
121 | 9.510079323819108569501613916191477479397E17Q, | |
122 | 1.063193817503280529676423936545854693915E16Q, | |
123 | 5.934143516050192600795972192791775226920E13Q, | |
124 | 2.168000911950620999091479265214368352883E11Q, | |
125 | 5.673775894803172808323058205986256928794E8Q, | |
126 | 1.080329960080981204840966206372671147224E6Q, | |
127 | 1.411951256636576283942477881535283304912E3Q, | |
e409716d | 128 | /* 1.000000000000000000000000000000000000000E0L */ |
87969c8c | 129 | }; |
130 | ||
131 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
132 | 0 <= 1/x <= .0625 | |
133 | Peak relative error 3.6e-36 */ | |
134 | #define NP16_IN 9 | |
135 | static const __float128 P16_IN[NP16_IN + 1] = { | |
136 | 5.143674369359646114999545149085139822905E-16Q, | |
137 | 4.836645664124562546056389268546233577376E-13Q, | |
138 | 1.730945562285804805325011561498453013673E-10Q, | |
139 | 3.047976856147077889834905908605310585810E-8Q, | |
140 | 2.855227609107969710407464739188141162386E-6Q, | |
141 | 1.439362407936705484122143713643023998457E-4Q, | |
142 | 3.774489768532936551500999699815873422073E-3Q, | |
143 | 4.723962172984642566142399678920790598426E-2Q, | |
144 | 2.359289678988743939925017240478818248735E-1Q, | |
145 | 3.032580002220628812728954785118117124520E-1Q, | |
146 | }; | |
147 | #define NP16_ID 9 | |
148 | static const __float128 P16_ID[NP16_ID + 1] = { | |
149 | 4.389268795186898018132945193912677177553E-15Q, | |
150 | 4.132671824807454334388868363256830961655E-12Q, | |
151 | 1.482133328179508835835963635130894413136E-9Q, | |
152 | 2.618941412861122118906353737117067376236E-7Q, | |
153 | 2.467854246740858470815714426201888034270E-5Q, | |
154 | 1.257192927368839847825938545925340230490E-3Q, | |
155 | 3.362739031941574274949719324644120720341E-2Q, | |
156 | 4.384458231338934105875343439265370178858E-1Q, | |
157 | 2.412830809841095249170909628197264854651E0Q, | |
158 | 4.176078204111348059102962617368214856874E0Q, | |
159 | /* 1.000000000000000000000000000000000000000E0 */ | |
160 | }; | |
161 | ||
162 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
163 | 0.0625 <= 1/x <= 0.125 | |
164 | Peak relative error 1.9e-36 */ | |
165 | #define NP8_16N 11 | |
166 | static const __float128 P8_16N[NP8_16N + 1] = { | |
167 | 2.984612480763362345647303274082071598135E-16Q, | |
168 | 1.923651877544126103941232173085475682334E-13Q, | |
169 | 4.881258879388869396043760693256024307743E-11Q, | |
170 | 6.368866572475045408480898921866869811889E-9Q, | |
171 | 4.684818344104910450523906967821090796737E-7Q, | |
172 | 2.005177298271593587095982211091300382796E-5Q, | |
173 | 4.979808067163957634120681477207147536182E-4Q, | |
174 | 6.946005761642579085284689047091173581127E-3Q, | |
175 | 5.074601112955765012750207555985299026204E-2Q, | |
176 | 1.698599455896180893191766195194231825379E-1Q, | |
177 | 1.957536905259237627737222775573623779638E-1Q, | |
178 | 2.991314703282528370270179989044994319374E-2Q, | |
179 | }; | |
180 | #define NP8_16D 10 | |
181 | static const __float128 P8_16D[NP8_16D + 1] = { | |
182 | 2.546869316918069202079580939942463010937E-15Q, | |
183 | 1.644650111942455804019788382157745229955E-12Q, | |
184 | 4.185430770291694079925607420808011147173E-10Q, | |
185 | 5.485331966975218025368698195861074143153E-8Q, | |
186 | 4.062884421686912042335466327098932678905E-6Q, | |
187 | 1.758139661060905948870523641319556816772E-4Q, | |
188 | 4.445143889306356207566032244985607493096E-3Q, | |
189 | 6.391901016293512632765621532571159071158E-2Q, | |
190 | 4.933040207519900471177016015718145795434E-1Q, | |
191 | 1.839144086168947712971630337250761842976E0Q, | |
192 | 2.715120873995490920415616716916149586579E0Q, | |
193 | /* 1.000000000000000000000000000000000000000E0 */ | |
194 | }; | |
195 | ||
196 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
197 | 0.125 <= 1/x <= 0.1875 | |
198 | Peak relative error 1.3e-36 */ | |
199 | #define NP5_8N 10 | |
200 | static const __float128 P5_8N[NP5_8N + 1] = { | |
201 | 2.837678373978003452653763806968237227234E-12Q, | |
202 | 9.726641165590364928442128579282742354806E-10Q, | |
203 | 1.284408003604131382028112171490633956539E-7Q, | |
204 | 8.524624695868291291250573339272194285008E-6Q, | |
205 | 3.111516908953172249853673787748841282846E-4Q, | |
206 | 6.423175156126364104172801983096596409176E-3Q, | |
207 | 7.430220589989104581004416356260692450652E-2Q, | |
208 | 4.608315409833682489016656279567605536619E-1Q, | |
209 | 1.396870223510964882676225042258855977512E0Q, | |
210 | 1.718500293904122365894630460672081526236E0Q, | |
211 | 5.465927698800862172307352821870223855365E-1Q | |
212 | }; | |
213 | #define NP5_8D 10 | |
214 | static const __float128 P5_8D[NP5_8D + 1] = { | |
215 | 2.421485545794616609951168511612060482715E-11Q, | |
216 | 8.329862750896452929030058039752327232310E-9Q, | |
217 | 1.106137992233383429630592081375289010720E-6Q, | |
218 | 7.405786153760681090127497796448503306939E-5Q, | |
219 | 2.740364785433195322492093333127633465227E-3Q, | |
220 | 5.781246470403095224872243564165254652198E-2Q, | |
221 | 6.927711353039742469918754111511109983546E-1Q, | |
222 | 4.558679283460430281188304515922826156690E0Q, | |
223 | 1.534468499844879487013168065728837900009E1Q, | |
224 | 2.313927430889218597919624843161569422745E1Q, | |
225 | 1.194506341319498844336768473218382828637E1Q, | |
226 | /* 1.000000000000000000000000000000000000000E0 */ | |
227 | }; | |
228 | ||
229 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
230 | Peak relative error 1.4e-36 | |
231 | 0.1875 <= 1/x <= 0.25 */ | |
232 | #define NP4_5N 10 | |
233 | static const __float128 P4_5N[NP4_5N + 1] = { | |
234 | 1.846029078268368685834261260420933914621E-10Q, | |
235 | 3.916295939611376119377869680335444207768E-8Q, | |
236 | 3.122158792018920627984597530935323997312E-6Q, | |
237 | 1.218073444893078303994045653603392272450E-4Q, | |
238 | 2.536420827983485448140477159977981844883E-3Q, | |
239 | 2.883011322006690823959367922241169171315E-2Q, | |
240 | 1.755255190734902907438042414495469810830E-1Q, | |
241 | 5.379317079922628599870898285488723736599E-1Q, | |
242 | 7.284904050194300773890303361501726561938E-1Q, | |
243 | 3.270110346613085348094396323925000362813E-1Q, | |
244 | 1.804473805689725610052078464951722064757E-2Q, | |
245 | }; | |
246 | #define NP4_5D 9 | |
247 | static const __float128 P4_5D[NP4_5D + 1] = { | |
248 | 1.575278146806816970152174364308980863569E-9Q, | |
249 | 3.361289173657099516191331123405675054321E-7Q, | |
250 | 2.704692281550877810424745289838790693708E-5Q, | |
251 | 1.070854930483999749316546199273521063543E-3Q, | |
252 | 2.282373093495295842598097265627962125411E-2Q, | |
253 | 2.692025460665354148328762368240343249830E-1Q, | |
254 | 1.739892942593664447220951225734811133759E0Q, | |
255 | 5.890727576752230385342377570386657229324E0Q, | |
256 | 9.517442287057841500750256954117735128153E0Q, | |
257 | 6.100616353935338240775363403030137736013E0Q, | |
258 | /* 1.000000000000000000000000000000000000000E0 */ | |
259 | }; | |
260 | ||
261 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
262 | Peak relative error 3.0e-36 | |
263 | 0.25 <= 1/x <= 0.3125 */ | |
264 | #define NP3r2_4N 9 | |
265 | static const __float128 P3r2_4N[NP3r2_4N + 1] = { | |
266 | 8.240803130988044478595580300846665863782E-8Q, | |
267 | 1.179418958381961224222969866406483744580E-5Q, | |
268 | 6.179787320956386624336959112503824397755E-4Q, | |
269 | 1.540270833608687596420595830747166658383E-2Q, | |
270 | 1.983904219491512618376375619598837355076E-1Q, | |
271 | 1.341465722692038870390470651608301155565E0Q, | |
272 | 4.617865326696612898792238245990854646057E0Q, | |
273 | 7.435574801812346424460233180412308000587E0Q, | |
274 | 4.671327027414635292514599201278557680420E0Q, | |
275 | 7.299530852495776936690976966995187714739E-1Q, | |
276 | }; | |
277 | #define NP3r2_4D 9 | |
278 | static const __float128 P3r2_4D[NP3r2_4D + 1] = { | |
279 | 7.032152009675729604487575753279187576521E-7Q, | |
280 | 1.015090352324577615777511269928856742848E-4Q, | |
281 | 5.394262184808448484302067955186308730620E-3Q, | |
282 | 1.375291438480256110455809354836988584325E-1Q, | |
283 | 1.836247144461106304788160919310404376670E0Q, | |
284 | 1.314378564254376655001094503090935880349E1Q, | |
285 | 4.957184590465712006934452500894672343488E1Q, | |
286 | 9.287394244300647738855415178790263465398E1Q, | |
287 | 7.652563275535900609085229286020552768399E1Q, | |
288 | 2.147042473003074533150718117770093209096E1Q, | |
289 | /* 1.000000000000000000000000000000000000000E0 */ | |
290 | }; | |
291 | ||
292 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
293 | Peak relative error 1.0e-35 | |
294 | 0.3125 <= 1/x <= 0.375 */ | |
295 | #define NP2r7_3r2N 9 | |
296 | static const __float128 P2r7_3r2N[NP2r7_3r2N + 1] = { | |
297 | 4.599033469240421554219816935160627085991E-7Q, | |
298 | 4.665724440345003914596647144630893997284E-5Q, | |
299 | 1.684348845667764271596142716944374892756E-3Q, | |
300 | 2.802446446884455707845985913454440176223E-2Q, | |
301 | 2.321937586453963310008279956042545173930E-1Q, | |
302 | 9.640277413988055668692438709376437553804E-1Q, | |
303 | 1.911021064710270904508663334033003246028E0Q, | |
304 | 1.600811610164341450262992138893970224971E0Q, | |
305 | 4.266299218652587901171386591543457861138E-1Q, | |
306 | 1.316470424456061252962568223251247207325E-2Q, | |
307 | }; | |
308 | #define NP2r7_3r2D 8 | |
309 | static const __float128 P2r7_3r2D[NP2r7_3r2D + 1] = { | |
310 | 3.924508608545520758883457108453520099610E-6Q, | |
311 | 4.029707889408829273226495756222078039823E-4Q, | |
312 | 1.484629715787703260797886463307469600219E-2Q, | |
313 | 2.553136379967180865331706538897231588685E-1Q, | |
314 | 2.229457223891676394409880026887106228740E0Q, | |
315 | 1.005708903856384091956550845198392117318E1Q, | |
316 | 2.277082659664386953166629360352385889558E1Q, | |
317 | 2.384726835193630788249826630376533988245E1Q, | |
318 | 9.700989749041320895890113781610939632410E0Q, | |
319 | /* 1.000000000000000000000000000000000000000E0 */ | |
320 | }; | |
321 | ||
322 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
323 | Peak relative error 1.7e-36 | |
324 | 0.3125 <= 1/x <= 0.4375 */ | |
325 | #define NP2r3_2r7N 9 | |
326 | static const __float128 P2r3_2r7N[NP2r3_2r7N + 1] = { | |
327 | 3.916766777108274628543759603786857387402E-6Q, | |
328 | 3.212176636756546217390661984304645137013E-4Q, | |
329 | 9.255768488524816445220126081207248947118E-3Q, | |
330 | 1.214853146369078277453080641911700735354E-1Q, | |
331 | 7.855163309847214136198449861311404633665E-1Q, | |
332 | 2.520058073282978403655488662066019816540E0Q, | |
333 | 3.825136484837545257209234285382183711466E0Q, | |
334 | 2.432569427554248006229715163865569506873E0Q, | |
335 | 4.877934835018231178495030117729800489743E-1Q, | |
336 | 1.109902737860249670981355149101343427885E-2Q, | |
337 | }; | |
338 | #define NP2r3_2r7D 8 | |
339 | static const __float128 P2r3_2r7D[NP2r3_2r7D + 1] = { | |
340 | 3.342307880794065640312646341190547184461E-5Q, | |
341 | 2.782182891138893201544978009012096558265E-3Q, | |
342 | 8.221304931614200702142049236141249929207E-2Q, | |
343 | 1.123728246291165812392918571987858010949E0Q, | |
344 | 7.740482453652715577233858317133423434590E0Q, | |
345 | 2.737624677567945952953322566311201919139E1Q, | |
346 | 4.837181477096062403118304137851260715475E1Q, | |
347 | 3.941098643468580791437772701093795299274E1Q, | |
348 | 1.245821247166544627558323920382547533630E1Q, | |
349 | /* 1.000000000000000000000000000000000000000E0 */ | |
350 | }; | |
351 | ||
352 | /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), | |
353 | Peak relative error 1.7e-35 | |
354 | 0.4375 <= 1/x <= 0.5 */ | |
355 | #define NP2_2r3N 8 | |
356 | static const __float128 P2_2r3N[NP2_2r3N + 1] = { | |
357 | 3.397930802851248553545191160608731940751E-4Q, | |
358 | 2.104020902735482418784312825637833698217E-2Q, | |
359 | 4.442291771608095963935342749477836181939E-1Q, | |
360 | 4.131797328716583282869183304291833754967E0Q, | |
361 | 1.819920169779026500146134832455189917589E1Q, | |
362 | 3.781779616522937565300309684282401791291E1Q, | |
363 | 3.459605449728864218972931220783543410347E1Q, | |
364 | 1.173594248397603882049066603238568316561E1Q, | |
365 | 9.455702270242780642835086549285560316461E-1Q, | |
366 | }; | |
367 | #define NP2_2r3D 8 | |
368 | static const __float128 P2_2r3D[NP2_2r3D + 1] = { | |
369 | 2.899568897241432883079888249845707400614E-3Q, | |
370 | 1.831107138190848460767699919531132426356E-1Q, | |
371 | 3.999350044057883839080258832758908825165E0Q, | |
372 | 3.929041535867957938340569419874195303712E1Q, | |
373 | 1.884245613422523323068802689915538908291E2Q, | |
374 | 4.461469948819229734353852978424629815929E2Q, | |
375 | 5.004998753999796821224085972610636347903E2Q, | |
376 | 2.386342520092608513170837883757163414100E2Q, | |
377 | 3.791322528149347975999851588922424189957E1Q, | |
378 | /* 1.000000000000000000000000000000000000000E0 */ | |
379 | }; | |
380 | ||
381 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
382 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
383 | Peak relative error 8.0e-36 | |
384 | 0 <= 1/x <= .0625 */ | |
385 | #define NQ16_IN 10 | |
386 | static const __float128 Q16_IN[NQ16_IN + 1] = { | |
387 | -3.917420835712508001321875734030357393421E-18Q, | |
388 | -4.440311387483014485304387406538069930457E-15Q, | |
389 | -1.951635424076926487780929645954007139616E-12Q, | |
390 | -4.318256438421012555040546775651612810513E-10Q, | |
391 | -5.231244131926180765270446557146989238020E-8Q, | |
392 | -3.540072702902043752460711989234732357653E-6Q, | |
393 | -1.311017536555269966928228052917534882984E-4Q, | |
394 | -2.495184669674631806622008769674827575088E-3Q, | |
395 | -2.141868222987209028118086708697998506716E-2Q, | |
396 | -6.184031415202148901863605871197272650090E-2Q, | |
397 | -1.922298704033332356899546792898156493887E-2Q, | |
398 | }; | |
399 | #define NQ16_ID 9 | |
400 | static const __float128 Q16_ID[NQ16_ID + 1] = { | |
401 | 3.820418034066293517479619763498400162314E-17Q, | |
402 | 4.340702810799239909648911373329149354911E-14Q, | |
403 | 1.914985356383416140706179933075303538524E-11Q, | |
404 | 4.262333682610888819476498617261895474330E-9Q, | |
405 | 5.213481314722233980346462747902942182792E-7Q, | |
406 | 3.585741697694069399299005316809954590558E-5Q, | |
407 | 1.366513429642842006385029778105539457546E-3Q, | |
408 | 2.745282599850704662726337474371355160594E-2Q, | |
409 | 2.637644521611867647651200098449903330074E-1Q, | |
410 | 1.006953426110765984590782655598680488746E0Q, | |
411 | /* 1.000000000000000000000000000000000000000E0 */ | |
412 | }; | |
413 | ||
414 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
415 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
416 | Peak relative error 1.9e-36 | |
417 | 0.0625 <= 1/x <= 0.125 */ | |
418 | #define NQ8_16N 11 | |
419 | static const __float128 Q8_16N[NQ8_16N + 1] = { | |
420 | -2.028630366670228670781362543615221542291E-17Q, | |
421 | -1.519634620380959966438130374006858864624E-14Q, | |
422 | -4.540596528116104986388796594639405114524E-12Q, | |
423 | -7.085151756671466559280490913558388648274E-10Q, | |
424 | -6.351062671323970823761883833531546885452E-8Q, | |
425 | -3.390817171111032905297982523519503522491E-6Q, | |
426 | -1.082340897018886970282138836861233213972E-4Q, | |
427 | -2.020120801187226444822977006648252379508E-3Q, | |
428 | -2.093169910981725694937457070649605557555E-2Q, | |
429 | -1.092176538874275712359269481414448063393E-1Q, | |
430 | -2.374790947854765809203590474789108718733E-1Q, | |
431 | -1.365364204556573800719985118029601401323E-1Q, | |
432 | }; | |
433 | #define NQ8_16D 11 | |
434 | static const __float128 Q8_16D[NQ8_16D + 1] = { | |
435 | 1.978397614733632533581207058069628242280E-16Q, | |
436 | 1.487361156806202736877009608336766720560E-13Q, | |
437 | 4.468041406888412086042576067133365913456E-11Q, | |
438 | 7.027822074821007443672290507210594648877E-9Q, | |
439 | 6.375740580686101224127290062867976007374E-7Q, | |
440 | 3.466887658320002225888644977076410421940E-5Q, | |
441 | 1.138625640905289601186353909213719596986E-3Q, | |
442 | 2.224470799470414663443449818235008486439E-2Q, | |
443 | 2.487052928527244907490589787691478482358E-1Q, | |
444 | 1.483927406564349124649083853892380899217E0Q, | |
445 | 4.182773513276056975777258788903489507705E0Q, | |
446 | 4.419665392573449746043880892524360870944E0Q, | |
447 | /* 1.000000000000000000000000000000000000000E0 */ | |
448 | }; | |
449 | ||
450 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
451 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
452 | Peak relative error 1.5e-35 | |
453 | 0.125 <= 1/x <= 0.1875 */ | |
454 | #define NQ5_8N 10 | |
455 | static const __float128 Q5_8N[NQ5_8N + 1] = { | |
456 | -3.656082407740970534915918390488336879763E-13Q, | |
457 | -1.344660308497244804752334556734121771023E-10Q, | |
458 | -1.909765035234071738548629788698150760791E-8Q, | |
459 | -1.366668038160120210269389551283666716453E-6Q, | |
460 | -5.392327355984269366895210704976314135683E-5Q, | |
461 | -1.206268245713024564674432357634540343884E-3Q, | |
462 | -1.515456784370354374066417703736088291287E-2Q, | |
463 | -1.022454301137286306933217746545237098518E-1Q, | |
464 | -3.373438906472495080504907858424251082240E-1Q, | |
465 | -4.510782522110845697262323973549178453405E-1Q, | |
466 | -1.549000892545288676809660828213589804884E-1Q, | |
467 | }; | |
468 | #define NQ5_8D 10 | |
469 | static const __float128 Q5_8D[NQ5_8D + 1] = { | |
470 | 3.565550843359501079050699598913828460036E-12Q, | |
471 | 1.321016015556560621591847454285330528045E-9Q, | |
472 | 1.897542728662346479999969679234270605975E-7Q, | |
473 | 1.381720283068706710298734234287456219474E-5Q, | |
474 | 5.599248147286524662305325795203422873725E-4Q, | |
475 | 1.305442352653121436697064782499122164843E-2Q, | |
476 | 1.750234079626943298160445750078631894985E-1Q, | |
477 | 1.311420542073436520965439883806946678491E0Q, | |
478 | 5.162757689856842406744504211089724926650E0Q, | |
479 | 9.527760296384704425618556332087850581308E0Q, | |
480 | 6.604648207463236667912921642545100248584E0Q, | |
481 | /* 1.000000000000000000000000000000000000000E0 */ | |
482 | }; | |
483 | ||
484 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
485 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
486 | Peak relative error 1.3e-35 | |
487 | 0.1875 <= 1/x <= 0.25 */ | |
488 | #define NQ4_5N 10 | |
489 | static const __float128 Q4_5N[NQ4_5N + 1] = { | |
490 | -4.079513568708891749424783046520200903755E-11Q, | |
491 | -9.326548104106791766891812583019664893311E-9Q, | |
492 | -8.016795121318423066292906123815687003356E-7Q, | |
493 | -3.372350544043594415609295225664186750995E-5Q, | |
494 | -7.566238665947967882207277686375417983917E-4Q, | |
495 | -9.248861580055565402130441618521591282617E-3Q, | |
496 | -6.033106131055851432267702948850231270338E-2Q, | |
497 | -1.966908754799996793730369265431584303447E-1Q, | |
498 | -2.791062741179964150755788226623462207560E-1Q, | |
499 | -1.255478605849190549914610121863534191666E-1Q, | |
500 | -4.320429862021265463213168186061696944062E-3Q, | |
501 | }; | |
502 | #define NQ4_5D 9 | |
503 | static const __float128 Q4_5D[NQ4_5D + 1] = { | |
504 | 3.978497042580921479003851216297330701056E-10Q, | |
505 | 9.203304163828145809278568906420772246666E-8Q, | |
506 | 8.059685467088175644915010485174545743798E-6Q, | |
507 | 3.490187375993956409171098277561669167446E-4Q, | |
508 | 8.189109654456872150100501732073810028829E-3Q, | |
509 | 1.072572867311023640958725265762483033769E-1Q, | |
510 | 7.790606862409960053675717185714576937994E-1Q, | |
511 | 3.016049768232011196434185423512777656328E0Q, | |
512 | 5.722963851442769787733717162314477949360E0Q, | |
513 | 4.510527838428473279647251350931380867663E0Q, | |
514 | /* 1.000000000000000000000000000000000000000E0 */ | |
515 | }; | |
516 | ||
517 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
518 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
519 | Peak relative error 2.1e-35 | |
520 | 0.25 <= 1/x <= 0.3125 */ | |
521 | #define NQ3r2_4N 9 | |
522 | static const __float128 Q3r2_4N[NQ3r2_4N + 1] = { | |
523 | -1.087480809271383885936921889040388133627E-8Q, | |
524 | -1.690067828697463740906962973479310170932E-6Q, | |
525 | -9.608064416995105532790745641974762550982E-5Q, | |
526 | -2.594198839156517191858208513873961837410E-3Q, | |
527 | -3.610954144421543968160459863048062977822E-2Q, | |
528 | -2.629866798251843212210482269563961685666E-1Q, | |
529 | -9.709186825881775885917984975685752956660E-1Q, | |
530 | -1.667521829918185121727268867619982417317E0Q, | |
531 | -1.109255082925540057138766105229900943501E0Q, | |
532 | -1.812932453006641348145049323713469043328E-1Q, | |
533 | }; | |
534 | #define NQ3r2_4D 9 | |
535 | static const __float128 Q3r2_4D[NQ3r2_4D + 1] = { | |
536 | 1.060552717496912381388763753841473407026E-7Q, | |
537 | 1.676928002024920520786883649102388708024E-5Q, | |
538 | 9.803481712245420839301400601140812255737E-4Q, | |
539 | 2.765559874262309494758505158089249012930E-2Q, | |
540 | 4.117921827792571791298862613287549140706E-1Q, | |
541 | 3.323769515244751267093378361930279161413E0Q, | |
542 | 1.436602494405814164724810151689705353670E1Q, | |
543 | 3.163087869617098638064881410646782408297E1Q, | |
544 | 3.198181264977021649489103980298349589419E1Q, | |
545 | 1.203649258862068431199471076202897823272E1Q, | |
546 | /* 1.000000000000000000000000000000000000000E0 */ | |
547 | }; | |
548 | ||
549 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
550 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
551 | Peak relative error 1.6e-36 | |
552 | 0.3125 <= 1/x <= 0.375 */ | |
553 | #define NQ2r7_3r2N 9 | |
554 | static const __float128 Q2r7_3r2N[NQ2r7_3r2N + 1] = { | |
555 | -1.723405393982209853244278760171643219530E-7Q, | |
556 | -2.090508758514655456365709712333460087442E-5Q, | |
557 | -9.140104013370974823232873472192719263019E-4Q, | |
558 | -1.871349499990714843332742160292474780128E-2Q, | |
559 | -1.948930738119938669637865956162512983416E-1Q, | |
560 | -1.048764684978978127908439526343174139788E0Q, | |
561 | -2.827714929925679500237476105843643064698E0Q, | |
562 | -3.508761569156476114276988181329773987314E0Q, | |
563 | -1.669332202790211090973255098624488308989E0Q, | |
564 | -1.930796319299022954013840684651016077770E-1Q, | |
565 | }; | |
566 | #define NQ2r7_3r2D 9 | |
567 | static const __float128 Q2r7_3r2D[NQ2r7_3r2D + 1] = { | |
568 | 1.680730662300831976234547482334347983474E-6Q, | |
569 | 2.084241442440551016475972218719621841120E-4Q, | |
570 | 9.445316642108367479043541702688736295579E-3Q, | |
571 | 2.044637889456631896650179477133252184672E-1Q, | |
572 | 2.316091982244297350829522534435350078205E0Q, | |
573 | 1.412031891783015085196708811890448488865E1Q, | |
574 | 4.583830154673223384837091077279595496149E1Q, | |
575 | 7.549520609270909439885998474045974122261E1Q, | |
576 | 5.697605832808113367197494052388203310638E1Q, | |
577 | 1.601496240876192444526383314589371686234E1Q, | |
578 | /* 1.000000000000000000000000000000000000000E0 */ | |
579 | }; | |
580 | ||
581 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
582 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
583 | Peak relative error 9.5e-36 | |
584 | 0.375 <= 1/x <= 0.4375 */ | |
585 | #define NQ2r3_2r7N 9 | |
586 | static const __float128 Q2r3_2r7N[NQ2r3_2r7N + 1] = { | |
587 | -8.603042076329122085722385914954878953775E-7Q, | |
588 | -7.701746260451647874214968882605186675720E-5Q, | |
589 | -2.407932004380727587382493696877569654271E-3Q, | |
590 | -3.403434217607634279028110636919987224188E-2Q, | |
591 | -2.348707332185238159192422084985713102877E-1Q, | |
592 | -7.957498841538254916147095255700637463207E-1Q, | |
593 | -1.258469078442635106431098063707934348577E0Q, | |
594 | -8.162415474676345812459353639449971369890E-1Q, | |
595 | -1.581783890269379690141513949609572806898E-1Q, | |
596 | -1.890595651683552228232308756569450822905E-3Q, | |
597 | }; | |
598 | #define NQ2r3_2r7D 8 | |
599 | static const __float128 Q2r3_2r7D[NQ2r3_2r7D + 1] = { | |
600 | 8.390017524798316921170710533381568175665E-6Q, | |
601 | 7.738148683730826286477254659973968763659E-4Q, | |
602 | 2.541480810958665794368759558791634341779E-2Q, | |
603 | 3.878879789711276799058486068562386244873E-1Q, | |
604 | 3.003783779325811292142957336802456109333E0Q, | |
605 | 1.206480374773322029883039064575464497400E1Q, | |
606 | 2.458414064785315978408974662900438351782E1Q, | |
607 | 2.367237826273668567199042088835448715228E1Q, | |
608 | 9.231451197519171090875569102116321676763E0Q, | |
609 | /* 1.000000000000000000000000000000000000000E0 */ | |
610 | }; | |
611 | ||
612 | /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), | |
613 | Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), | |
614 | Peak relative error 1.4e-36 | |
615 | 0.4375 <= 1/x <= 0.5 */ | |
616 | #define NQ2_2r3N 9 | |
617 | static const __float128 Q2_2r3N[NQ2_2r3N + 1] = { | |
618 | -5.552507516089087822166822364590806076174E-6Q, | |
619 | -4.135067659799500521040944087433752970297E-4Q, | |
620 | -1.059928728869218962607068840646564457980E-2Q, | |
621 | -1.212070036005832342565792241385459023801E-1Q, | |
622 | -6.688350110633603958684302153362735625156E-1Q, | |
623 | -1.793587878197360221340277951304429821582E0Q, | |
624 | -2.225407682237197485644647380483725045326E0Q, | |
625 | -1.123402135458940189438898496348239744403E0Q, | |
626 | -1.679187241566347077204805190763597299805E-1Q, | |
627 | -1.458550613639093752909985189067233504148E-3Q, | |
628 | }; | |
629 | #define NQ2_2r3D 8 | |
630 | static const __float128 Q2_2r3D[NQ2_2r3D + 1] = { | |
631 | 5.415024336507980465169023996403597916115E-5Q, | |
632 | 4.179246497380453022046357404266022870788E-3Q, | |
633 | 1.136306384261959483095442402929502368598E-1Q, | |
634 | 1.422640343719842213484515445393284072830E0Q, | |
635 | 8.968786703393158374728850922289204805764E0Q, | |
636 | 2.914542473339246127533384118781216495934E1Q, | |
637 | 4.781605421020380669870197378210457054685E1Q, | |
638 | 3.693865837171883152382820584714795072937E1Q, | |
639 | 1.153220502744204904763115556224395893076E1Q, | |
640 | /* 1.000000000000000000000000000000000000000E0 */ | |
641 | }; | |
642 | ||
643 | ||
644 | /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ | |
645 | ||
646 | static __float128 | |
647 | neval (__float128 x, const __float128 *p, int n) | |
648 | { | |
649 | __float128 y; | |
650 | ||
651 | p += n; | |
652 | y = *p--; | |
653 | do | |
654 | { | |
655 | y = y * x + *p--; | |
656 | } | |
657 | while (--n > 0); | |
658 | return y; | |
659 | } | |
660 | ||
661 | ||
662 | /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ | |
663 | ||
664 | static __float128 | |
665 | deval (__float128 x, const __float128 *p, int n) | |
666 | { | |
667 | __float128 y; | |
668 | ||
669 | p += n; | |
670 | y = x + *p--; | |
671 | do | |
672 | { | |
673 | y = y * x + *p--; | |
674 | } | |
675 | while (--n > 0); | |
676 | return y; | |
677 | } | |
678 | ||
679 | ||
680 | /* Bessel function of the first kind, order one. */ | |
681 | ||
682 | __float128 | |
683 | j1q (__float128 x) | |
684 | { | |
685 | __float128 xx, xinv, z, p, q, c, s, cc, ss; | |
686 | ||
687 | if (! finiteq (x)) | |
688 | { | |
689 | if (x != x) | |
c98f0ea6 | 690 | return x + x; |
87969c8c | 691 | else |
e409716d | 692 | return 0; |
87969c8c | 693 | } |
e409716d | 694 | if (x == 0) |
87969c8c | 695 | return x; |
696 | xx = fabsq (x); | |
c98f0ea6 | 697 | if (xx <= 0x1p-58Q) |
698 | { | |
699 | __float128 ret = x * 0.5Q; | |
700 | math_check_force_underflow (ret); | |
701 | if (ret == 0) | |
702 | errno = ERANGE; | |
703 | return ret; | |
704 | } | |
e409716d | 705 | if (xx <= 2) |
87969c8c | 706 | { |
707 | /* 0 <= x <= 2 */ | |
708 | z = xx * xx; | |
709 | p = xx * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D); | |
710 | p += 0.5Q * xx; | |
711 | if (x < 0) | |
712 | p = -p; | |
713 | return p; | |
714 | } | |
715 | ||
c98f0ea6 | 716 | /* X = x - 3 pi/4 |
717 | cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4) | |
718 | = 1/sqrt(2) * (-cos(x) + sin(x)) | |
719 | sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4) | |
720 | = -1/sqrt(2) * (sin(x) + cos(x)) | |
721 | cf. Fdlibm. */ | |
722 | sincosq (xx, &s, &c); | |
723 | ss = -s - c; | |
724 | cc = s - c; | |
e409716d | 725 | if (xx <= FLT128_MAX / 2) |
c98f0ea6 | 726 | { |
727 | z = cosq (xx + xx); | |
728 | if ((s * c) > 0) | |
729 | cc = z / ss; | |
730 | else | |
731 | ss = z / cc; | |
732 | } | |
733 | ||
734 | if (xx > 0x1p256Q) | |
735 | { | |
736 | z = ONEOSQPI * cc / sqrtq (xx); | |
737 | if (x < 0) | |
738 | z = -z; | |
739 | return z; | |
740 | } | |
741 | ||
e409716d | 742 | xinv = 1 / xx; |
87969c8c | 743 | z = xinv * xinv; |
744 | if (xinv <= 0.25) | |
745 | { | |
746 | if (xinv <= 0.125) | |
747 | { | |
748 | if (xinv <= 0.0625) | |
749 | { | |
750 | p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); | |
751 | q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); | |
752 | } | |
753 | else | |
754 | { | |
755 | p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); | |
756 | q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); | |
757 | } | |
758 | } | |
759 | else if (xinv <= 0.1875) | |
760 | { | |
761 | p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); | |
762 | q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); | |
763 | } | |
764 | else | |
765 | { | |
766 | p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); | |
767 | q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); | |
768 | } | |
769 | } /* .25 */ | |
770 | else /* if (xinv <= 0.5) */ | |
771 | { | |
772 | if (xinv <= 0.375) | |
773 | { | |
774 | if (xinv <= 0.3125) | |
775 | { | |
776 | p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); | |
777 | q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); | |
778 | } | |
779 | else | |
780 | { | |
781 | p = neval (z, P2r7_3r2N, NP2r7_3r2N) | |
782 | / deval (z, P2r7_3r2D, NP2r7_3r2D); | |
783 | q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) | |
784 | / deval (z, Q2r7_3r2D, NQ2r7_3r2D); | |
785 | } | |
786 | } | |
787 | else if (xinv <= 0.4375) | |
788 | { | |
789 | p = neval (z, P2r3_2r7N, NP2r3_2r7N) | |
790 | / deval (z, P2r3_2r7D, NP2r3_2r7D); | |
791 | q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) | |
792 | / deval (z, Q2r3_2r7D, NQ2r3_2r7D); | |
793 | } | |
794 | else | |
795 | { | |
796 | p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); | |
797 | q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); | |
798 | } | |
799 | } | |
e409716d | 800 | p = 1 + z * p; |
87969c8c | 801 | q = z * q; |
802 | q = q * xinv + 0.375Q * xinv; | |
87969c8c | 803 | z = ONEOSQPI * (p * cc - q * ss) / sqrtq (xx); |
804 | if (x < 0) | |
805 | z = -z; | |
806 | return z; | |
807 | } | |
808 | ||
809 | ||
e409716d | 810 | |
87969c8c | 811 | /* Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2) |
812 | Peak relative error 6.2e-38 | |
813 | 0 <= x <= 2 */ | |
814 | #define NY0_2N 7 | |
e409716d | 815 | static const __float128 Y0_2N[NY0_2N + 1] = { |
87969c8c | 816 | -6.804415404830253804408698161694720833249E19Q, |
817 | 1.805450517967019908027153056150465849237E19Q, | |
818 | -8.065747497063694098810419456383006737312E17Q, | |
819 | 1.401336667383028259295830955439028236299E16Q, | |
820 | -1.171654432898137585000399489686629680230E14Q, | |
821 | 5.061267920943853732895341125243428129150E11Q, | |
822 | -1.096677850566094204586208610960870217970E9Q, | |
823 | 9.541172044989995856117187515882879304461E5Q, | |
824 | }; | |
825 | #define NY0_2D 7 | |
e409716d | 826 | static const __float128 Y0_2D[NY0_2D + 1] = { |
87969c8c | 827 | 3.470629591820267059538637461549677594549E20Q, |
828 | 4.120796439009916326855848107545425217219E18Q, | |
829 | 2.477653371652018249749350657387030814542E16Q, | |
830 | 9.954678543353888958177169349272167762797E13Q, | |
831 | 2.957927997613630118216218290262851197754E11Q, | |
832 | 6.748421382188864486018861197614025972118E8Q, | |
833 | 1.173453425218010888004562071020305709319E6Q, | |
834 | 1.450335662961034949894009554536003377187E3Q, | |
835 | /* 1.000000000000000000000000000000000000000E0 */ | |
836 | }; | |
837 | ||
838 | ||
839 | /* Bessel function of the second kind, order one. */ | |
840 | ||
841 | __float128 | |
842 | y1q (__float128 x) | |
843 | { | |
844 | __float128 xx, xinv, z, p, q, c, s, cc, ss; | |
845 | ||
846 | if (! finiteq (x)) | |
c98f0ea6 | 847 | return 1 / (x + x * x); |
e409716d | 848 | if (x <= 0) |
87969c8c | 849 | { |
e409716d | 850 | if (x < 0) |
87969c8c | 851 | return (zero / (zero * x)); |
c98f0ea6 | 852 | return -1 / zero; /* -inf and divide by zero exception. */ |
87969c8c | 853 | } |
854 | xx = fabsq (x); | |
89a213c9 | 855 | if (xx <= 0x1p-114) |
c98f0ea6 | 856 | { |
857 | z = -TWOOPI / x; | |
858 | if (isinfq (z)) | |
859 | errno = ERANGE; | |
860 | return z; | |
861 | } | |
e409716d | 862 | if (xx <= 2) |
863 | { | |
87969c8c | 864 | /* 0 <= x <= 2 */ |
e409716d | 865 | SET_RESTORE_ROUNDF128 (FE_TONEAREST); |
87969c8c | 866 | z = xx * xx; |
867 | p = xx * neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D); | |
868 | p = -TWOOPI / xx + p; | |
869 | p = TWOOPI * logq (x) * j1q (x) + p; | |
870 | return p; | |
871 | } | |
872 | ||
c98f0ea6 | 873 | /* X = x - 3 pi/4 |
874 | cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4) | |
875 | = 1/sqrt(2) * (-cos(x) + sin(x)) | |
876 | sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4) | |
877 | = -1/sqrt(2) * (sin(x) + cos(x)) | |
878 | cf. Fdlibm. */ | |
879 | sincosq (xx, &s, &c); | |
880 | ss = -s - c; | |
881 | cc = s - c; | |
e409716d | 882 | if (xx <= FLT128_MAX / 2) |
c98f0ea6 | 883 | { |
884 | z = cosq (xx + xx); | |
885 | if ((s * c) > 0) | |
886 | cc = z / ss; | |
887 | else | |
888 | ss = z / cc; | |
889 | } | |
890 | ||
891 | if (xx > 0x1p256Q) | |
892 | return ONEOSQPI * ss / sqrtq (xx); | |
893 | ||
e409716d | 894 | xinv = 1 / xx; |
87969c8c | 895 | z = xinv * xinv; |
896 | if (xinv <= 0.25) | |
897 | { | |
898 | if (xinv <= 0.125) | |
899 | { | |
900 | if (xinv <= 0.0625) | |
901 | { | |
902 | p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); | |
903 | q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); | |
904 | } | |
905 | else | |
906 | { | |
907 | p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); | |
908 | q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); | |
909 | } | |
910 | } | |
911 | else if (xinv <= 0.1875) | |
912 | { | |
913 | p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); | |
914 | q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); | |
915 | } | |
916 | else | |
917 | { | |
918 | p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); | |
919 | q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); | |
920 | } | |
921 | } /* .25 */ | |
922 | else /* if (xinv <= 0.5) */ | |
923 | { | |
924 | if (xinv <= 0.375) | |
925 | { | |
926 | if (xinv <= 0.3125) | |
927 | { | |
928 | p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); | |
929 | q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); | |
930 | } | |
931 | else | |
932 | { | |
933 | p = neval (z, P2r7_3r2N, NP2r7_3r2N) | |
934 | / deval (z, P2r7_3r2D, NP2r7_3r2D); | |
935 | q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) | |
936 | / deval (z, Q2r7_3r2D, NQ2r7_3r2D); | |
937 | } | |
938 | } | |
939 | else if (xinv <= 0.4375) | |
940 | { | |
941 | p = neval (z, P2r3_2r7N, NP2r3_2r7N) | |
942 | / deval (z, P2r3_2r7D, NP2r3_2r7D); | |
943 | q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) | |
944 | / deval (z, Q2r3_2r7D, NQ2r3_2r7D); | |
945 | } | |
946 | else | |
947 | { | |
948 | p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); | |
949 | q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); | |
950 | } | |
951 | } | |
e409716d | 952 | p = 1 + z * p; |
87969c8c | 953 | q = z * q; |
954 | q = q * xinv + 0.375Q * xinv; | |
87969c8c | 955 | z = ONEOSQPI * (p * ss + q * cc) / sqrtq (xx); |
956 | return z; | |
957 | } |