Mercurial > hg > CbC > CbC_gcc
comparison libquadmath/math/j1q.c @ 68:561a7518be6b
update gcc-4.6
author | Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp> |
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date | Sun, 21 Aug 2011 07:07:55 +0900 |
parents | |
children | 04ced10e8804 |
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67:f6334be47118 | 68:561a7518be6b |
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1 /* j1l.c | |
2 * | |
3 * Bessel function of order one | |
4 * | |
5 * | |
6 * | |
7 * SYNOPSIS: | |
8 * | |
9 * long double x, y, j1l(); | |
10 * | |
11 * y = j1l( x ); | |
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 * | |
55 * double x, y, y1l(); | |
56 * | |
57 * y = y1l( x ); | |
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 | |
95 License along with this library; if not, write to the Free Software | |
96 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ | |
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; | |
104 static const __float128 zero = 0.0Q; | |
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, | |
128 /* 1.000000000000000000000000000000000000000E0Q */ | |
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) | |
690 return x; | |
691 else | |
692 return 0.0Q; | |
693 } | |
694 if (x == 0.0Q) | |
695 return x; | |
696 xx = fabsq (x); | |
697 if (xx <= 2.0Q) | |
698 { | |
699 /* 0 <= x <= 2 */ | |
700 z = xx * xx; | |
701 p = xx * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D); | |
702 p += 0.5Q * xx; | |
703 if (x < 0) | |
704 p = -p; | |
705 return p; | |
706 } | |
707 | |
708 xinv = 1.0Q / xx; | |
709 z = xinv * xinv; | |
710 if (xinv <= 0.25) | |
711 { | |
712 if (xinv <= 0.125) | |
713 { | |
714 if (xinv <= 0.0625) | |
715 { | |
716 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); | |
717 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); | |
718 } | |
719 else | |
720 { | |
721 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); | |
722 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); | |
723 } | |
724 } | |
725 else if (xinv <= 0.1875) | |
726 { | |
727 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); | |
728 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); | |
729 } | |
730 else | |
731 { | |
732 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); | |
733 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); | |
734 } | |
735 } /* .25 */ | |
736 else /* if (xinv <= 0.5) */ | |
737 { | |
738 if (xinv <= 0.375) | |
739 { | |
740 if (xinv <= 0.3125) | |
741 { | |
742 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); | |
743 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); | |
744 } | |
745 else | |
746 { | |
747 p = neval (z, P2r7_3r2N, NP2r7_3r2N) | |
748 / deval (z, P2r7_3r2D, NP2r7_3r2D); | |
749 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) | |
750 / deval (z, Q2r7_3r2D, NQ2r7_3r2D); | |
751 } | |
752 } | |
753 else if (xinv <= 0.4375) | |
754 { | |
755 p = neval (z, P2r3_2r7N, NP2r3_2r7N) | |
756 / deval (z, P2r3_2r7D, NP2r3_2r7D); | |
757 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) | |
758 / deval (z, Q2r3_2r7D, NQ2r3_2r7D); | |
759 } | |
760 else | |
761 { | |
762 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); | |
763 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); | |
764 } | |
765 } | |
766 p = 1.0Q + z * p; | |
767 q = z * q; | |
768 q = q * xinv + 0.375Q * xinv; | |
769 /* X = x - 3 pi/4 | |
770 cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4) | |
771 = 1/sqrt(2) * (-cos(x) + sin(x)) | |
772 sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4) | |
773 = -1/sqrt(2) * (sin(x) + cos(x)) | |
774 cf. Fdlibm. */ | |
775 sincosq (xx, &s, &c); | |
776 ss = -s - c; | |
777 cc = s - c; | |
778 z = cosq (xx + xx); | |
779 if ((s * c) > 0) | |
780 cc = z / ss; | |
781 else | |
782 ss = z / cc; | |
783 z = ONEOSQPI * (p * cc - q * ss) / sqrtq (xx); | |
784 if (x < 0) | |
785 z = -z; | |
786 return z; | |
787 } | |
788 | |
789 | |
790 /* Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2) | |
791 Peak relative error 6.2e-38 | |
792 0 <= x <= 2 */ | |
793 #define NY0_2N 7 | |
794 static __float128 Y0_2N[NY0_2N + 1] = { | |
795 -6.804415404830253804408698161694720833249E19Q, | |
796 1.805450517967019908027153056150465849237E19Q, | |
797 -8.065747497063694098810419456383006737312E17Q, | |
798 1.401336667383028259295830955439028236299E16Q, | |
799 -1.171654432898137585000399489686629680230E14Q, | |
800 5.061267920943853732895341125243428129150E11Q, | |
801 -1.096677850566094204586208610960870217970E9Q, | |
802 9.541172044989995856117187515882879304461E5Q, | |
803 }; | |
804 #define NY0_2D 7 | |
805 static __float128 Y0_2D[NY0_2D + 1] = { | |
806 3.470629591820267059538637461549677594549E20Q, | |
807 4.120796439009916326855848107545425217219E18Q, | |
808 2.477653371652018249749350657387030814542E16Q, | |
809 9.954678543353888958177169349272167762797E13Q, | |
810 2.957927997613630118216218290262851197754E11Q, | |
811 6.748421382188864486018861197614025972118E8Q, | |
812 1.173453425218010888004562071020305709319E6Q, | |
813 1.450335662961034949894009554536003377187E3Q, | |
814 /* 1.000000000000000000000000000000000000000E0 */ | |
815 }; | |
816 | |
817 | |
818 /* Bessel function of the second kind, order one. */ | |
819 | |
820 __float128 | |
821 y1q (__float128 x) | |
822 { | |
823 __float128 xx, xinv, z, p, q, c, s, cc, ss; | |
824 | |
825 if (! finiteq (x)) | |
826 { | |
827 if (x != x) | |
828 return x; | |
829 else | |
830 return 0.0Q; | |
831 } | |
832 if (x <= 0.0Q) | |
833 { | |
834 if (x < 0.0Q) | |
835 return (zero / (zero * x)); | |
836 return -HUGE_VALQ + x; | |
837 } | |
838 xx = fabsq (x); | |
839 if (xx <= 2.0Q) | |
840 { | |
841 /* 0 <= x <= 2 */ | |
842 z = xx * xx; | |
843 p = xx * neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D); | |
844 p = -TWOOPI / xx + p; | |
845 p = TWOOPI * logq (x) * j1q (x) + p; | |
846 return p; | |
847 } | |
848 | |
849 xinv = 1.0Q / xx; | |
850 z = xinv * xinv; | |
851 if (xinv <= 0.25) | |
852 { | |
853 if (xinv <= 0.125) | |
854 { | |
855 if (xinv <= 0.0625) | |
856 { | |
857 p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); | |
858 q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); | |
859 } | |
860 else | |
861 { | |
862 p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); | |
863 q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); | |
864 } | |
865 } | |
866 else if (xinv <= 0.1875) | |
867 { | |
868 p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); | |
869 q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); | |
870 } | |
871 else | |
872 { | |
873 p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); | |
874 q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); | |
875 } | |
876 } /* .25 */ | |
877 else /* if (xinv <= 0.5) */ | |
878 { | |
879 if (xinv <= 0.375) | |
880 { | |
881 if (xinv <= 0.3125) | |
882 { | |
883 p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); | |
884 q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); | |
885 } | |
886 else | |
887 { | |
888 p = neval (z, P2r7_3r2N, NP2r7_3r2N) | |
889 / deval (z, P2r7_3r2D, NP2r7_3r2D); | |
890 q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) | |
891 / deval (z, Q2r7_3r2D, NQ2r7_3r2D); | |
892 } | |
893 } | |
894 else if (xinv <= 0.4375) | |
895 { | |
896 p = neval (z, P2r3_2r7N, NP2r3_2r7N) | |
897 / deval (z, P2r3_2r7D, NP2r3_2r7D); | |
898 q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) | |
899 / deval (z, Q2r3_2r7D, NQ2r3_2r7D); | |
900 } | |
901 else | |
902 { | |
903 p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); | |
904 q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); | |
905 } | |
906 } | |
907 p = 1.0Q + z * p; | |
908 q = z * q; | |
909 q = q * xinv + 0.375Q * xinv; | |
910 /* X = x - 3 pi/4 | |
911 cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4) | |
912 = 1/sqrt(2) * (-cos(x) + sin(x)) | |
913 sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4) | |
914 = -1/sqrt(2) * (sin(x) + cos(x)) | |
915 cf. Fdlibm. */ | |
916 sincosq (xx, &s, &c); | |
917 ss = -s - c; | |
918 cc = s - c; | |
919 z = cosq (xx + xx); | |
920 if ((s * c) > 0) | |
921 cc = z / ss; | |
922 else | |
923 ss = z / cc; | |
924 z = ONEOSQPI * (p * ss + q * cc) / sqrtq (xx); | |
925 return z; | |
926 } |