sysdeps/ieee754/ldbl-128ibm/e_j1l.c

   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    <http://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
  28 /* 1 / sqrt(pi) */
  29 static const long double ONEOSQPI = 5.6418958354775628694807945156077258584405E-1L;
  30 /* 2 / pi */
  31 static const long double TWOOPI = 6.3661977236758134307553505349005744813784E-1L;
  32 static const long double zero = 0;
  33
  34 /* J1(x) = .5x + x x^2 R(x^2)
  35    Peak relative error 1.9e-35
  36    0 <= x <= 2  */
  37 #define NJ0_2N 6
  38 static const long double J0_2N[NJ0_2N + 1] = {
  39  -5.943799577386942855938508697619735179660E16L,
  40   1.812087021305009192259946997014044074711E15L,
  41  -2.761698314264509665075127515729146460895E13L,
  42   2.091089497823600978949389109350658815972E11L,
  43  -8.546413231387036372945453565654130054307E8L,
  44   1.797229225249742247475464052741320612261E6L,
  45  -1.559552840946694171346552770008812083969E3L
  46 };
  47 #define NJ0_2D 6
  48 static const long double J0_2D[NJ0_2D + 1] = {
  49   9.510079323819108569501613916191477479397E17L,
  50   1.063193817503280529676423936545854693915E16L,
  51   5.934143516050192600795972192791775226920E13L,
  52   2.168000911950620999091479265214368352883E11L,
  53   5.673775894803172808323058205986256928794E8L,
  54   1.080329960080981204840966206372671147224E6L,
  55   1.411951256636576283942477881535283304912E3L,
  56  /* 1.000000000000000000000000000000000000000E0L */
  57 };
  58
  59 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
  60    0 <= 1/x <= .0625
  61    Peak relative error 3.6e-36  */
  62 #define NP16_IN 9
  63 static const long double P16_IN[NP16_IN + 1] = {
  64   5.143674369359646114999545149085139822905E-16L,
  65   4.836645664124562546056389268546233577376E-13L,
  66   1.730945562285804805325011561498453013673E-10L,
  67   3.047976856147077889834905908605310585810E-8L,
  68   2.855227609107969710407464739188141162386E-6L,
  69   1.439362407936705484122143713643023998457E-4L,
  70   3.774489768532936551500999699815873422073E-3L,
  71   4.723962172984642566142399678920790598426E-2L,
  72   2.359289678988743939925017240478818248735E-1L,
  73   3.032580002220628812728954785118117124520E-1L,
  74 };
  75 #define NP16_ID 9
  76 static const long double P16_ID[NP16_ID + 1] = {
  77   4.389268795186898018132945193912677177553E-15L,
  78   4.132671824807454334388868363256830961655E-12L,
  79   1.482133328179508835835963635130894413136E-9L,
  80   2.618941412861122118906353737117067376236E-7L,
  81   2.467854246740858470815714426201888034270E-5L,
  82   1.257192927368839847825938545925340230490E-3L,
  83   3.362739031941574274949719324644120720341E-2L,
  84   4.384458231338934105875343439265370178858E-1L,
  85   2.412830809841095249170909628197264854651E0L,
  86   4.176078204111348059102962617368214856874E0L,
  87  /* 1.000000000000000000000000000000000000000E0 */
  88 };
  89
  90 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
  91     0.0625 <= 1/x <= 0.125
  92     Peak relative error 1.9e-36  */
  93 #define NP8_16N 11
  94 static const long double P8_16N[NP8_16N + 1] = {
  95   2.984612480763362345647303274082071598135E-16L,
  96   1.923651877544126103941232173085475682334E-13L,
  97   4.881258879388869396043760693256024307743E-11L,
  98   6.368866572475045408480898921866869811889E-9L,
  99   4.684818344104910450523906967821090796737E-7L,
 100   2.005177298271593587095982211091300382796E-5L,
 101   4.979808067163957634120681477207147536182E-4L,
 102   6.946005761642579085284689047091173581127E-3L,
 103   5.074601112955765012750207555985299026204E-2L,
 104   1.698599455896180893191766195194231825379E-1L,
 105   1.957536905259237627737222775573623779638E-1L,
 106   2.991314703282528370270179989044994319374E-2L,
 107 };
 108 #define NP8_16D 10
 109 static const long double P8_16D[NP8_16D + 1] = {
 110   2.546869316918069202079580939942463010937E-15L,
 111   1.644650111942455804019788382157745229955E-12L,
 112   4.185430770291694079925607420808011147173E-10L,
 113   5.485331966975218025368698195861074143153E-8L,
 114   4.062884421686912042335466327098932678905E-6L,
 115   1.758139661060905948870523641319556816772E-4L,
 116   4.445143889306356207566032244985607493096E-3L,
 117   6.391901016293512632765621532571159071158E-2L,
 118   4.933040207519900471177016015718145795434E-1L,
 119   1.839144086168947712971630337250761842976E0L,
 120   2.715120873995490920415616716916149586579E0L,
 121  /* 1.000000000000000000000000000000000000000E0 */
 122 };
 123
 124 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
 125   0.125 <= 1/x <= 0.1875
 126   Peak relative error 1.3e-36  */
 127 #define NP5_8N 10
 128 static const long double P5_8N[NP5_8N + 1] = {
 129   2.837678373978003452653763806968237227234E-12L,
 130   9.726641165590364928442128579282742354806E-10L,
 131   1.284408003604131382028112171490633956539E-7L,
 132   8.524624695868291291250573339272194285008E-6L,
 133   3.111516908953172249853673787748841282846E-4L,
 134   6.423175156126364104172801983096596409176E-3L,
 135   7.430220589989104581004416356260692450652E-2L,
 136   4.608315409833682489016656279567605536619E-1L,
 137   1.396870223510964882676225042258855977512E0L,
 138   1.718500293904122365894630460672081526236E0L,
 139   5.465927698800862172307352821870223855365E-1L
 140 };
 141 #define NP5_8D 10
 142 static const long double P5_8D[NP5_8D + 1] = {
 143   2.421485545794616609951168511612060482715E-11L,
 144   8.329862750896452929030058039752327232310E-9L,
 145   1.106137992233383429630592081375289010720E-6L,
 146   7.405786153760681090127497796448503306939E-5L,
 147   2.740364785433195322492093333127633465227E-3L,
 148   5.781246470403095224872243564165254652198E-2L,
 149   6.927711353039742469918754111511109983546E-1L,
 150   4.558679283460430281188304515922826156690E0L,
 151   1.534468499844879487013168065728837900009E1L,
 152   2.313927430889218597919624843161569422745E1L,
 153   1.194506341319498844336768473218382828637E1L,
 154  /* 1.000000000000000000000000000000000000000E0 */
 155 };
 156
 157 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
 158    Peak relative error 1.4e-36
 159    0.1875 <= 1/x <= 0.25  */
 160 #define NP4_5N 10
 161 static const long double P4_5N[NP4_5N + 1] = {
 162   1.846029078268368685834261260420933914621E-10L,
 163   3.916295939611376119377869680335444207768E-8L,
 164   3.122158792018920627984597530935323997312E-6L,
 165   1.218073444893078303994045653603392272450E-4L,
 166   2.536420827983485448140477159977981844883E-3L,
 167   2.883011322006690823959367922241169171315E-2L,
 168   1.755255190734902907438042414495469810830E-1L,
 169   5.379317079922628599870898285488723736599E-1L,
 170   7.284904050194300773890303361501726561938E-1L,
 171   3.270110346613085348094396323925000362813E-1L,
 172   1.804473805689725610052078464951722064757E-2L,
 173 };
 174 #define NP4_5D 9
 175 static const long double P4_5D[NP4_5D + 1] = {
 176   1.575278146806816970152174364308980863569E-9L,
 177   3.361289173657099516191331123405675054321E-7L,
 178   2.704692281550877810424745289838790693708E-5L,
 179   1.070854930483999749316546199273521063543E-3L,
 180   2.282373093495295842598097265627962125411E-2L,
 181   2.692025460665354148328762368240343249830E-1L,
 182   1.739892942593664447220951225734811133759E0L,
 183   5.890727576752230385342377570386657229324E0L,
 184   9.517442287057841500750256954117735128153E0L,
 185   6.100616353935338240775363403030137736013E0L,
 186  /* 1.000000000000000000000000000000000000000E0 */
 187 };
 188
 189 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
 190    Peak relative error 3.0e-36
 191    0.25 <= 1/x <= 0.3125  */
 192 #define NP3r2_4N 9
 193 static const long double P3r2_4N[NP3r2_4N + 1] = {
 194   8.240803130988044478595580300846665863782E-8L,
 195   1.179418958381961224222969866406483744580E-5L,
 196   6.179787320956386624336959112503824397755E-4L,
 197   1.540270833608687596420595830747166658383E-2L,
 198   1.983904219491512618376375619598837355076E-1L,
 199   1.341465722692038870390470651608301155565E0L,
 200   4.617865326696612898792238245990854646057E0L,
 201   7.435574801812346424460233180412308000587E0L,
 202   4.671327027414635292514599201278557680420E0L,
 203   7.299530852495776936690976966995187714739E-1L,
 204 };
 205 #define NP3r2_4D 9
 206 static const long double P3r2_4D[NP3r2_4D + 1] = {
 207   7.032152009675729604487575753279187576521E-7L,
 208   1.015090352324577615777511269928856742848E-4L,
 209   5.394262184808448484302067955186308730620E-3L,
 210   1.375291438480256110455809354836988584325E-1L,
 211   1.836247144461106304788160919310404376670E0L,
 212   1.314378564254376655001094503090935880349E1L,
 213   4.957184590465712006934452500894672343488E1L,
 214   9.287394244300647738855415178790263465398E1L,
 215   7.652563275535900609085229286020552768399E1L,
 216   2.147042473003074533150718117770093209096E1L,
 217  /* 1.000000000000000000000000000000000000000E0 */
 218 };
 219
 220 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
 221    Peak relative error 1.0e-35
 222    0.3125 <= 1/x <= 0.375  */
 223 #define NP2r7_3r2N 9
 224 static const long double P2r7_3r2N[NP2r7_3r2N + 1] = {
 225   4.599033469240421554219816935160627085991E-7L,
 226   4.665724440345003914596647144630893997284E-5L,
 227   1.684348845667764271596142716944374892756E-3L,
 228   2.802446446884455707845985913454440176223E-2L,
 229   2.321937586453963310008279956042545173930E-1L,
 230   9.640277413988055668692438709376437553804E-1L,
 231   1.911021064710270904508663334033003246028E0L,
 232   1.600811610164341450262992138893970224971E0L,
 233   4.266299218652587901171386591543457861138E-1L,
 234   1.316470424456061252962568223251247207325E-2L,
 235 };
 236 #define NP2r7_3r2D 8
 237 static const long double P2r7_3r2D[NP2r7_3r2D + 1] = {
 238   3.924508608545520758883457108453520099610E-6L,
 239   4.029707889408829273226495756222078039823E-4L,
 240   1.484629715787703260797886463307469600219E-2L,
 241   2.553136379967180865331706538897231588685E-1L,
 242   2.229457223891676394409880026887106228740E0L,
 243   1.005708903856384091956550845198392117318E1L,
 244   2.277082659664386953166629360352385889558E1L,
 245   2.384726835193630788249826630376533988245E1L,
 246   9.700989749041320895890113781610939632410E0L,
 247  /* 1.000000000000000000000000000000000000000E0 */
 248 };
 249
 250 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
 251    Peak relative error 1.7e-36
 252    0.3125 <= 1/x <= 0.4375  */
 253 #define NP2r3_2r7N 9
 254 static const long double P2r3_2r7N[NP2r3_2r7N + 1] = {
 255   3.916766777108274628543759603786857387402E-6L,
 256   3.212176636756546217390661984304645137013E-4L,
 257   9.255768488524816445220126081207248947118E-3L,
 258   1.214853146369078277453080641911700735354E-1L,
 259   7.855163309847214136198449861311404633665E-1L,
 260   2.520058073282978403655488662066019816540E0L,
 261   3.825136484837545257209234285382183711466E0L,
 262   2.432569427554248006229715163865569506873E0L,
 263   4.877934835018231178495030117729800489743E-1L,
 264   1.109902737860249670981355149101343427885E-2L,
 265 };
 266 #define NP2r3_2r7D 8
 267 static const long double P2r3_2r7D[NP2r3_2r7D + 1] = {
 268   3.342307880794065640312646341190547184461E-5L,
 269   2.782182891138893201544978009012096558265E-3L,
 270   8.221304931614200702142049236141249929207E-2L,
 271   1.123728246291165812392918571987858010949E0L,
 272   7.740482453652715577233858317133423434590E0L,
 273   2.737624677567945952953322566311201919139E1L,
 274   4.837181477096062403118304137851260715475E1L,
 275   3.941098643468580791437772701093795299274E1L,
 276   1.245821247166544627558323920382547533630E1L,
 277  /* 1.000000000000000000000000000000000000000E0 */
 278 };
 279
 280 /* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2),
 281    Peak relative error 1.7e-35
 282    0.4375 <= 1/x <= 0.5  */
 283 #define NP2_2r3N 8
 284 static const long double P2_2r3N[NP2_2r3N + 1] = {
 285   3.397930802851248553545191160608731940751E-4L,
 286   2.104020902735482418784312825637833698217E-2L,
 287   4.442291771608095963935342749477836181939E-1L,
 288   4.131797328716583282869183304291833754967E0L,
 289   1.819920169779026500146134832455189917589E1L,
 290   3.781779616522937565300309684282401791291E1L,
 291   3.459605449728864218972931220783543410347E1L,
 292   1.173594248397603882049066603238568316561E1L,
 293   9.455702270242780642835086549285560316461E-1L,
 294 };
 295 #define NP2_2r3D 8
 296 static const long double P2_2r3D[NP2_2r3D + 1] = {
 297   2.899568897241432883079888249845707400614E-3L,
 298   1.831107138190848460767699919531132426356E-1L,
 299   3.999350044057883839080258832758908825165E0L,
 300   3.929041535867957938340569419874195303712E1L,
 301   1.884245613422523323068802689915538908291E2L,
 302   4.461469948819229734353852978424629815929E2L,
 303   5.004998753999796821224085972610636347903E2L,
 304   2.386342520092608513170837883757163414100E2L,
 305   3.791322528149347975999851588922424189957E1L,
 306  /* 1.000000000000000000000000000000000000000E0 */
 307 };
 308
 309 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 310    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 311    Peak relative error 8.0e-36
 312    0 <= 1/x <= .0625  */
 313 #define NQ16_IN 10
 314 static const long double Q16_IN[NQ16_IN + 1] = {
 315   -3.917420835712508001321875734030357393421E-18L,
 316   -4.440311387483014485304387406538069930457E-15L,
 317   -1.951635424076926487780929645954007139616E-12L,
 318   -4.318256438421012555040546775651612810513E-10L,
 319   -5.231244131926180765270446557146989238020E-8L,
 320   -3.540072702902043752460711989234732357653E-6L,
 321   -1.311017536555269966928228052917534882984E-4L,
 322   -2.495184669674631806622008769674827575088E-3L,
 323   -2.141868222987209028118086708697998506716E-2L,
 324   -6.184031415202148901863605871197272650090E-2L,
 325   -1.922298704033332356899546792898156493887E-2L,
 326 };
 327 #define NQ16_ID 9
 328 static const long double Q16_ID[NQ16_ID + 1] = {
 329   3.820418034066293517479619763498400162314E-17L,
 330   4.340702810799239909648911373329149354911E-14L,
 331   1.914985356383416140706179933075303538524E-11L,
 332   4.262333682610888819476498617261895474330E-9L,
 333   5.213481314722233980346462747902942182792E-7L,
 334   3.585741697694069399299005316809954590558E-5L,
 335   1.366513429642842006385029778105539457546E-3L,
 336   2.745282599850704662726337474371355160594E-2L,
 337   2.637644521611867647651200098449903330074E-1L,
 338   1.006953426110765984590782655598680488746E0L,
 339  /* 1.000000000000000000000000000000000000000E0 */
 340  };
 341
 342 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 343    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 344    Peak relative error 1.9e-36
 345    0.0625 <= 1/x <= 0.125  */
 346 #define NQ8_16N 11
 347 static const long double Q8_16N[NQ8_16N + 1] = {
 348   -2.028630366670228670781362543615221542291E-17L,
 349   -1.519634620380959966438130374006858864624E-14L,
 350   -4.540596528116104986388796594639405114524E-12L,
 351   -7.085151756671466559280490913558388648274E-10L,
 352   -6.351062671323970823761883833531546885452E-8L,
 353   -3.390817171111032905297982523519503522491E-6L,
 354   -1.082340897018886970282138836861233213972E-4L,
 355   -2.020120801187226444822977006648252379508E-3L,
 356   -2.093169910981725694937457070649605557555E-2L,
 357   -1.092176538874275712359269481414448063393E-1L,
 358   -2.374790947854765809203590474789108718733E-1L,
 359   -1.365364204556573800719985118029601401323E-1L,
 360 };
 361 #define NQ8_16D 11
 362 static const long double Q8_16D[NQ8_16D + 1] = {
 363   1.978397614733632533581207058069628242280E-16L,
 364   1.487361156806202736877009608336766720560E-13L,
 365   4.468041406888412086042576067133365913456E-11L,
 366   7.027822074821007443672290507210594648877E-9L,
 367   6.375740580686101224127290062867976007374E-7L,
 368   3.466887658320002225888644977076410421940E-5L,
 369   1.138625640905289601186353909213719596986E-3L,
 370   2.224470799470414663443449818235008486439E-2L,
 371   2.487052928527244907490589787691478482358E-1L,
 372   1.483927406564349124649083853892380899217E0L,
 373   4.182773513276056975777258788903489507705E0L,
 374   4.419665392573449746043880892524360870944E0L,
 375  /* 1.000000000000000000000000000000000000000E0 */
 376 };
 377
 378 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 379    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 380    Peak relative error 1.5e-35
 381    0.125 <= 1/x <= 0.1875  */
 382 #define NQ5_8N 10
 383 static const long double Q5_8N[NQ5_8N + 1] = {
 384   -3.656082407740970534915918390488336879763E-13L,
 385   -1.344660308497244804752334556734121771023E-10L,
 386   -1.909765035234071738548629788698150760791E-8L,
 387   -1.366668038160120210269389551283666716453E-6L,
 388   -5.392327355984269366895210704976314135683E-5L,
 389   -1.206268245713024564674432357634540343884E-3L,
 390   -1.515456784370354374066417703736088291287E-2L,
 391   -1.022454301137286306933217746545237098518E-1L,
 392   -3.373438906472495080504907858424251082240E-1L,
 393   -4.510782522110845697262323973549178453405E-1L,
 394   -1.549000892545288676809660828213589804884E-1L,
 395 };
 396 #define NQ5_8D 10
 397 static const long double Q5_8D[NQ5_8D + 1] = {
 398   3.565550843359501079050699598913828460036E-12L,
 399   1.321016015556560621591847454285330528045E-9L,
 400   1.897542728662346479999969679234270605975E-7L,
 401   1.381720283068706710298734234287456219474E-5L,
 402   5.599248147286524662305325795203422873725E-4L,
 403   1.305442352653121436697064782499122164843E-2L,
 404   1.750234079626943298160445750078631894985E-1L,
 405   1.311420542073436520965439883806946678491E0L,
 406   5.162757689856842406744504211089724926650E0L,
 407   9.527760296384704425618556332087850581308E0L,
 408   6.604648207463236667912921642545100248584E0L,
 409  /* 1.000000000000000000000000000000000000000E0 */
 410 };
 411
 412 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 413    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 414    Peak relative error 1.3e-35
 415    0.1875 <= 1/x <= 0.25  */
 416 #define NQ4_5N 10
 417 static const long double Q4_5N[NQ4_5N + 1] = {
 418   -4.079513568708891749424783046520200903755E-11L,
 419   -9.326548104106791766891812583019664893311E-9L,
 420   -8.016795121318423066292906123815687003356E-7L,
 421   -3.372350544043594415609295225664186750995E-5L,
 422   -7.566238665947967882207277686375417983917E-4L,
 423   -9.248861580055565402130441618521591282617E-3L,
 424   -6.033106131055851432267702948850231270338E-2L,
 425   -1.966908754799996793730369265431584303447E-1L,
 426   -2.791062741179964150755788226623462207560E-1L,
 427   -1.255478605849190549914610121863534191666E-1L,
 428   -4.320429862021265463213168186061696944062E-3L,
 429 };
 430 #define NQ4_5D 9
 431 static const long double Q4_5D[NQ4_5D + 1] = {
 432   3.978497042580921479003851216297330701056E-10L,
 433   9.203304163828145809278568906420772246666E-8L,
 434   8.059685467088175644915010485174545743798E-6L,
 435   3.490187375993956409171098277561669167446E-4L,
 436   8.189109654456872150100501732073810028829E-3L,
 437   1.072572867311023640958725265762483033769E-1L,
 438   7.790606862409960053675717185714576937994E-1L,
 439   3.016049768232011196434185423512777656328E0L,
 440   5.722963851442769787733717162314477949360E0L,
 441   4.510527838428473279647251350931380867663E0L,
 442  /* 1.000000000000000000000000000000000000000E0 */
 443 };
 444
 445 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 446    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 447    Peak relative error 2.1e-35
 448    0.25 <= 1/x <= 0.3125  */
 449 #define NQ3r2_4N 9
 450 static const long double Q3r2_4N[NQ3r2_4N + 1] = {
 451   -1.087480809271383885936921889040388133627E-8L,
 452   -1.690067828697463740906962973479310170932E-6L,
 453   -9.608064416995105532790745641974762550982E-5L,
 454   -2.594198839156517191858208513873961837410E-3L,
 455   -3.610954144421543968160459863048062977822E-2L,
 456   -2.629866798251843212210482269563961685666E-1L,
 457   -9.709186825881775885917984975685752956660E-1L,
 458   -1.667521829918185121727268867619982417317E0L,
 459   -1.109255082925540057138766105229900943501E0L,
 460   -1.812932453006641348145049323713469043328E-1L,
 461 };
 462 #define NQ3r2_4D 9
 463 static const long double Q3r2_4D[NQ3r2_4D + 1] = {
 464   1.060552717496912381388763753841473407026E-7L,
 465   1.676928002024920520786883649102388708024E-5L,
 466   9.803481712245420839301400601140812255737E-4L,
 467   2.765559874262309494758505158089249012930E-2L,
 468   4.117921827792571791298862613287549140706E-1L,
 469   3.323769515244751267093378361930279161413E0L,
 470   1.436602494405814164724810151689705353670E1L,
 471   3.163087869617098638064881410646782408297E1L,
 472   3.198181264977021649489103980298349589419E1L,
 473   1.203649258862068431199471076202897823272E1L,
 474  /* 1.000000000000000000000000000000000000000E0  */
 475 };
 476
 477 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 478    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 479    Peak relative error 1.6e-36
 480    0.3125 <= 1/x <= 0.375  */
 481 #define NQ2r7_3r2N 9
 482 static const long double Q2r7_3r2N[NQ2r7_3r2N + 1] = {
 483   -1.723405393982209853244278760171643219530E-7L,
 484   -2.090508758514655456365709712333460087442E-5L,
 485   -9.140104013370974823232873472192719263019E-4L,
 486   -1.871349499990714843332742160292474780128E-2L,
 487   -1.948930738119938669637865956162512983416E-1L,
 488   -1.048764684978978127908439526343174139788E0L,
 489   -2.827714929925679500237476105843643064698E0L,
 490   -3.508761569156476114276988181329773987314E0L,
 491   -1.669332202790211090973255098624488308989E0L,
 492   -1.930796319299022954013840684651016077770E-1L,
 493 };
 494 #define NQ2r7_3r2D 9
 495 static const long double Q2r7_3r2D[NQ2r7_3r2D + 1] = {
 496   1.680730662300831976234547482334347983474E-6L,
 497   2.084241442440551016475972218719621841120E-4L,
 498   9.445316642108367479043541702688736295579E-3L,
 499   2.044637889456631896650179477133252184672E-1L,
 500   2.316091982244297350829522534435350078205E0L,
 501   1.412031891783015085196708811890448488865E1L,
 502   4.583830154673223384837091077279595496149E1L,
 503   7.549520609270909439885998474045974122261E1L,
 504   5.697605832808113367197494052388203310638E1L,
 505   1.601496240876192444526383314589371686234E1L,
 506   /* 1.000000000000000000000000000000000000000E0 */
 507 };
 508
 509 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 510    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 511    Peak relative error 9.5e-36
 512    0.375 <= 1/x <= 0.4375  */
 513 #define NQ2r3_2r7N 9
 514 static const long double Q2r3_2r7N[NQ2r3_2r7N + 1] = {
 515   -8.603042076329122085722385914954878953775E-7L,
 516   -7.701746260451647874214968882605186675720E-5L,
 517   -2.407932004380727587382493696877569654271E-3L,
 518   -3.403434217607634279028110636919987224188E-2L,
 519   -2.348707332185238159192422084985713102877E-1L,
 520   -7.957498841538254916147095255700637463207E-1L,
 521   -1.258469078442635106431098063707934348577E0L,
 522   -8.162415474676345812459353639449971369890E-1L,
 523   -1.581783890269379690141513949609572806898E-1L,
 524   -1.890595651683552228232308756569450822905E-3L,
 525 };
 526 #define NQ2r3_2r7D 8
 527 static const long double Q2r3_2r7D[NQ2r3_2r7D + 1] = {
 528   8.390017524798316921170710533381568175665E-6L,
 529   7.738148683730826286477254659973968763659E-4L,
 530   2.541480810958665794368759558791634341779E-2L,
 531   3.878879789711276799058486068562386244873E-1L,
 532   3.003783779325811292142957336802456109333E0L,
 533   1.206480374773322029883039064575464497400E1L,
 534   2.458414064785315978408974662900438351782E1L,
 535   2.367237826273668567199042088835448715228E1L,
 536   9.231451197519171090875569102116321676763E0L,
 537  /* 1.000000000000000000000000000000000000000E0 */
 538 };
 539
 540 /* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
 541    Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)),
 542    Peak relative error 1.4e-36
 543    0.4375 <= 1/x <= 0.5  */
 544 #define NQ2_2r3N 9
 545 static const long double Q2_2r3N[NQ2_2r3N + 1] = {
 546   -5.552507516089087822166822364590806076174E-6L,
 547   -4.135067659799500521040944087433752970297E-4L,
 548   -1.059928728869218962607068840646564457980E-2L,
 549   -1.212070036005832342565792241385459023801E-1L,
 550   -6.688350110633603958684302153362735625156E-1L,
 551   -1.793587878197360221340277951304429821582E0L,
 552   -2.225407682237197485644647380483725045326E0L,
 553   -1.123402135458940189438898496348239744403E0L,
 554   -1.679187241566347077204805190763597299805E-1L,
 555   -1.458550613639093752909985189067233504148E-3L,
 556 };
 557 #define NQ2_2r3D 8
 558 static const long double Q2_2r3D[NQ2_2r3D + 1] = {
 559   5.415024336507980465169023996403597916115E-5L,
 560   4.179246497380453022046357404266022870788E-3L,
 561   1.136306384261959483095442402929502368598E-1L,
 562   1.422640343719842213484515445393284072830E0L,
 563   8.968786703393158374728850922289204805764E0L,
 564   2.914542473339246127533384118781216495934E1L,
 565   4.781605421020380669870197378210457054685E1L,
 566   3.693865837171883152382820584714795072937E1L,
 567   1.153220502744204904763115556224395893076E1L,
 568   /* 1.000000000000000000000000000000000000000E0 */
 569 };
 570
 571
 572 /* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
 573
 574 static long double
 575 neval (long double x, const long double *p, int n)
 576 {
 577   long double y;
 578
 579   p += n;
 580   y = *p--;
 581   do
 582     {
 583       y = y * x + *p--;
 584     }
 585   while (--n > 0);
 586   return y;
 587 }
 588
 589
 590 /* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
 591
 592 static long double
 593 deval (long double x, const long double *p, int n)
 594 {
 595   long double y;
 596
 597   p += n;
 598   y = x + *p--;
 599   do
 600     {
 601       y = y * x + *p--;
 602     }
 603   while (--n > 0);
 604   return y;
 605 }
 606
 607
 608 /* Bessel function of the first kind, order one.  */
 609
 610 long double
 611 __ieee754_j1l (long double x)
 612 {
 613   long double xx, xinv, z, p, q, c, s, cc, ss;
 614
 615   if (! isfinite (x))
 616     {
 617       if (x != x)
 618         return x + x;
 619       else
 620         return 0;
 621     }
 622   if (x == 0)
 623     return x;
 624   xx = fabsl (x);
 625   if (xx <= 0x1p-58L)
 626     {
 627       long double ret = x * 0.5L;
 628       math_check_force_underflow (ret);
 629       if (ret == 0)
 630         __set_errno (ERANGE);
 631       return ret;
 632     }
 633   if (xx <= 2)
 634     {
 635       /* 0 <= x <= 2 */
 636       z = xx * xx;
 637       p = xx * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D);
 638       p += 0.5L * xx;
 639       if (x < 0)
 640         p = -p;
 641       return p;
 642     }
 643
 644   /* X = x - 3 pi/4
 645      cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4)
 646      = 1/sqrt(2) * (-cos(x) + sin(x))
 647      sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4)
 648      = -1/sqrt(2) * (sin(x) + cos(x))
 649      cf. Fdlibm.  */
 650   __sincosl (xx, &s, &c);
 651   ss = -s - c;
 652   cc = s - c;
 653   if (xx <= LDBL_MAX / 2)
 654     {
 655       z = __cosl (xx + xx);
 656       if ((s * c) > 0)
 657         cc = z / ss;
 658       else
 659         ss = z / cc;
 660     }
 661
 662   if (xx > 0x1p256L)
 663     {
 664       z = ONEOSQPI * cc / sqrtl (xx);
 665       if (x < 0)
 666         z = -z;
 667       return z;
 668     }
 669
 670   xinv = 1 / xx;
 671   z = xinv * xinv;
 672   if (xinv <= 0.25)
 673     {
 674       if (xinv <= 0.125)
 675         {
 676           if (xinv <= 0.0625)
 677             {
 678               p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
 679               q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
 680             }
 681           else
 682             {
 683               p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
 684               q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
 685             }
 686         }
 687       else if (xinv <= 0.1875)
 688         {
 689           p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
 690           q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
 691         }
 692       else
 693         {
 694           p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
 695           q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
 696         }
 697     }                           /* .25 */
 698   else /* if (xinv <= 0.5) */
 699     {
 700       if (xinv <= 0.375)
 701         {
 702           if (xinv <= 0.3125)
 703             {
 704               p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
 705               q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
 706             }
 707           else
 708             {
 709               p = neval (z, P2r7_3r2N, NP2r7_3r2N)
 710                   / deval (z, P2r7_3r2D, NP2r7_3r2D);
 711               q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
 712                   / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
 713             }
 714         }
 715       else if (xinv <= 0.4375)
 716         {
 717           p = neval (z, P2r3_2r7N, NP2r3_2r7N)
 718               / deval (z, P2r3_2r7D, NP2r3_2r7D);
 719           q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
 720               / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
 721         }
 722       else
 723         {
 724           p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
 725           q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
 726         }
 727     }
 728   p = 1 + z * p;
 729   q = z * q;
 730   q = q * xinv + 0.375L * xinv;
 731   z = ONEOSQPI * (p * cc - q * ss) / sqrtl (xx);
 732   if (x < 0)
 733     z = -z;
 734   return z;
 735 }
 736 strong_alias (__ieee754_j1l, __j1l_finite)
 737
 738
 739 /* Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2)
 740    Peak relative error 6.2e-38
 741    0 <= x <= 2   */
 742 #define NY0_2N 7
 743 static const long double Y0_2N[NY0_2N + 1] = {
 744   -6.804415404830253804408698161694720833249E19L,
 745   1.805450517967019908027153056150465849237E19L,
 746   -8.065747497063694098810419456383006737312E17L,
 747   1.401336667383028259295830955439028236299E16L,
 748   -1.171654432898137585000399489686629680230E14L,
 749   5.061267920943853732895341125243428129150E11L,
 750   -1.096677850566094204586208610960870217970E9L,
 751   9.541172044989995856117187515882879304461E5L,
 752 };
 753 #define NY0_2D 7
 754 static const long double Y0_2D[NY0_2D + 1] = {
 755   3.470629591820267059538637461549677594549E20L,
 756   4.120796439009916326855848107545425217219E18L,
 757   2.477653371652018249749350657387030814542E16L,
 758   9.954678543353888958177169349272167762797E13L,
 759   2.957927997613630118216218290262851197754E11L,
 760   6.748421382188864486018861197614025972118E8L,
 761   1.173453425218010888004562071020305709319E6L,
 762   1.450335662961034949894009554536003377187E3L,
 763   /* 1.000000000000000000000000000000000000000E0 */
 764 };
 765
 766
 767 /* Bessel function of the second kind, order one.  */
 768
 769 long double
 770 __ieee754_y1l (long double x)
 771 {
 772   long double xx, xinv, z, p, q, c, s, cc, ss;
 773
 774   if (! isfinite (x))
 775     return 1 / (x + x * x);
 776   if (x <= 0)
 777     {
 778       if (x < 0)
 779         return (zero / (zero * x));
 780       return -1 / zero; /* -inf and divide by zero exception.  */
 781     }
 782   xx = fabsl (x);
 783   if (xx <= 0x1p-114)
 784     {
 785       z = -TWOOPI / x;
 786       if (isinf (z))
 787         __set_errno (ERANGE);
 788       return z;
 789     }
 790   if (xx <= 2)
 791     {
 792       /* 0 <= x <= 2 */
 793       SET_RESTORE_ROUNDL (FE_TONEAREST);
 794       z = xx * xx;
 795       p = xx * neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D);
 796       p = -TWOOPI / xx + p;
 797       p = TWOOPI * __ieee754_logl (x) * __ieee754_j1l (x) + p;
 798       return p;
 799     }
 800
 801   /* X = x - 3 pi/4
 802      cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4)
 803      = 1/sqrt(2) * (-cos(x) + sin(x))
 804      sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4)
 805      = -1/sqrt(2) * (sin(x) + cos(x))
 806      cf. Fdlibm.  */
 807   __sincosl (xx, &s, &c);
 808   ss = -s - c;
 809   cc = s - c;
 810   if (xx <= LDBL_MAX / 2)
 811     {
 812       z = __cosl (xx + xx);
 813       if ((s * c) > 0)
 814         cc = z / ss;
 815       else
 816         ss = z / cc;
 817     }
 818
 819   if (xx > 0x1p256L)
 820     return ONEOSQPI * ss / sqrtl (xx);
 821
 822   xinv = 1 / xx;
 823   z = xinv * xinv;
 824   if (xinv <= 0.25)
 825     {
 826       if (xinv <= 0.125)
 827         {
 828           if (xinv <= 0.0625)
 829             {
 830               p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
 831               q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
 832             }
 833           else
 834             {
 835               p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
 836               q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
 837             }
 838         }
 839       else if (xinv <= 0.1875)
 840         {
 841           p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
 842           q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
 843         }
 844       else
 845         {
 846           p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
 847           q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
 848         }
 849     }                           /* .25 */
 850   else /* if (xinv <= 0.5) */
 851     {
 852       if (xinv <= 0.375)
 853         {
 854           if (xinv <= 0.3125)
 855             {
 856               p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
 857               q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
 858             }
 859           else
 860             {
 861               p = neval (z, P2r7_3r2N, NP2r7_3r2N)
 862                   / deval (z, P2r7_3r2D, NP2r7_3r2D);
 863               q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
 864                   / deval (z, Q2r7_3r2D, NQ2r7_3r2D);
 865             }
 866         }
 867       else if (xinv <= 0.4375)
 868         {
 869           p = neval (z, P2r3_2r7N, NP2r3_2r7N)
 870               / deval (z, P2r3_2r7D, NP2r3_2r7D);
 871           q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
 872               / deval (z, Q2r3_2r7D, NQ2r3_2r7D);
 873         }
 874       else
 875         {
 876           p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
 877           q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
 878         }
 879     }
 880   p = 1 + z * p;
 881   q = z * q;
 882   q = q * xinv + 0.375L * xinv;
 883   z = ONEOSQPI * (p * ss + q * cc) / sqrtl (xx);
 884   return z;
 885 }
 886 strong_alias (__ieee754_y1l, __y1l_finite)