usr/src/lib/libm/common/LD/j0l.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21
  22 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29
  30 /*
  31  * Floating point Bessel's function of the first and second kinds
  32  * of order zero: j0(x),y0(x);
  33  *
  34  * Special cases:
  35  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  36  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  37  */
  38
  39 #pragma weak __j0l = j0l
  40 #pragma weak __y0l = y0l
  41
  42 #include "libm.h"
  43
  44 #include "longdouble.h"
  45
  46 #include <math.h>
  47 #if defined(__SUNPRO_C)
  48 #include <sunmath.h>
  49 #endif
  50
  51 #define GENERIC long double
  52 static GENERIC
  53 zero    = 0.0L,
  54 small   = 1.0e-9L,
  55 tiny    = 1.0e-38L,
  56 one     = 1.0L,
  57 five    = 5.0L,
  58 eight   = 8.0L,
  59 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
  60 tpi     = 0.636619772367581343075535053490057448L;
  61
  62 static GENERIC pzero(), qzero();
  63 static GENERIC r0[7] = {
  64   -2.499999999999999999999999999999998934492e-0001L,
  65    1.272657927360049786327618451133763714880e-0002L,
  66   -2.694499763712963276900636693400659600898e-0004L,
  67    2.724877475058977576903234070919616447883e-0006L,
  68   -1.432617103214330236967477495393076320281e-0008L,
  69    3.823248804080079168706683540513792224471e-0011L,
  70   -4.183174277567983647337568504286313665065e-0014L,
  71 };
  72 static GENERIC s0[7] = {
  73    1.0e0L,
  74    1.159368290559800854689526195462884666395e-0002L,
  75    6.629397597394973383009743876169946772559e-0005L,
  76    2.426779981394054406305431142501735094340e-0007L,
  77    6.097663491248511069094400469635449749883e-0010L,
  78    1.017019133340929220238747413216052224036e-0012L,
  79    9.012593179306197579518374581969371278481e-0016L,
  80 };
  81
  82 GENERIC
  83 j0l(x) GENERIC x; {
  84         GENERIC z, s, c, ss, cc, r, u, v;
  85         int i;
  86
  87         if (isnanl(x))
  88                 return (x+x);
  89         x = fabsl(x);
  90         if (x > 1.28L) {
  91                 if (!finitel(x))
  92                         return (zero);
  93                 s = sinl(x);
  94                 c = cosl(x);
  95         /*
  96          * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
  97          * where x0 = x-pi/4
  98          *      Better formula:
  99          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 100          *                      = 1/sqrt(2) * (cos(x) + sin(x))
 101          *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
 102          *                      = 1/sqrt(2) * (sin(x) - cos(x))
 103          * To avoid cancellation, use
 104          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 105          * to compute the worse one.
 106          */
 107                 if (x > 1.0e2450L) {    /* x+x may overflow */
 108                         ss = s-c;
 109                         cc = s+c;
 110                 } else if (signbitl(s) != signbitl(c)) {
 111                         ss = s - c;
 112                         cc = -cosl(x+x)/ss;
 113                 } else {
 114                         cc = s + c;
 115                         ss = -cosl(x+x)/cc;
 116                 }
 117         /*
 118          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 119          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 120          */
 121                 if (x > 1.0e120L)
 122                         return (invsqrtpi*cc)/sqrtl(x);
 123                 u = pzero(x); v = qzero(x);
 124                 return (invsqrtpi*(u*cc-v*ss)/sqrtl(x));
 125         }
 126         if (x <= small) {
 127             if (x <= tiny)
 128                         return (one-x);
 129             else
 130                         return (one-x*x*0.25L);
 131         }
 132         z = x*x;
 133         r = r0[6]; s = s0[6];
 134         for (i = 5; i >= 0; i--) {
 135             r = r*z + r0[i];
 136             s = s*z + s0[i];
 137         }
 138         return (one+z*(r/s));
 139 }
 140
 141 static GENERIC u0[8] = {
 142   -7.380429510868722527434392794848301631220e-0002L,
 143    1.766855559625940791857536949301981816513e-0001L,
 144   -1.386470722701047923235553251240162839408e-0002L,
 145    3.520149242724811578636970811631224862615e-0004L,
 146   -3.978599663243790049853642275624951870025e-0006L,
 147    2.228801153263957224547222556806915479763e-0008L,
 148   -6.121246764298785018658597179498837316177e-0011L,
 149    6.677103629722678833475965810525587396596e-0014L,
 150 };
 151 static GENERIC v0[8] = {
 152    1.0e0L,
 153    1.247164416539111311571676766127767127970e-0002L,
 154    7.829144749639791500052900281489367443576e-0005L,
 155    3.247126540422245330511218321013360336606e-0007L,
 156    9.750516724789499678567062572549568447869e-0010L,
 157    2.156713223173591212250543390258458098776e-0012L,
 158    3.322169561597890004231482431236452752624e-0015L,
 159    2.821213295314000924252226486305726805093e-0018L,
 160 };
 161
 162 GENERIC
 163 y0l(x) GENERIC x; {
 164         GENERIC z, d, s, c, ss, cc, u, v;
 165         int i;
 166
 167         if (isnanl(x))
 168                 return (x+x);
 169         if (x <= zero) {
 170                 if (x == zero)
 171                     d = -one/(x-x);
 172                 else
 173                     d = zero/(x-x);
 174         }
 175         if (x > 1.28L) {
 176                 if (!finitel(x))
 177                         return (zero);
 178                 s = sinl(x);
 179                 c = cosl(x);
 180         /*
 181          * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
 182          * where x0 = x-pi/4
 183          *      Better formula:
 184          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 185          *                      = 1/sqrt(2) * (cos(x) + sin(x))
 186          *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
 187          *                      = 1/sqrt(2) * (sin(x) - cos(x))
 188          * To avoid cancellation, use
 189          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 190          * to compute the worse one.
 191          */
 192                 if (x > 1.0e2450L) {    /* x+x may overflow */
 193                         ss = s-c;
 194                         cc = s+c;
 195                 } else if (signbitl(s) != signbitl(c)) {
 196                         ss = s - c;
 197                         cc = -cosl(x+x)/ss;
 198                 } else {
 199                         cc = s + c;
 200                         ss = -cosl(x+x)/cc;
 201                 }
 202         /*
 203          * j0(x) = 1/sqrt(pi*x) * (P(0,x)*cc - Q(0,x)*ss)
 204          * y0(x) = 1/sqrt(pi*x) * (P(0,x)*ss + Q(0,x)*cc)
 205          */
 206                 if (x > 1.0e120L)
 207                         return (invsqrtpi*ss)/sqrtl(x);
 208                 return (invsqrtpi*(pzero(x)*ss+qzero(x)*cc)/sqrtl(x));
 209
 210         }
 211         if (x <= tiny) {
 212             return (u0[0] + tpi*logl(x));
 213         }
 214         z = x*x;
 215         u = u0[7]; v = v0[7];
 216         for (i = 6; i >= 0; i--) {
 217             u = u*z + u0[i];
 218             v = v*z + v0[i];
 219         }
 220         return (u/v + tpi*(j0l(x)*logl(x)));
 221 }
 222
 223 static GENERIC pr0[12] = {      /* [16 -- inf] */
 224    9.999999999999999999999999999999999997515e-0001L,
 225    1.065981615377273376425365823967550598358e+0003L,
 226    4.390991200927588978306374718984240719130e+0005L,
 227    9.072086218607986711847069407339321363103e+0007L,
 228    1.022552886177375367408408501046461671528e+0010L,
 229    6.420766912243658241570635854089597269031e+0011L,
 230    2.206451725126933913591080211081242266908e+0013L,
 231    3.928369596816895077363705478743346298368e+0014L,
 232    3.258159928874124597286701119721482876596e+0015L,
 233    1.025715808134188978860679130140685101348e+0016L,
 234    7.537170874795721255796001687024031280685e+0015L,
 235   -1.579413901450157332307745586004207687796e+0014L,
 236 };
 237 static GENERIC ps0[11] = {
 238    1.0e0L,
 239    1.066051927877273376425365823967550512687e+0003L,
 240    4.391739647168381592399173804329266353038e+0005L,
 241    9.075162261801343671805658294123888867884e+0007L,
 242    1.023186118519904751819581912075985995058e+0010L,
 243    6.427861860414223746340515376512730275061e+0011L,
 244    2.210861503237823589735481303627993406235e+0013L,
 245    3.943247335784292905915956840901818177989e+0014L,
 246    3.283720976777545142150200110647270004481e+0015L,
 247    1.045346918812754048903645641538728986759e+0016L,
 248    8.043455468065618900750599584291193680463e+0015L,
 249 };
 250 static GENERIC pr1[12] = {      /* [8 -- 16] */
 251    9.999999999999999999999784422701108683618e-0001L,
 252    6.796098532948334207755488692777907062894e+0002L,
 253    1.840036112605722168824530758797169836042e+0005L,
 254    2.598490483191916637264894340635847598122e+0007L,
 255    2.105774863242707025525730249472054578523e+0009L,
 256    1.015822044230542426666314997796944979959e+0011L,
 257    2.931557457008110436764077699944189071875e+0012L,
 258    4.962885121125457633655259224179322808824e+0013L,
 259    4.705424055148223269155430598563351566279e+0014L,
 260    2.294439854910747229152056080910427001110e+0015L,
 261    4.905531843137486691500950019322475458629e+0015L,
 262    3.187543169710339218793442542845735994565e+0015L,
 263 };
 264 static GENERIC ps1[14] = {
 265    1.0e0L,
 266    6.796801657948334207754571576066758180288e+0002L,
 267    1.840512891201300567325421059826676366447e+0005L,
 268    2.599777028312918975306252167127695075221e+0007L,
 269    2.107582572771047636846811284634244892537e+0009L,
 270    1.017275794694156108975782763889979940348e+0011L,
 271    2.938487645192463845428059755454762316011e+0012L,
 272    4.982512164735557054521042916182317924466e+0013L,
 273    4.737639900153703274792677468264564361437e+0014L,
 274    2.323398719123742743524249528275097100646e+0015L,
 275    5.033419107069210577868909797896984419391e+0015L,
 276    3.409036105931068609601317076759804716059e+0015L,
 277    7.505655364352679737585745147753521662166e+0013L,
 278   -9.976837153983688250780198248297109118313e+0012L,
 279 };
 280 static GENERIC pr2[12] = {      /* [5 -- 8 ] */
 281    9.999999999999999937857236789277366320220e-0001L,
 282    3.692848765268649571651602420376358849214e+0002L,
 283    5.373022067535476576926715900057760985410e+0004L,
 284    4.038738891191314969971504035057219430725e+0006L,
 285    1.728285706306940523397385566659762646999e+0008L,
 286    4.375400819645889911158688737206054788534e+0009L,
 287    6.598950418204912408375591217782088567076e+0010L,
 288    5.827182039183238492480275401520072793783e+0011L,
 289    2.884222642913492390887572414999490975844e+0012L,
 290    7.373278873797767721932837830628688632775e+0012L,
 291    8.338295457568973761205077964397969230489e+0012L,
 292    2.911383183467288345772308817209806922143e+0012L,
 293 };
 294 static GENERIC ps2[14] = {
 295    1.0e0L,
 296    3.693551890268649477288896267171993213102e+0002L,
 297    5.375607880998361502474715133828068514297e+0004L,
 298    4.042477764024108249744998862572786367328e+0006L,
 299    1.731069838737016956685839588670132939513e+0008L,
 300    4.387147674049898778738226585935491417728e+0009L,
 301    6.628058659620653765349556940567715258165e+0010L,
 302    5.869659904164177740471685856367322160664e+0011L,
 303    2.919839445622817017058977559638969436383e+0012L,
 304    7.535314897696671402628203718612309253907e+0012L,
 305    8.696355561452933775773309859748610658935e+0012L,
 306    3.216155103141537221173601557697083216257e+0012L,
 307    4.756857081068942248246880159213789086363e+0010L,
 308   -3.496356619666608032231074866481472824067e+0009L,
 309 };
 310 static GENERIC pr3[13] = {      /* [3.5 -- 5 ] */
 311    9.999999999999916693107285612398196588247e-0001L,
 312    2.263975921282917721194425320484974336945e+0002L,
 313    1.994358386744245848889492762781484199966e+0004L,
 314    8.980067458430542243559962493831661323168e+0005L,
 315    2.282213787521372663705567756420087553508e+0007L,
 316    3.409784374889063618250288699908375135923e+0008L,
 317    3.024380857401448589254343517589811711108e+0009L,
 318    1.571110368046740246895071721443082286379e+0010L,
 319    4.603187020243604632153685300463160593768e+0010L,
 320    7.087196453409712719449549280664058793403e+0010L,
 321    5.046196021776346356803687409644239065041e+0010L,
 322    1.287758439080165765709154276618854799932e+0010L,
 323    5.900679773415023433787846658096813590784e+0008L,
 324 };
 325 static GENERIC ps3[13] = {
 326    1.0e0L,
 327    2.264679046282855061328604619231774747116e+0002L,
 328    1.995939523988944553755653255389812103448e+0004L,
 329    8.993853144706348727038389967490183236820e+0005L,
 330    2.288326099634588843906989983704795468773e+0007L,
 331    3.424967100255240885169240956804790118282e+0008L,
 332    3.046311797972463991368023759640028910016e+0009L,
 333    1.589614961932826812790222479700797224003e+0010L,
 334    4.692406624527744816497089139325073939927e+0010L,
 335    7.320486495902008912866462849073108323948e+0010L,
 336    5.345945972828978289935309597742981360994e+0010L,
 337    1.444033091910423754121309915092247171008e+0010L,
 338    7.987714685115314668378957273824383610525e+0008L,
 339 };
 340 static GENERIC pr4[13] = {      /* [2.5, 3.5] */
 341    9.999999999986736677961118722747757712260e-0001L,
 342    1.453824980703800559037873123568378845663e+0002L,
 343    8.097327216430682288267610447006508661032e+0003L,
 344    2.273847252038264370231169686380192662135e+0005L,
 345    3.561056728046211111354759998976985449622e+0006L,
 346    3.244933588800096378434627029369680378599e+0007L,
 347    1.740112392860717950376210038908476792588e+0008L,
 348    5.426170187455893285197878563881579269524e+0008L,
 349    9.490107486454362321004377336020526281371e+0008L,
 350    8.688872439428470049801714121070005313806e+0008L,
 351    3.673315853166437222811910656900123215515e+0008L,
 352    5.577770470359303305164877446339693270239e+0007L,
 353    1.540438642031689641308197880181291865714e+0006L,
 354 };
 355 static GENERIC ps4[13] = {      /* [2.5, 3.5] */
 356    1.0e0L,
 357    1.454528105698159439773035951959131799816e+0002L,
 358    8.107442215200392397172179900434987859618e+0003L,
 359    2.279390393778242887574177096606328994140e+0005L,
 360    3.576251625592252008424781111770934135844e+0006L,
 361    3.267909499056932631405942058670933813863e+0007L,
 362    1.760021515330805537499778238099704648805e+0008L,
 363    5.525553787667353981242060222587465726729e+0008L,
 364    9.769870295912820457889384082671269328511e+0008L,
 365    9.110582071004774279226905629624018008454e+0008L,
 366    3.981857678621955599371967680343918454345e+0008L,
 367    6.482404686230769399073192961667697036706e+0007L,
 368    2.210046943095878402443535460329391782298e+0006L,
 369 };
 370 static GENERIC pr5[13] = {      /* [1.777..., 2.5] */
 371    9.999999999114986107951817871144655880699e-0001L,
 372    9.252583736048588342568344570315435947614e+0001L,
 373    3.218726757856078715214631502407386264637e+0003L,
 374    5.554009964621111656479588505862577040831e+0004L,
 375    5.269993115643664338253196944523510290175e+0005L,
 376    2.874613773778430691192912190618220544575e+0006L,
 377    9.133538151103658353874146919613442436035e+0006L,
 378    1.673067041410338922825193013077354249193e+0007L,
 379    1.706913873848398011744790289200151840498e+0007L,
 380    9.067766583853288534551600235576747618679e+0006L,
 381    2.216746733457884568532695355036338655872e+0006L,
 382    1.945753880802872541235703812722344514405e+0005L,
 383    3.132374412921948071539195638885330951749e+0003L,
 384 };
 385 static GENERIC ps5[13] = {      /* [1.777..., 2.5] */
 386    1.0e0L,
 387    9.259614983862181118883831670990340052982e+0001L,
 388    3.225125275462903384842124075132609290304e+0003L,
 389    5.575705362829101545292760055941855246492e+0004L,
 390    5.306049863037087855496170121958448492522e+0005L,
 391    2.907060758873509564309729903109018597215e+0006L,
 392    9.298059206584995898298257827131208539289e+0006L,
 393    1.720391071006963176836108026556547062980e+0007L,
 394    1.782614812922865190479394509487941920612e+0007L,
 395    9.708016389605273153536452032839879950155e+0006L,
 396    2.476495084688170096480215640962175140027e+0006L,
 397    2.363200660365585759668077790194604917187e+0005L,
 398    4.803239569848196077121203575704356936731e+0003L,
 399 };
 400 static GENERIC pr6[13] = {      /* [1.28, 1.777...] */
 401    9.999999969777095495998606925524322559556e-0001L,
 402    5.825486719466194430503283824096872219216e+0001L,
 403    1.248155491637757281915184824965379905380e+0003L,
 404    1.302093199842358609321338417071710477615e+0004L,
 405    7.353835804186292782840961999810543016039e+0004L,
 406    2.356471661113686180549195092555751341757e+0005L,
 407    4.350553267429009581632987060942780847101e+0005L,
 408    4.588762661876600638719159826652389418235e+0005L,
 409    2.675796398548523436544221045225290128611e+0005L,
 410    8.077649557108971388298292919988449940464e+0004L,
 411    1.117640459221306873519068741664054573776e+0004L,
 412    5.544400072396814695175787511557757885585e+0002L,
 413    5.072550541191480498431289089905822910718e+0000L,
 414 };
 415 static GENERIC ps6[13] = {      /* [1.28, 1.777...] */
 416    1.0e0L,
 417    5.832517925357165050639075848183613063291e+0001L,
 418    1.252144364743592128171256104364976466898e+0003L,
 419    1.310300234342216813579118022415585740772e+0004L,
 420    7.434667697093812197817292154032863632923e+0004L,
 421    2.398706595587719165726469002404004614711e+0005L,
 422    4.472737517625103157004869372427480602511e+0005L,
 423    4.786313523337761975294171429067037723611e+0005L,
 424    2.851161872872731228472536061865365370192e+0005L,
 425    8.891648269899148412331918021801385815586e+0004L,
 426    1.297097489535351517572978123584751042287e+0004L,
 427    7.096761640545975756202184143400469812618e+0002L,
 428    8.378049338590233325977702401733340820351e+0000L,
 429 };
 430 static GENERIC sixteen = 16.0L;
 431 static GENERIC huge    = 1.0e30L;
 432
 433 static GENERIC pzero(x)
 434 GENERIC x;
 435 {
 436         GENERIC s, r, t, z;
 437         int i;
 438         if (x > huge)
 439                 return (one);
 440         t = one/x; z = t*t;
 441         if (x > sixteen) {
 442             r = z*pr0[11]+pr0[10]; s = ps0[10];
 443             for (i = 9; i >= 0; i--) {
 444                 r = z*r + pr0[i];
 445                 s = z*s + ps0[i];
 446             }
 447         } else if (x > eight) {
 448             r = pr1[11]; s = ps1[11]+z*(ps1[12]+z*ps1[13]);
 449             for (i = 10; i >= 0; i--) {
 450                 r = z*r + pr1[i];
 451                 s = z*s + ps1[i];
 452             }
 453         } else if (x > five) {  /* x > 5.0 */
 454             r = pr2[11]; s = ps2[11]+z*(ps2[12]+z*ps2[13]);
 455             for (i = 10; i >= 0; i--) {
 456                 r = z*r + pr2[i];
 457                 s = z*s + ps2[i];
 458             }
 459         } else if (x > 3.5L) {
 460             r = pr3[12]; s = ps3[12];
 461             for (i = 11; i >= 0; i--) {
 462                 r = z*r + pr3[i];
 463                 s = z*s + ps3[i];
 464             }
 465         } else if (x > 2.5L) {
 466             r = pr4[12]; s = ps4[12];
 467             for (i = 11; i >= 0; i--) {
 468                 r = z*r + pr4[i];
 469                 s = z*s + ps4[i];
 470             }
 471         } else if (x > (1.0L/0.5625L)) {
 472             r = pr5[12]; s = ps5[12];
 473             for (i = 11; i >= 0; i--) {
 474                 r = z*r + pr5[i];
 475                 s = z*s + ps5[i];
 476             }
 477         } else {        /* assume x > 1.28 */
 478             r = pr6[12]; s = ps6[12];
 479             for (i = 11; i >= 0; i--) {
 480                 r = z*r + pr6[i];
 481                 s = z*s + ps6[i];
 482             }
 483         }
 484         return (r/s);
 485 }
 486
 487
 488 static GENERIC qr0[12] = {      /* [16, inf] */
 489   -1.249999999999999999999999999999999972972e-0001L,
 490   -1.425179595545670577414395762503991596897e+0002L,
 491   -6.312499645625970845534460257936222407219e+0004L,
 492   -1.411374326457208384315121243698814446848e+0007L,
 493   -1.735034212758873581410984757860787252842e+0009L,
 494   -1.199777647512789489421826342485055280680e+0011L,
 495   -4.596025334081655714499860409699100373644e+0012L,
 496   -9.262525628201284107792924477031653399187e+0013L,
 497   -8.858394728685039245344398842180662867639e+0014L,
 498   -3.267527953687534887623740622709505972113e+0015L,
 499   -2.664222971186311967587129347029450062019e+0015L,
 500    3.442464060723987869585180095344504100204e+0014L,
 501 };
 502 static GENERIC qs0[11] = {
 503    1.0e0L,
 504    1.140729613936536461931516610003185687881e+0003L,
 505    5.056665510442299351009198186490085803580e+0005L,
 506    1.132041763825642787943941650522718199115e+0008L,
 507    1.394570111872581606392620678214246479767e+0010L,
 508    9.677945218152264789534431079563744378421e+0011L,
 509    3.731140327851536828225143058896348502096e+0013L,
 510    7.612785951064869291722846681020881676410e+0014L,
 511    7.476077016406764891730191004811863975940e+0015L,
 512    2.951246482613592035421503427100393831709e+0016L,
 513    3.108361803691811711136854587074302034901e+0016L,
 514 };
 515 static GENERIC qr1[12] = {      /* [8, 16 ] */
 516   -1.249999999999999999997949010383433818157e-0001L,
 517   -9.051215166393822640636752244895124126934e+0001L,
 518   -2.620782703428148837671179031904208303947e+0004L,
 519   -3.975571261553504457766177974508785790884e+0006L,
 520   -3.479029330759311306270072218074074994090e+0008L,
 521   -1.823955008124268573036216746186239829089e+0010L,
 522   -5.765932697111801375765156029221568664435e+0011L,
 523   -1.079843680798742592954002192417934779114e+0013L,
 524   -1.146893630504592739082205764611581332897e+0014L,
 525   -6.367016059683898464936104447282880704182e+0014L,
 526   -1.583109041961213490464459111903484209098e+0015L,
 527   -1.230149555764242473103128650135795639412e+0015L,
 528 };
 529 static GENERIC qs1[14] = {
 530    1.0e0L,
 531    7.246831508115058112438579847778014458432e+0002L,
 532    2.100854184439168518399383786306927037611e+0005L,
 533    3.192636418837951507430188285940994235122e+0007L,
 534    2.801558443383354674538443461124434216152e+0009L,
 535    1.475026997664373739293483927250653467487e+0011L,
 536    4.694486824913954608552363821799927145318e+0012L,
 537    8.890350100919200250838438709601547334021e+0013L,
 538    9.626844429082905144874701068760469752067e+0014L,
 539    5.541110744600460773528263862687521642140e+0015L,
 540    1.486500494789452556727470329232123096563e+0016L,
 541    1.415840104845959400365430773732093899210e+0016L,
 542    1.780866095241517418081312567239682336483e+0015L,
 543   -2.359230917384889357887631544079990129494e+0014L,
 544 };
 545 static GENERIC qr2[12] = {      /* [5, 8] */
 546   -1.249999999999999531937744362527772181614e-0001L,
 547   -4.944373897356969774839375977239241573966e+0001L,
 548   -7.728449175433465285314261650078450473909e+0003L,
 549   -6.262574329612752346336901434651220705903e+0005L,
 550   -2.900948220220943306027235217424380672732e+0007L,
 551   -7.988719647634192770463917157562874119535e+0008L,
 552   -1.318228171927181389547760026626357012375e+0010L,
 553   -1.282439773983029245309263271945424928196e+0011L,
 554   -7.050925570827818040186149940257918845138e+0011L,
 555   -2.021751882573871990004205616874202684429e+0012L,
 556   -2.592939962400668552384333900573812635658e+0012L,
 557   -1.038267109518891262840601514932972850326e+0012L,
 558 };
 559 static GENERIC qs2[14] = {
 560    1.0e0L,
 561    3.961358492885570003202784022894248952116e+0002L,
 562    6.205788738864701882828752634586510926968e+0004L,
 563    5.045715603932670286550673813011764406749e+0006L,
 564    2.349248611362658323353343389430968751429e+0008L,
 565    6.520244524415828635917683553721880063911e+0009L,
 566    1.089111211223507719337067159886281887722e+0011L,
 567    1.080406000905359867958779409414903018610e+0012L,
 568    6.135645280895514703514154680623769562148e+0012L,
 569    1.862433040246625874245867151368643668215e+0013L,
 570    2.667780805786648888840777888702193708994e+0013L,
 571    1.394401107289087774765300711809313112824e+0013L,
 572    1.093247500616320375562898297156722445484e+0012L,
 573   -7.228875530378928722826604216491493780775e+0010L,
 574 };
 575 static GENERIC qr3[13] = {      /* [3.5 5] */
 576   -1.249999999999473067748420379578481661075e-0001L,
 577   -3.044549048635289351913574324803250977998e+0001L,
 578   -2.890081140649769078496693003524681440869e+0003L,
 579   -1.404922456817202235879343275330529107684e+0005L,
 580   -3.862746614385573443518177403617349281869e+0006L,
 581   -6.257517309110249049201133708911155047689e+0007L,
 582   -6.031451330920839916987079782727323477520e+0008L,
 583   -3.411542405173830611454025765755854382346e+0009L,
 584   -1.089392478149726672133014498723021526099e+0010L,
 585   -1.824934078420210941290140903415956782726e+0010L,
 586   -1.400780278304358710423481070486939531139e+0010L,
 587   -3.716484136064917363926635716743771092093e+0009L,
 588   -1.397591075296425529970434890954904331580e+0008L,
 589 };
 590 static GENERIC qs3[13] = {
 591    1.0e0L,
 592    2.441498613904962049391000187014945858042e+0002L,
 593    2.326188882072370711500164222341514337043e+0004L,
 594    1.137138213121231338494977104659239578165e+0006L,
 595    3.152918070735662728722998452605364253517e+0007L,
 596    5.172877993426507259314270488444013595108e+0008L,
 597    5.083086439731669807455961078856470774115e+0009L,
 598    2.961842732066434123119325521139476909941e+0010L,
 599    9.912185866862440735829781856081353151390e+0010L,
 600    1.793560561251622234430564181567297983598e+0011L,
 601    1.577090119341228122525265108497940403073e+0011L,
 602    5.509910306780166194333889999985463681636e+0010L,
 603    4.761691134078874491202320181517936758141e+0009L,
 604 };
 605 static GENERIC qr4[13] = {      /* [2.5 3.5] */
 606   -1.249999999928567734339745043490705340835e-0001L,
 607   -1.967201748731419063051601624435565528481e+0001L,
 608   -1.186329146714562236407099740615528170707e+0003L,
 609   -3.607736959222941810356301491152457934060e+0004L,
 610   -6.119200717978104904932828468575194267125e+0005L,
 611   -6.037847781158358226670305078652205586384e+0006L,
 612   -3.503558153336140359700536720393565984740e+0007L,
 613   -1.180196478268225718757218523746787309773e+0008L,
 614   -2.221860232085134915841426363505169680528e+0008L,
 615   -2.173372505452747585296176761701746236760e+0008L,
 616   -9.649364865061237558517730539506568013963e+0007L,
 617   -1.465429227847933034546039640094862650385e+0007L,
 618   -3.083003197920262085170581866246663380607e+0005L,
 619 };
 620 static GENERIC qs4[13] = {      /* [2.5 3.5] */
 621    1.0e0L,
 622    1.579620773732259142752614142139986854055e+0002L,
 623    9.581372220329138733203879503753685054968e+0003L,
 624    2.939598672379108095776114131010825885308e+0005L,
 625    5.052183049314542218630341818692588448168e+0006L,
 626    5.083497695595206639433839326338971980149e+0007L,
 627    3.036385361800553388049719014005099206516e+0008L,
 628    1.067826481452753409910563785161661492137e+0009L,
 629    2.145644125557118044720741775125319669272e+0009L,
 630    2.324115615959719949363946673491552216799e+0009L,
 631    1.223262962112070757966959855619847011146e+0009L,
 632    2.569765553318495423738478585947110270709e+0008L,
 633    1.354744744299227127897905787732636565504e+0007L,
 634 };
 635 static GENERIC qr5[13] = {      /* [1.777.., 2.5] */
 636   -1.249999995936639697637680428174576069971e-0001L,
 637   -1.260846055371311453485891923426489068315e+0001L,
 638   -4.772398467544467480801174330290141578895e+0002L,
 639   -8.939852599990298486613760833996490599724e+0003L,
 640   -9.184070787149542050979542226446134243197e+0004L,
 641   -5.406038945018274458362637897739280435171e+0005L,
 642   -1.845896544705190261018653728678171084418e+0006L,
 643   -3.613616990680809501878667570653308071547e+0006L,
 644   -3.908782978135693252252557720414348623779e+0006L,
 645   -2.173711022517323927109138170588442768176e+0006L,
 646   -5.431253130679918485836408549007856244495e+0005L,
 647   -4.591098546452684510082591587275940765959e+0004L,
 648   -5.244711364168207806835520057792229646578e+0002L,
 649 };
 650 static GENERIC qs5[13] = {      /* [1.777.., 2.5] */
 651    1.0e0L,
 652    1.014536210851290878350892750972474861447e+0002L,
 653    3.875547510687135314064434160096139681076e+0003L,
 654    7.361913122670079814955259281995617732580e+0004L,
 655    7.720288944218771126581086539585529314636e+0005L,
 656    4.681529554446752496404431433608306558038e+0006L,
 657    1.667882621940503925455031252308367745820e+0007L,
 658    3.469403153761399881888272620855305156241e+0007L,
 659    4.096992047964210711867089384719947863019e+0007L,
 660    2.596804755829217449311530735959560630554e+0007L,
 661    7.983933774697889238154465064019410763845e+0006L,
 662    9.818133816979900819087242425280757938152e+0005L,
 663    3.061083930868694396013541535670745443560e+0004L,
 664 };
 665
 666 static GENERIC qr6[13] = {      /* [1.28, 1.777..] */
 667   -1.249999881577289001807137282824929082771e-0001L,
 668   -7.998273510053110759610810594119533619282e+0000L,
 669   -1.872481955335172543369089617771565632719e+0002L,
 670   -2.122116786726300805079874003303799646812e+0003L,
 671   -1.293850285839529282503178263484773478457e+0004L,
 672   -4.445024742266316181033354192262529356093e+0004L,
 673   -8.730161378334357767668344467356505347070e+0004L,
 674   -9.706222895172078442801444972505315054736e+0004L,
 675   -5.896325518259858270165531513618195321041e+0004L,
 676   -1.823172034368108822276420827074668832233e+0004L,
 677   -2.509304178635055926638833040337472387175e+0003L,
 678   -1.156608965715779237316769828941729964099e+0002L,
 679   -7.028005789650731396887346826397785210442e-0001L,
 680 };
 681 static GENERIC qs6[13] = {      /* [1.28, 1.777..] */
 682    1.0e0L,
 683    6.457211085058064845601261321277721075900e+0001L,
 684    1.534005216588011210342824555136008682950e+0003L,
 685    1.777217999176441782593357660462379097171e+0004L,
 686    1.118372652642469468091084810263231199696e+0005L,
 687    4.015242433858461813142365748386473605294e+0005L,
 688    8.377081045517098645448616514388280497673e+0005L,
 689    1.011495020008010352575398009604164287337e+0006L,
 690    6.886722075290430568652227875200208955970e+0005L,
 691    2.504735189948021472047157148613171956537e+0005L,
 692    4.408138920171044846941001844352009817062e+0004L,
 693    3.105572178072115145673058722853640854884e+0003L,
 694    5.588294821118916113437396504573817033678e+0001L,
 695 };
 696 static GENERIC qzero(x)
 697 GENERIC x;
 698 {
 699         GENERIC s, r, t, z;
 700         int i;
 701         if (x > huge)
 702                 return (-0.125L/x);
 703         t = one/x; z = t*t;
 704         if (x > sixteen) {
 705             r = z*qr0[11]+qr0[10]; s = qs0[10];
 706             for (i = 9; i >= 0; i--) {
 707                 r = z*r + qr0[i];
 708                 s = z*s + qs0[i];
 709             }
 710         } else if (x > eight) {
 711             r = qr1[11]; s = qs1[11]+z*(qs1[12]+z*qs1[13]);
 712             for (i = 10; i >= 0; i--) {
 713                 r = z*r + qr1[i];
 714                 s = z*s + qs1[i];
 715             }
 716         } else if (x > five) {  /* assume x > 5.0 */
 717             r = qr2[11]; s = qs2[11]+z*(qs2[12]+z*qs2[13]);
 718             for (i = 10; i >= 0; i--) {
 719                 r = z*r + qr2[i];
 720                 s = z*s + qs2[i];
 721             }
 722         } else if (x > 3.5L) {
 723             r = qr3[12]; s = qs3[12];
 724             for (i = 11; i >= 0; i--) {
 725                 r = z*r + qr3[i];
 726                 s = z*s + qs3[i];
 727             }
 728         } else if (x > 2.5L) {
 729             r = qr4[12]; s = qs4[12];
 730             for (i = 11; i >= 0; i--) {
 731                 r = z*r + qr4[i];
 732                 s = z*s + qs4[i];
 733             }
 734         } else if (x > (1.0L/0.5625L)) {
 735             r = qr5[12]; s = qs5[12];
 736             for (i = 11; i >= 0; i--) {
 737                 r = z*r + qr5[i];
 738                 s = z*s + qs5[i];
 739             }
 740         } else {        /* assume x > 1.28 */
 741             r = qr6[12]; s = qs6[12];
 742             for (i = 11; i >= 0; i--) {
 743                 r = z*r + qr6[i];
 744                 s = z*s + qs6[i];
 745             }
 746         }
 747         return (t*(r/s));
 748 }