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 /*
  27  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  28  * Use is subject to license terms.
  29  */
  30 
  31 /*
  32  * Floating point Bessel's function of the first and second kinds
  33  * of order zero: j0(x),y0(x);
  34  *
  35  * Special cases:
  36  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  37  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  38  */
  39 
  40 #pragma weak __j0l = j0l
  41 #pragma weak __y0l = y0l
  42 
  43 #include "libm.h"
  44 
  45 #include "longdouble.h"
  46 
  47 #include <math.h>
  48 #if defined(__SUNPRO_C)
  49 #include <sunmath.h>
  50 #endif
  51 
  52 #define GENERIC long double
  53 
  54 static GENERIC zero = 0.0L,
  55         small = 1.0e-9L,
  56         tiny = 1.0e-38L,
  57         one = 1.0L,
  58         five = 5.0L,
  59         eight = 8.0L,
  60         invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
  61         tpi = 0.636619772367581343075535053490057448L;
  62 
  63 static GENERIC pzero(), qzero();
  64 
  65 static GENERIC r0[7] = {
  66         -2.499999999999999999999999999999998934492e-0001L,
  67         1.272657927360049786327618451133763714880e-0002L,
  68         -2.694499763712963276900636693400659600898e-0004L,
  69         2.724877475058977576903234070919616447883e-0006L,
  70         -1.432617103214330236967477495393076320281e-0008L,
  71         3.823248804080079168706683540513792224471e-0011L,
  72         -4.183174277567983647337568504286313665065e-0014L,
  73 };
  74 
  75 static GENERIC s0[7] = {
  76         1.0e0L,
  77         1.159368290559800854689526195462884666395e-0002L,
  78         6.629397597394973383009743876169946772559e-0005L,
  79         2.426779981394054406305431142501735094340e-0007L,
  80         6.097663491248511069094400469635449749883e-0010L,
  81         1.017019133340929220238747413216052224036e-0012L,
  82         9.012593179306197579518374581969371278481e-0016L,
  83 };
  84 
  85 GENERIC
  86 j0l(GENERIC x)
  87 {
  88         GENERIC z, s, c, ss, cc, r, u, v;
  89         int i;
  90 
  91         if (isnanl(x))
  92                 return (x + x);
  93 
  94         x = fabsl(x);
  95 
  96         if (x > 1.28L) {
  97                 if (!finitel(x))
  98                         return (zero);
  99 
 100                 s = sinl(x);
 101                 c = cosl(x);
 102 
 103                 /* BEGIN CSTYLED */
 104                 /*
 105                  * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
 106                  * where x0 = x-pi/4
 107                  *      Better formula:
 108                  *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 109                  *                      = 1/sqrt(2) * (cos(x) + sin(x))
 110                  *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
 111                  *                      = 1/sqrt(2) * (sin(x) - cos(x))
 112                  * To avoid cancellation, use
 113                  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 114                  * to compute the worse one.
 115                  */
 116                 /* END CSTYLED */
 117                 if (x > 1.0e2450L) { /* x+x may overflow */
 118                         ss = s - c;
 119                         cc = s + c;
 120                 } else if (signbitl(s) != signbitl(c)) {
 121                         ss = s - c;
 122                         cc = -cosl(x + x) / ss;
 123                 } else {
 124                         cc = s + c;
 125                         ss = -cosl(x + x) / cc;
 126                 }
 127 
 128                 /*
 129                  * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 130                  * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 131                  */
 132                 if (x > 1.0e120L)
 133                         return ((invsqrtpi * cc) / sqrtl(x));
 134 
 135                 u = pzero(x);
 136                 v = qzero(x);
 137                 return (invsqrtpi * (u * cc - v * ss) / sqrtl(x));
 138         }
 139 
 140         if (x <= small) {
 141                 if (x <= tiny)
 142                         return (one - x);
 143                 else
 144                         return (one - x * x * 0.25L);
 145         }
 146 
 147         z = x * x;
 148         r = r0[6];
 149         s = s0[6];
 150 
 151         for (i = 5; i >= 0; i--) {
 152                 r = r * z + r0[i];
 153                 s = s * z + s0[i];
 154         }
 155 
 156         return (one + z * (r / s));
 157 }
 158 
 159 static GENERIC u0[8] = {
 160         -7.380429510868722527434392794848301631220e-0002L,
 161         1.766855559625940791857536949301981816513e-0001L,
 162         -1.386470722701047923235553251240162839408e-0002L,
 163         3.520149242724811578636970811631224862615e-0004L,
 164         -3.978599663243790049853642275624951870025e-0006L,
 165         2.228801153263957224547222556806915479763e-0008L,
 166         -6.121246764298785018658597179498837316177e-0011L,
 167         6.677103629722678833475965810525587396596e-0014L,
 168 };
 169 
 170 static GENERIC v0[8] = {
 171         1.0e0L,
 172         1.247164416539111311571676766127767127970e-0002L,
 173         7.829144749639791500052900281489367443576e-0005L,
 174         3.247126540422245330511218321013360336606e-0007L,
 175         9.750516724789499678567062572549568447869e-0010L,
 176         2.156713223173591212250543390258458098776e-0012L,
 177         3.322169561597890004231482431236452752624e-0015L,
 178         2.821213295314000924252226486305726805093e-0018L,
 179 };
 180 
 181 GENERIC
 182 y0l(GENERIC x)
 183 {
 184         GENERIC z, s, c, ss, cc, u, v;
 185         GENERIC d __unused;
 186         int i;
 187 
 188         if (isnanl(x))
 189                 return (x + x);
 190 
 191         if (x <= zero) {
 192                 if (x == zero)
 193                         d = -one / (x - x);
 194                 else
 195                         d = zero / (x - x);
 196         }
 197 
 198         if (x > 1.28L) {
 199                 if (!finitel(x))
 200                         return (zero);
 201 
 202                 s = sinl(x);
 203                 c = cosl(x);
 204 
 205                 /* BEGIN CSTYLED */
 206                 /*
 207                  * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
 208                  * where x0 = x-pi/4
 209                  *      Better formula:
 210                  *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 211                  *                      = 1/sqrt(2) * (cos(x) + sin(x))
 212                  *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
 213                  *                      = 1/sqrt(2) * (sin(x) - cos(x))
 214                  * To avoid cancellation, use
 215                  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 216                  * to compute the worse one.
 217                  */
 218                 /* END CSTYLED */
 219                 if (x > 1.0e2450L) { /* x+x may overflow */
 220                         ss = s - c;
 221                         cc = s + c;
 222                 } else if (signbitl(s) != signbitl(c)) {
 223                         ss = s - c;
 224                         cc = -cosl(x + x) / ss;
 225                 } else {
 226                         cc = s + c;
 227                         ss = -cosl(x + x) / cc;
 228                 }
 229 
 230                 /*
 231                  * j0(x) = 1/sqrt(pi*x) * (P(0,x)*cc - Q(0,x)*ss)
 232                  * y0(x) = 1/sqrt(pi*x) * (P(0,x)*ss + Q(0,x)*cc)
 233                  */
 234                 if (x > 1.0e120L)
 235                         return ((invsqrtpi * ss) / sqrtl(x));
 236 
 237                 return (invsqrtpi * (pzero(x) * ss + qzero(x) * cc) / sqrtl(x));
 238         }
 239 
 240         if (x <= tiny)
 241                 return (u0[0] + tpi * logl(x));
 242 
 243         z = x * x;
 244         u = u0[7];
 245         v = v0[7];
 246 
 247         for (i = 6; i >= 0; i--) {
 248                 u = u * z + u0[i];
 249                 v = v * z + v0[i];
 250         }
 251 
 252         return (u / v + tpi * (j0l(x) * logl(x)));
 253 }
 254 
 255 static GENERIC pr0[12] = {              /* [16 -- inf] */
 256         9.999999999999999999999999999999999997515e-0001L,
 257         1.065981615377273376425365823967550598358e+0003L,
 258         4.390991200927588978306374718984240719130e+0005L,
 259         9.072086218607986711847069407339321363103e+0007L,
 260         1.022552886177375367408408501046461671528e+0010L,
 261         6.420766912243658241570635854089597269031e+0011L,
 262         2.206451725126933913591080211081242266908e+0013L,
 263         3.928369596816895077363705478743346298368e+0014L,
 264         3.258159928874124597286701119721482876596e+0015L,
 265         1.025715808134188978860679130140685101348e+0016L,
 266         7.537170874795721255796001687024031280685e+0015L,
 267         -1.579413901450157332307745586004207687796e+0014L,
 268 };
 269 
 270 static GENERIC ps0[11] = {
 271         1.0e0L,
 272         1.066051927877273376425365823967550512687e+0003L,
 273         4.391739647168381592399173804329266353038e+0005L,
 274         9.075162261801343671805658294123888867884e+0007L,
 275         1.023186118519904751819581912075985995058e+0010L,
 276         6.427861860414223746340515376512730275061e+0011L,
 277         2.210861503237823589735481303627993406235e+0013L,
 278         3.943247335784292905915956840901818177989e+0014L,
 279         3.283720976777545142150200110647270004481e+0015L,
 280         1.045346918812754048903645641538728986759e+0016L,
 281         8.043455468065618900750599584291193680463e+0015L,
 282 };
 283 
 284 static GENERIC pr1[12] = {              /* [8 -- 16] */
 285         9.999999999999999999999784422701108683618e-0001L,
 286         6.796098532948334207755488692777907062894e+0002L,
 287         1.840036112605722168824530758797169836042e+0005L,
 288         2.598490483191916637264894340635847598122e+0007L,
 289         2.105774863242707025525730249472054578523e+0009L,
 290         1.015822044230542426666314997796944979959e+0011L,
 291         2.931557457008110436764077699944189071875e+0012L,
 292         4.962885121125457633655259224179322808824e+0013L,
 293         4.705424055148223269155430598563351566279e+0014L,
 294         2.294439854910747229152056080910427001110e+0015L,
 295         4.905531843137486691500950019322475458629e+0015L,
 296         3.187543169710339218793442542845735994565e+0015L,
 297 };
 298 
 299 static GENERIC ps1[14] = {
 300         1.0e0L,
 301         6.796801657948334207754571576066758180288e+0002L,
 302         1.840512891201300567325421059826676366447e+0005L,
 303         2.599777028312918975306252167127695075221e+0007L,
 304         2.107582572771047636846811284634244892537e+0009L,
 305         1.017275794694156108975782763889979940348e+0011L,
 306         2.938487645192463845428059755454762316011e+0012L,
 307         4.982512164735557054521042916182317924466e+0013L,
 308         4.737639900153703274792677468264564361437e+0014L,
 309         2.323398719123742743524249528275097100646e+0015L,
 310         5.033419107069210577868909797896984419391e+0015L,
 311         3.409036105931068609601317076759804716059e+0015L,
 312         7.505655364352679737585745147753521662166e+0013L,
 313         -9.976837153983688250780198248297109118313e+0012L,
 314 };
 315 
 316 static GENERIC pr2[12] = {              /* [5 -- 8 ] */
 317         9.999999999999999937857236789277366320220e-0001L,
 318         3.692848765268649571651602420376358849214e+0002L,
 319         5.373022067535476576926715900057760985410e+0004L,
 320         4.038738891191314969971504035057219430725e+0006L,
 321         1.728285706306940523397385566659762646999e+0008L,
 322         4.375400819645889911158688737206054788534e+0009L,
 323         6.598950418204912408375591217782088567076e+0010L,
 324         5.827182039183238492480275401520072793783e+0011L,
 325         2.884222642913492390887572414999490975844e+0012L,
 326         7.373278873797767721932837830628688632775e+0012L,
 327         8.338295457568973761205077964397969230489e+0012L,
 328         2.911383183467288345772308817209806922143e+0012L,
 329 };
 330 
 331 static GENERIC ps2[14] = {
 332         1.0e0L,
 333         3.693551890268649477288896267171993213102e+0002L,
 334         5.375607880998361502474715133828068514297e+0004L,
 335         4.042477764024108249744998862572786367328e+0006L,
 336         1.731069838737016956685839588670132939513e+0008L,
 337         4.387147674049898778738226585935491417728e+0009L,
 338         6.628058659620653765349556940567715258165e+0010L,
 339         5.869659904164177740471685856367322160664e+0011L,
 340         2.919839445622817017058977559638969436383e+0012L,
 341         7.535314897696671402628203718612309253907e+0012L,
 342         8.696355561452933775773309859748610658935e+0012L,
 343         3.216155103141537221173601557697083216257e+0012L,
 344         4.756857081068942248246880159213789086363e+0010L,
 345         -3.496356619666608032231074866481472824067e+0009L,
 346 };
 347 
 348 static GENERIC pr3[13] = {              /* [3.5 -- 5 ] */
 349         9.999999999999916693107285612398196588247e-0001L,
 350         2.263975921282917721194425320484974336945e+0002L,
 351         1.994358386744245848889492762781484199966e+0004L,
 352         8.980067458430542243559962493831661323168e+0005L,
 353         2.282213787521372663705567756420087553508e+0007L,
 354         3.409784374889063618250288699908375135923e+0008L,
 355         3.024380857401448589254343517589811711108e+0009L,
 356         1.571110368046740246895071721443082286379e+0010L,
 357         4.603187020243604632153685300463160593768e+0010L,
 358         7.087196453409712719449549280664058793403e+0010L,
 359         5.046196021776346356803687409644239065041e+0010L,
 360         1.287758439080165765709154276618854799932e+0010L,
 361         5.900679773415023433787846658096813590784e+0008L,
 362 };
 363 
 364 static GENERIC ps3[13] = {
 365         1.0e0L,
 366         2.264679046282855061328604619231774747116e+0002L,
 367         1.995939523988944553755653255389812103448e+0004L,
 368         8.993853144706348727038389967490183236820e+0005L,
 369         2.288326099634588843906989983704795468773e+0007L,
 370         3.424967100255240885169240956804790118282e+0008L,
 371         3.046311797972463991368023759640028910016e+0009L,
 372         1.589614961932826812790222479700797224003e+0010L,
 373         4.692406624527744816497089139325073939927e+0010L,
 374         7.320486495902008912866462849073108323948e+0010L,
 375         5.345945972828978289935309597742981360994e+0010L,
 376         1.444033091910423754121309915092247171008e+0010L,
 377         7.987714685115314668378957273824383610525e+0008L,
 378 };
 379 
 380 static GENERIC pr4[13] = {              /* [2.5, 3.5] */
 381         9.999999999986736677961118722747757712260e-0001L,
 382         1.453824980703800559037873123568378845663e+0002L,
 383         8.097327216430682288267610447006508661032e+0003L,
 384         2.273847252038264370231169686380192662135e+0005L,
 385         3.561056728046211111354759998976985449622e+0006L,
 386         3.244933588800096378434627029369680378599e+0007L,
 387         1.740112392860717950376210038908476792588e+0008L,
 388         5.426170187455893285197878563881579269524e+0008L,
 389         9.490107486454362321004377336020526281371e+0008L,
 390         8.688872439428470049801714121070005313806e+0008L,
 391         3.673315853166437222811910656900123215515e+0008L,
 392         5.577770470359303305164877446339693270239e+0007L,
 393         1.540438642031689641308197880181291865714e+0006L,
 394 };
 395 
 396 static GENERIC ps4[13] = {              /* [2.5, 3.5] */
 397         1.0e0L,
 398         1.454528105698159439773035951959131799816e+0002L,
 399         8.107442215200392397172179900434987859618e+0003L,
 400         2.279390393778242887574177096606328994140e+0005L,
 401         3.576251625592252008424781111770934135844e+0006L,
 402         3.267909499056932631405942058670933813863e+0007L,
 403         1.760021515330805537499778238099704648805e+0008L,
 404         5.525553787667353981242060222587465726729e+0008L,
 405         9.769870295912820457889384082671269328511e+0008L,
 406         9.110582071004774279226905629624018008454e+0008L,
 407         3.981857678621955599371967680343918454345e+0008L,
 408         6.482404686230769399073192961667697036706e+0007L,
 409         2.210046943095878402443535460329391782298e+0006L,
 410 };
 411 
 412 static GENERIC pr5[13] = {              /* [1.777..., 2.5] */
 413         9.999999999114986107951817871144655880699e-0001L,
 414         9.252583736048588342568344570315435947614e+0001L,
 415         3.218726757856078715214631502407386264637e+0003L,
 416         5.554009964621111656479588505862577040831e+0004L,
 417         5.269993115643664338253196944523510290175e+0005L,
 418         2.874613773778430691192912190618220544575e+0006L,
 419         9.133538151103658353874146919613442436035e+0006L,
 420         1.673067041410338922825193013077354249193e+0007L,
 421         1.706913873848398011744790289200151840498e+0007L,
 422         9.067766583853288534551600235576747618679e+0006L,
 423         2.216746733457884568532695355036338655872e+0006L,
 424         1.945753880802872541235703812722344514405e+0005L,
 425         3.132374412921948071539195638885330951749e+0003L,
 426 };
 427 
 428 static GENERIC ps5[13] = {              /* [1.777..., 2.5] */
 429         1.0e0L,
 430         9.259614983862181118883831670990340052982e+0001L,
 431         3.225125275462903384842124075132609290304e+0003L,
 432         5.575705362829101545292760055941855246492e+0004L,
 433         5.306049863037087855496170121958448492522e+0005L,
 434         2.907060758873509564309729903109018597215e+0006L,
 435         9.298059206584995898298257827131208539289e+0006L,
 436         1.720391071006963176836108026556547062980e+0007L,
 437         1.782614812922865190479394509487941920612e+0007L,
 438         9.708016389605273153536452032839879950155e+0006L,
 439         2.476495084688170096480215640962175140027e+0006L,
 440         2.363200660365585759668077790194604917187e+0005L,
 441         4.803239569848196077121203575704356936731e+0003L,
 442 };
 443 
 444 static GENERIC pr6[13] = {              /* [1.28, 1.777...] */
 445         9.999999969777095495998606925524322559556e-0001L,
 446         5.825486719466194430503283824096872219216e+0001L,
 447         1.248155491637757281915184824965379905380e+0003L,
 448         1.302093199842358609321338417071710477615e+0004L,
 449         7.353835804186292782840961999810543016039e+0004L,
 450         2.356471661113686180549195092555751341757e+0005L,
 451         4.350553267429009581632987060942780847101e+0005L,
 452         4.588762661876600638719159826652389418235e+0005L,
 453         2.675796398548523436544221045225290128611e+0005L,
 454         8.077649557108971388298292919988449940464e+0004L,
 455         1.117640459221306873519068741664054573776e+0004L,
 456         5.544400072396814695175787511557757885585e+0002L,
 457         5.072550541191480498431289089905822910718e+0000L,
 458 };
 459 
 460 static GENERIC ps6[13] = {              /* [1.28, 1.777...] */
 461         1.0e0L,
 462         5.832517925357165050639075848183613063291e+0001L,
 463         1.252144364743592128171256104364976466898e+0003L,
 464         1.310300234342216813579118022415585740772e+0004L,
 465         7.434667697093812197817292154032863632923e+0004L,
 466         2.398706595587719165726469002404004614711e+0005L,
 467         4.472737517625103157004869372427480602511e+0005L,
 468         4.786313523337761975294171429067037723611e+0005L,
 469         2.851161872872731228472536061865365370192e+0005L,
 470         8.891648269899148412331918021801385815586e+0004L,
 471         1.297097489535351517572978123584751042287e+0004L,
 472         7.096761640545975756202184143400469812618e+0002L,
 473         8.378049338590233325977702401733340820351e+0000L,
 474 };
 475 
 476 static GENERIC sixteen = 16.0L;
 477 static GENERIC huge = 1.0e30L;
 478 
 479 static GENERIC
 480 pzero(x)
 481     GENERIC x;
 482 {
 483         GENERIC s, r, t, z;
 484         int i;
 485 
 486         if (x > huge)
 487                 return (one);
 488 
 489         t = one / x;
 490         z = t * t;
 491 
 492         if (x > sixteen) {
 493                 r = z * pr0[11] + pr0[10];
 494                 s = ps0[10];
 495 
 496                 for (i = 9; i >= 0; i--) {
 497                         r = z * r + pr0[i];
 498                         s = z * s + ps0[i];
 499                 }
 500         } else if (x > eight) {
 501                 r = pr1[11];
 502                 s = ps1[11] + z * (ps1[12] + z * ps1[13]);
 503 
 504                 for (i = 10; i >= 0; i--) {
 505                         r = z * r + pr1[i];
 506                         s = z * s + ps1[i];
 507                 }
 508         } else if (x > five) {               /* x > 5.0 */
 509                 r = pr2[11];
 510                 s = ps2[11] + z * (ps2[12] + z * ps2[13]);
 511 
 512                 for (i = 10; i >= 0; i--) {
 513                         r = z * r + pr2[i];
 514                         s = z * s + ps2[i];
 515                 }
 516         } else if (x > 3.5L) {
 517                 r = pr3[12];
 518                 s = ps3[12];
 519 
 520                 for (i = 11; i >= 0; i--) {
 521                         r = z * r + pr3[i];
 522                         s = z * s + ps3[i];
 523                 }
 524         } else if (x > 2.5L) {
 525                 r = pr4[12];
 526                 s = ps4[12];
 527 
 528                 for (i = 11; i >= 0; i--) {
 529                         r = z * r + pr4[i];
 530                         s = z * s + ps4[i];
 531                 }
 532         } else if (x > (1.0L / 0.5625L)) {
 533                 r = pr5[12];
 534                 s = ps5[12];
 535 
 536                 for (i = 11; i >= 0; i--) {
 537                         r = z * r + pr5[i];
 538                         s = z * s + ps5[i];
 539                 }
 540         } else {                        /* assume x > 1.28 */
 541                 r = pr6[12];
 542                 s = ps6[12];
 543 
 544                 for (i = 11; i >= 0; i--) {
 545                         r = z * r + pr6[i];
 546                         s = z * s + ps6[i];
 547                 }
 548         }
 549 
 550         return (r / s);
 551 }
 552 static GENERIC qr0[12] = {              /* [16, inf] */
 553         -1.249999999999999999999999999999999972972e-0001L,
 554         -1.425179595545670577414395762503991596897e+0002L,
 555         -6.312499645625970845534460257936222407219e+0004L,
 556         -1.411374326457208384315121243698814446848e+0007L,
 557         -1.735034212758873581410984757860787252842e+0009L,
 558         -1.199777647512789489421826342485055280680e+0011L,
 559         -4.596025334081655714499860409699100373644e+0012L,
 560         -9.262525628201284107792924477031653399187e+0013L,
 561         -8.858394728685039245344398842180662867639e+0014L,
 562         -3.267527953687534887623740622709505972113e+0015L,
 563         -2.664222971186311967587129347029450062019e+0015L,
 564         3.442464060723987869585180095344504100204e+0014L,
 565 };
 566 
 567 static GENERIC qs0[11] = {
 568         1.0e0L,
 569         1.140729613936536461931516610003185687881e+0003L,
 570         5.056665510442299351009198186490085803580e+0005L,
 571         1.132041763825642787943941650522718199115e+0008L,
 572         1.394570111872581606392620678214246479767e+0010L,
 573         9.677945218152264789534431079563744378421e+0011L,
 574         3.731140327851536828225143058896348502096e+0013L,
 575         7.612785951064869291722846681020881676410e+0014L,
 576         7.476077016406764891730191004811863975940e+0015L,
 577         2.951246482613592035421503427100393831709e+0016L,
 578         3.108361803691811711136854587074302034901e+0016L,
 579 };
 580 
 581 static GENERIC qr1[12] = {              /* [8, 16 ] */
 582         -1.249999999999999999997949010383433818157e-0001L,
 583         -9.051215166393822640636752244895124126934e+0001L,
 584         -2.620782703428148837671179031904208303947e+0004L,
 585         -3.975571261553504457766177974508785790884e+0006L,
 586         -3.479029330759311306270072218074074994090e+0008L,
 587         -1.823955008124268573036216746186239829089e+0010L,
 588         -5.765932697111801375765156029221568664435e+0011L,
 589         -1.079843680798742592954002192417934779114e+0013L,
 590         -1.146893630504592739082205764611581332897e+0014L,
 591         -6.367016059683898464936104447282880704182e+0014L,
 592         -1.583109041961213490464459111903484209098e+0015L,
 593         -1.230149555764242473103128650135795639412e+0015L,
 594 };
 595 
 596 static GENERIC qs1[14] = {
 597         1.0e0L,
 598         7.246831508115058112438579847778014458432e+0002L,
 599         2.100854184439168518399383786306927037611e+0005L,
 600         3.192636418837951507430188285940994235122e+0007L,
 601         2.801558443383354674538443461124434216152e+0009L,
 602         1.475026997664373739293483927250653467487e+0011L,
 603         4.694486824913954608552363821799927145318e+0012L,
 604         8.890350100919200250838438709601547334021e+0013L,
 605         9.626844429082905144874701068760469752067e+0014L,
 606         5.541110744600460773528263862687521642140e+0015L,
 607         1.486500494789452556727470329232123096563e+0016L,
 608         1.415840104845959400365430773732093899210e+0016L,
 609         1.780866095241517418081312567239682336483e+0015L,
 610         -2.359230917384889357887631544079990129494e+0014L,
 611 };
 612 
 613 static GENERIC qr2[12] = {              /* [5, 8] */
 614         -1.249999999999999531937744362527772181614e-0001L,
 615         -4.944373897356969774839375977239241573966e+0001L,
 616         -7.728449175433465285314261650078450473909e+0003L,
 617         -6.262574329612752346336901434651220705903e+0005L,
 618         -2.900948220220943306027235217424380672732e+0007L,
 619         -7.988719647634192770463917157562874119535e+0008L,
 620         -1.318228171927181389547760026626357012375e+0010L,
 621         -1.282439773983029245309263271945424928196e+0011L,
 622         -7.050925570827818040186149940257918845138e+0011L,
 623         -2.021751882573871990004205616874202684429e+0012L,
 624         -2.592939962400668552384333900573812635658e+0012L,
 625         -1.038267109518891262840601514932972850326e+0012L,
 626 };
 627 
 628 static GENERIC qs2[14] = {
 629         1.0e0L,
 630         3.961358492885570003202784022894248952116e+0002L,
 631         6.205788738864701882828752634586510926968e+0004L,
 632         5.045715603932670286550673813011764406749e+0006L,
 633         2.349248611362658323353343389430968751429e+0008L,
 634         6.520244524415828635917683553721880063911e+0009L,
 635         1.089111211223507719337067159886281887722e+0011L,
 636         1.080406000905359867958779409414903018610e+0012L,
 637         6.135645280895514703514154680623769562148e+0012L,
 638         1.862433040246625874245867151368643668215e+0013L,
 639         2.667780805786648888840777888702193708994e+0013L,
 640         1.394401107289087774765300711809313112824e+0013L,
 641         1.093247500616320375562898297156722445484e+0012L,
 642         -7.228875530378928722826604216491493780775e+0010L,
 643 };
 644 
 645 static GENERIC qr3[13] = {              /* [3.5 5] */
 646         -1.249999999999473067748420379578481661075e-0001L,
 647         -3.044549048635289351913574324803250977998e+0001L,
 648         -2.890081140649769078496693003524681440869e+0003L,
 649         -1.404922456817202235879343275330529107684e+0005L,
 650         -3.862746614385573443518177403617349281869e+0006L,
 651         -6.257517309110249049201133708911155047689e+0007L,
 652         -6.031451330920839916987079782727323477520e+0008L,
 653         -3.411542405173830611454025765755854382346e+0009L,
 654         -1.089392478149726672133014498723021526099e+0010L,
 655         -1.824934078420210941290140903415956782726e+0010L,
 656         -1.400780278304358710423481070486939531139e+0010L,
 657         -3.716484136064917363926635716743771092093e+0009L,
 658         -1.397591075296425529970434890954904331580e+0008L,
 659 };
 660 
 661 static GENERIC qs3[13] = {
 662         1.0e0L,
 663         2.441498613904962049391000187014945858042e+0002L,
 664         2.326188882072370711500164222341514337043e+0004L,
 665         1.137138213121231338494977104659239578165e+0006L,
 666         3.152918070735662728722998452605364253517e+0007L,
 667         5.172877993426507259314270488444013595108e+0008L,
 668         5.083086439731669807455961078856470774115e+0009L,
 669         2.961842732066434123119325521139476909941e+0010L,
 670         9.912185866862440735829781856081353151390e+0010L,
 671         1.793560561251622234430564181567297983598e+0011L,
 672         1.577090119341228122525265108497940403073e+0011L,
 673         5.509910306780166194333889999985463681636e+0010L,
 674         4.761691134078874491202320181517936758141e+0009L,
 675 };
 676 
 677 static GENERIC qr4[13] = {              /* [2.5 3.5] */
 678         -1.249999999928567734339745043490705340835e-0001L,
 679         -1.967201748731419063051601624435565528481e+0001L,
 680         -1.186329146714562236407099740615528170707e+0003L,
 681         -3.607736959222941810356301491152457934060e+0004L,
 682         -6.119200717978104904932828468575194267125e+0005L,
 683         -6.037847781158358226670305078652205586384e+0006L,
 684         -3.503558153336140359700536720393565984740e+0007L,
 685         -1.180196478268225718757218523746787309773e+0008L,
 686         -2.221860232085134915841426363505169680528e+0008L,
 687         -2.173372505452747585296176761701746236760e+0008L,
 688         -9.649364865061237558517730539506568013963e+0007L,
 689         -1.465429227847933034546039640094862650385e+0007L,
 690         -3.083003197920262085170581866246663380607e+0005L,
 691 };
 692 
 693 static GENERIC qs4[13] = {              /* [2.5 3.5] */
 694         1.0e0L,
 695         1.579620773732259142752614142139986854055e+0002L,
 696         9.581372220329138733203879503753685054968e+0003L,
 697         2.939598672379108095776114131010825885308e+0005L,
 698         5.052183049314542218630341818692588448168e+0006L,
 699         5.083497695595206639433839326338971980149e+0007L,
 700         3.036385361800553388049719014005099206516e+0008L,
 701         1.067826481452753409910563785161661492137e+0009L,
 702         2.145644125557118044720741775125319669272e+0009L,
 703         2.324115615959719949363946673491552216799e+0009L,
 704         1.223262962112070757966959855619847011146e+0009L,
 705         2.569765553318495423738478585947110270709e+0008L,
 706         1.354744744299227127897905787732636565504e+0007L,
 707 };
 708 
 709 static GENERIC qr5[13] = {              /* [1.777.., 2.5] */
 710         -1.249999995936639697637680428174576069971e-0001L,
 711         -1.260846055371311453485891923426489068315e+0001L,
 712         -4.772398467544467480801174330290141578895e+0002L,
 713         -8.939852599990298486613760833996490599724e+0003L,
 714         -9.184070787149542050979542226446134243197e+0004L,
 715         -5.406038945018274458362637897739280435171e+0005L,
 716         -1.845896544705190261018653728678171084418e+0006L,
 717         -3.613616990680809501878667570653308071547e+0006L,
 718         -3.908782978135693252252557720414348623779e+0006L,
 719         -2.173711022517323927109138170588442768176e+0006L,
 720         -5.431253130679918485836408549007856244495e+0005L,
 721         -4.591098546452684510082591587275940765959e+0004L,
 722         -5.244711364168207806835520057792229646578e+0002L,
 723 };
 724 
 725 static GENERIC qs5[13] = {              /* [1.777.., 2.5] */
 726         1.0e0L,
 727         1.014536210851290878350892750972474861447e+0002L,
 728         3.875547510687135314064434160096139681076e+0003L,
 729         7.361913122670079814955259281995617732580e+0004L,
 730         7.720288944218771126581086539585529314636e+0005L,
 731         4.681529554446752496404431433608306558038e+0006L,
 732         1.667882621940503925455031252308367745820e+0007L,
 733         3.469403153761399881888272620855305156241e+0007L,
 734         4.096992047964210711867089384719947863019e+0007L,
 735         2.596804755829217449311530735959560630554e+0007L,
 736         7.983933774697889238154465064019410763845e+0006L,
 737         9.818133816979900819087242425280757938152e+0005L,
 738         3.061083930868694396013541535670745443560e+0004L,
 739 };
 740 
 741 static GENERIC qr6[13] = {              /* [1.28, 1.777..] */
 742         -1.249999881577289001807137282824929082771e-0001L,
 743         -7.998273510053110759610810594119533619282e+0000L,
 744         -1.872481955335172543369089617771565632719e+0002L,
 745         -2.122116786726300805079874003303799646812e+0003L,
 746         -1.293850285839529282503178263484773478457e+0004L,
 747         -4.445024742266316181033354192262529356093e+0004L,
 748         -8.730161378334357767668344467356505347070e+0004L,
 749         -9.706222895172078442801444972505315054736e+0004L,
 750         -5.896325518259858270165531513618195321041e+0004L,
 751         -1.823172034368108822276420827074668832233e+0004L,
 752         -2.509304178635055926638833040337472387175e+0003L,
 753         -1.156608965715779237316769828941729964099e+0002L,
 754         -7.028005789650731396887346826397785210442e-0001L,
 755 };
 756 
 757 static GENERIC qs6[13] = {              /* [1.28, 1.777..] */
 758         1.0e0L,
 759         6.457211085058064845601261321277721075900e+0001L,
 760         1.534005216588011210342824555136008682950e+0003L,
 761         1.777217999176441782593357660462379097171e+0004L,
 762         1.118372652642469468091084810263231199696e+0005L,
 763         4.015242433858461813142365748386473605294e+0005L,
 764         8.377081045517098645448616514388280497673e+0005L,
 765         1.011495020008010352575398009604164287337e+0006L,
 766         6.886722075290430568652227875200208955970e+0005L,
 767         2.504735189948021472047157148613171956537e+0005L,
 768         4.408138920171044846941001844352009817062e+0004L,
 769         3.105572178072115145673058722853640854884e+0003L,
 770         5.588294821118916113437396504573817033678e+0001L,
 771 };
 772 
 773 static GENERIC
 774 qzero(x)
 775     GENERIC x;
 776 {
 777         GENERIC s, r, t, z;
 778         int i;
 779 
 780         if (x > huge)
 781                 return (-0.125L / x);
 782 
 783         t = one / x;
 784         z = t * t;
 785 
 786         if (x > sixteen) {
 787                 r = z * qr0[11] + qr0[10];
 788                 s = qs0[10];
 789 
 790                 for (i = 9; i >= 0; i--) {
 791                         r = z * r + qr0[i];
 792                         s = z * s + qs0[i];
 793                 }
 794         } else if (x > eight) {
 795                 r = qr1[11];
 796                 s = qs1[11] + z * (qs1[12] + z * qs1[13]);
 797 
 798                 for (i = 10; i >= 0; i--) {
 799                         r = z * r + qr1[i];
 800                         s = z * s + qs1[i];
 801                 }
 802         } else if (x > five) {               /* assume x > 5.0 */
 803                 r = qr2[11];
 804                 s = qs2[11] + z * (qs2[12] + z * qs2[13]);
 805 
 806                 for (i = 10; i >= 0; i--) {
 807                         r = z * r + qr2[i];
 808                         s = z * s + qs2[i];
 809                 }
 810         } else if (x > 3.5L) {
 811                 r = qr3[12];
 812                 s = qs3[12];
 813 
 814                 for (i = 11; i >= 0; i--) {
 815                         r = z * r + qr3[i];
 816                         s = z * s + qs3[i];
 817                 }
 818         } else if (x > 2.5L) {
 819                 r = qr4[12];
 820                 s = qs4[12];
 821 
 822                 for (i = 11; i >= 0; i--) {
 823                         r = z * r + qr4[i];
 824                         s = z * s + qs4[i];
 825                 }
 826         } else if (x > (1.0L / 0.5625L)) {
 827                 r = qr5[12];
 828                 s = qs5[12];
 829 
 830                 for (i = 11; i >= 0; i--) {
 831                         r = z * r + qr5[i];
 832                         s = z * s + qs5[i];
 833                 }
 834         } else {                        /* assume x > 1.28 */
 835                 r = qr6[12];
 836                 s = qs6[12];
 837 
 838                 for (i = 11; i >= 0; i--) {
 839                         r = z * r + qr6[i];
 840                         s = z * s + qs6[i];
 841                 }
 842         }
 843 
 844         return (t * (r / s));
 845 }