libstdc++-v3/testsuite/tr1/5_numerical_facilities/special_functions/02_assoc_legendre/check_value.cc

   1 // { dg-do run { target c++11 } }
   2 // { dg-options "-D__STDCPP_WANT_MATH_SPEC_FUNCS__ -ffp-contract=off" }
   3
   4 // Copyright (C) 2016-2018 Free Software Foundation, Inc.
   5 //
   6 // This file is part of the GNU ISO C++ Library.  This library is free
   7 // software; you can redistribute it and/or modify it under the
   8 // terms of the GNU General Public License as published by the
   9 // Free Software Foundation; either version 3, or (at your option)
  10 // any later version.
  11 //
  12 // This library is distributed in the hope that it will be useful,
  13 // but WITHOUT ANY WARRANTY; without even the implied warranty of
  14 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  15 // GNU General Public License for more details.
  16 //
  17 // You should have received a copy of the GNU General Public License along
  18 // with this library; see the file COPYING3.  If not see
  19 // <http://www.gnu.org/licenses/>.
  20
  21 //  assoc_legendre
  22 //  Compare against values generated by the GNU Scientific Library.
  23 //  The GSL can be found on the web: http://www.gnu.org/software/gsl/
  24 #include <limits>
  25 #include <tr1/cmath>
  26 #if defined(__TEST_DEBUG)
  27 #  include <iostream>
  28 #  define VERIFY(A) \
  29   if (!(A)) \
  30     { \
  31       std::cout << "line " << __LINE__ \
  32         << "  max_abs_frac = " << max_abs_frac \
  33         << '\n'; \
  34     }
  35 #else
  36 #  include <testsuite_hooks.h>
  37 #endif
  38 #include <specfun_testcase.h>
  39
  40 // Test data for l=0, m=0.
  41 // max(|f - f_GSL|): 0.0000000000000000 at index 0
  42 // max(|f - f_GSL| / |f_GSL|): 0.0000000000000000
  43 // mean(f - f_GSL): 0.0000000000000000
  44 // variance(f - f_GSL): 0.0000000000000000
  45 // stddev(f - f_GSL): 0.0000000000000000
  46 const testcase_assoc_legendre<double>
  47 data001[21] =
  48 {
  49   { 1.0000000000000000, 0, 0,
  50           -1.0000000000000000, 0.0 },
  51   { 1.0000000000000000, 0, 0,
  52           -0.90000000000000002, 0.0 },
  53   { 1.0000000000000000, 0, 0,
  54           -0.80000000000000004, 0.0 },
  55   { 1.0000000000000000, 0, 0,
  56           -0.69999999999999996, 0.0 },
  57   { 1.0000000000000000, 0, 0,
  58           -0.59999999999999998, 0.0 },
  59   { 1.0000000000000000, 0, 0,
  60           -0.50000000000000000, 0.0 },
  61   { 1.0000000000000000, 0, 0,
  62           -0.39999999999999991, 0.0 },
  63   { 1.0000000000000000, 0, 0,
  64           -0.29999999999999993, 0.0 },
  65   { 1.0000000000000000, 0, 0,
  66           -0.19999999999999996, 0.0 },
  67   { 1.0000000000000000, 0, 0,
  68           -0.099999999999999978, 0.0 },
  69   { 1.0000000000000000, 0, 0,
  70           0.0000000000000000, 0.0 },
  71   { 1.0000000000000000, 0, 0,
  72           0.10000000000000009, 0.0 },
  73   { 1.0000000000000000, 0, 0,
  74           0.20000000000000018, 0.0 },
  75   { 1.0000000000000000, 0, 0,
  76           0.30000000000000004, 0.0 },
  77   { 1.0000000000000000, 0, 0,
  78           0.40000000000000013, 0.0 },
  79   { 1.0000000000000000, 0, 0,
  80           0.50000000000000000, 0.0 },
  81   { 1.0000000000000000, 0, 0,
  82           0.60000000000000009, 0.0 },
  83   { 1.0000000000000000, 0, 0,
  84           0.70000000000000018, 0.0 },
  85   { 1.0000000000000000, 0, 0,
  86           0.80000000000000004, 0.0 },
  87   { 1.0000000000000000, 0, 0,
  88           0.90000000000000013, 0.0 },
  89   { 1.0000000000000000, 0, 0,
  90           1.0000000000000000, 0.0 },
  91 };
  92 const double toler001 = 2.5000000000000020e-13;
  93
  94 // Test data for l=1, m=0.
  95 // max(|f - f_GSL|): 0.0000000000000000 at index 0
  96 // max(|f - f_GSL| / |f_GSL|): 0.0000000000000000
  97 // mean(f - f_GSL): 0.0000000000000000
  98 // variance(f - f_GSL): 0.0000000000000000
  99 // stddev(f - f_GSL): 0.0000000000000000
 100 const testcase_assoc_legendre<double>
 101 data002[21] =
 102 {
 103   { -1.0000000000000000, 1, 0,
 104           -1.0000000000000000, 0.0 },
 105   { -0.90000000000000002, 1, 0,
 106           -0.90000000000000002, 0.0 },
 107   { -0.80000000000000004, 1, 0,
 108           -0.80000000000000004, 0.0 },
 109   { -0.69999999999999996, 1, 0,
 110           -0.69999999999999996, 0.0 },
 111   { -0.59999999999999998, 1, 0,
 112           -0.59999999999999998, 0.0 },
 113   { -0.50000000000000000, 1, 0,
 114           -0.50000000000000000, 0.0 },
 115   { -0.39999999999999991, 1, 0,
 116           -0.39999999999999991, 0.0 },
 117   { -0.29999999999999993, 1, 0,
 118           -0.29999999999999993, 0.0 },
 119   { -0.19999999999999996, 1, 0,
 120           -0.19999999999999996, 0.0 },
 121   { -0.099999999999999978, 1, 0,
 122           -0.099999999999999978, 0.0 },
 123   { 0.0000000000000000, 1, 0,
 124           0.0000000000000000, 0.0 },
 125   { 0.10000000000000009, 1, 0,
 126           0.10000000000000009, 0.0 },
 127   { 0.20000000000000018, 1, 0,
 128           0.20000000000000018, 0.0 },
 129   { 0.30000000000000004, 1, 0,
 130           0.30000000000000004, 0.0 },
 131   { 0.40000000000000013, 1, 0,
 132           0.40000000000000013, 0.0 },
 133   { 0.50000000000000000, 1, 0,
 134           0.50000000000000000, 0.0 },
 135   { 0.60000000000000009, 1, 0,
 136           0.60000000000000009, 0.0 },
 137   { 0.70000000000000018, 1, 0,
 138           0.70000000000000018, 0.0 },
 139   { 0.80000000000000004, 1, 0,
 140           0.80000000000000004, 0.0 },
 141   { 0.90000000000000013, 1, 0,
 142           0.90000000000000013, 0.0 },
 143   { 1.0000000000000000, 1, 0,
 144           1.0000000000000000, 0.0 },
 145 };
 146 const double toler002 = 2.5000000000000020e-13;
 147
 148 // Test data for l=1, m=1.
 149 // max(|f - f_GSL|): 2.2204460492503131e-16 at index 12
 150 // max(|f - f_GSL| / |f_GSL|): 2.4227052612512390e-16
 151 // mean(f - f_GSL): -1.0573552615477681e-17
 152 // variance(f - f_GSL): 5.8695007828944331e-36
 153 // stddev(f - f_GSL): 2.4227052612512388e-18
 154 const testcase_assoc_legendre<double>
 155 data003[21] =
 156 {
 157   { 0.0000000000000000, 1, 1,
 158           -1.0000000000000000, 0.0 },
 159   { 0.43588989435406728, 1, 1,
 160           -0.90000000000000002, 0.0 },
 161   { 0.59999999999999987, 1, 1,
 162           -0.80000000000000004, 0.0 },
 163   { 0.71414284285428509, 1, 1,
 164           -0.69999999999999996, 0.0 },
 165   { 0.80000000000000004, 1, 1,
 166           -0.59999999999999998, 0.0 },
 167   { 0.86602540378443860, 1, 1,
 168           -0.50000000000000000, 0.0 },
 169   { 0.91651513899116799, 1, 1,
 170           -0.39999999999999991, 0.0 },
 171   { 0.95393920141694555, 1, 1,
 172           -0.29999999999999993, 0.0 },
 173   { 0.97979589711327120, 1, 1,
 174           -0.19999999999999996, 0.0 },
 175   { 0.99498743710661997, 1, 1,
 176           -0.099999999999999978, 0.0 },
 177   { 1.0000000000000000, 1, 1,
 178           0.0000000000000000, 0.0 },
 179   { 0.99498743710661997, 1, 1,
 180           0.10000000000000009, 0.0 },
 181   { 0.97979589711327120, 1, 1,
 182           0.20000000000000018, 0.0 },
 183   { 0.95393920141694577, 1, 1,
 184           0.30000000000000004, 0.0 },
 185   { 0.91651513899116788, 1, 1,
 186           0.40000000000000013, 0.0 },
 187   { 0.86602540378443860, 1, 1,
 188           0.50000000000000000, 0.0 },
 189   { 0.79999999999999993, 1, 1,
 190           0.60000000000000009, 0.0 },
 191   { 0.71414284285428475, 1, 1,
 192           0.70000000000000018, 0.0 },
 193   { 0.59999999999999987, 1, 1,
 194           0.80000000000000004, 0.0 },
 195   { 0.43588989435406711, 1, 1,
 196           0.90000000000000013, 0.0 },
 197   { 0.0000000000000000, 1, 1,
 198           1.0000000000000000, 0.0 },
 199 };
 200 const double toler003 = 2.5000000000000020e-13;
 201
 202 // Test data for l=2, m=0.
 203 // max(|f - f_GSL|): 1.1102230246251565e-16 at index 17
 204 // max(|f - f_GSL| / |f_GSL|): 1.3877787807814482e-15
 205 // mean(f - f_GSL): 1.8503717077085941e-17
 206 // variance(f - f_GSL): 1.7975346147614202e-35
 207 // stddev(f - f_GSL): 4.2397342071896678e-18
 208 const testcase_assoc_legendre<double>
 209 data004[21] =
 210 {
 211   { 1.0000000000000000, 2, 0,
 212           -1.0000000000000000, 0.0 },
 213   { 0.71500000000000008, 2, 0,
 214           -0.90000000000000002, 0.0 },
 215   { 0.46000000000000019, 2, 0,
 216           -0.80000000000000004, 0.0 },
 217   { 0.23499999999999988, 2, 0,
 218           -0.69999999999999996, 0.0 },
 219   { 0.039999999999999925, 2, 0,
 220           -0.59999999999999998, 0.0 },
 221   { -0.12500000000000000, 2, 0,
 222           -0.50000000000000000, 0.0 },
 223   { -0.26000000000000012, 2, 0,
 224           -0.39999999999999991, 0.0 },
 225   { -0.36500000000000005, 2, 0,
 226           -0.29999999999999993, 0.0 },
 227   { -0.44000000000000006, 2, 0,
 228           -0.19999999999999996, 0.0 },
 229   { -0.48499999999999999, 2, 0,
 230           -0.099999999999999978, 0.0 },
 231   { -0.50000000000000000, 2, 0,
 232           0.0000000000000000, 0.0 },
 233   { -0.48499999999999999, 2, 0,
 234           0.10000000000000009, 0.0 },
 235   { -0.43999999999999989, 2, 0,
 236           0.20000000000000018, 0.0 },
 237   { -0.36499999999999999, 2, 0,
 238           0.30000000000000004, 0.0 },
 239   { -0.25999999999999984, 2, 0,
 240           0.40000000000000013, 0.0 },
 241   { -0.12500000000000000, 2, 0,
 242           0.50000000000000000, 0.0 },
 243   { 0.040000000000000147, 2, 0,
 244           0.60000000000000009, 0.0 },
 245   { 0.23500000000000032, 2, 0,
 246           0.70000000000000018, 0.0 },
 247   { 0.46000000000000019, 2, 0,
 248           0.80000000000000004, 0.0 },
 249   { 0.71500000000000030, 2, 0,
 250           0.90000000000000013, 0.0 },
 251   { 1.0000000000000000, 2, 0,
 252           1.0000000000000000, 0.0 },
 253 };
 254 const double toler004 = 2.5000000000000020e-13;
 255
 256 // Test data for l=2, m=1.
 257 // max(|f - f_GSL|): 2.2204460492503131e-16 at index 3
 258 // max(|f - f_GSL| / |f_GSL|): 3.7770554319736585e-16
 259 // mean(f - f_GSL): 1.0573552615477681e-17
 260 // variance(f - f_GSL): 5.8695007828944331e-36
 261 // stddev(f - f_GSL): 2.4227052612512388e-18
 262 const testcase_assoc_legendre<double>
 263 data005[21] =
 264 {
 265   { -0.0000000000000000, 2, 1,
 266           -1.0000000000000000, 0.0 },
 267   { -1.1769027147559816, 2, 1,
 268           -0.90000000000000002, 0.0 },
 269   { -1.4399999999999999, 2, 1,
 270           -0.80000000000000004, 0.0 },
 271   { -1.4996999699939983, 2, 1,
 272           -0.69999999999999996, 0.0 },
 273   { -1.4399999999999999, 2, 1,
 274           -0.59999999999999998, 0.0 },
 275   { -1.2990381056766580, 2, 1,
 276           -0.50000000000000000, 0.0 },
 277   { -1.0998181667894014, 2, 1,
 278           -0.39999999999999991, 0.0 },
 279   { -0.85854528127525076, 2, 1,
 280           -0.29999999999999993, 0.0 },
 281   { -0.58787753826796263, 2, 1,
 282           -0.19999999999999996, 0.0 },
 283   { -0.29849623113198592, 2, 1,
 284           -0.099999999999999978, 0.0 },
 285   { 0.0000000000000000, 2, 1,
 286           0.0000000000000000, 0.0 },
 287   { 0.29849623113198626, 2, 1,
 288           0.10000000000000009, 0.0 },
 289   { 0.58787753826796330, 2, 1,
 290           0.20000000000000018, 0.0 },
 291   { 0.85854528127525132, 2, 1,
 292           0.30000000000000004, 0.0 },
 293   { 1.0998181667894018, 2, 1,
 294           0.40000000000000013, 0.0 },
 295   { 1.2990381056766580, 2, 1,
 296           0.50000000000000000, 0.0 },
 297   { 1.4400000000000002, 2, 1,
 298           0.60000000000000009, 0.0 },
 299   { 1.4996999699939983, 2, 1,
 300           0.70000000000000018, 0.0 },
 301   { 1.4399999999999999, 2, 1,
 302           0.80000000000000004, 0.0 },
 303   { 1.1769027147559812, 2, 1,
 304           0.90000000000000013, 0.0 },
 305   { 0.0000000000000000, 2, 1,
 306           1.0000000000000000, 0.0 },
 307 };
 308 const double toler005 = 2.5000000000000020e-13;
 309
 310 // Test data for l=2, m=2.
 311 // max(|f - f_GSL|): 1.3322676295501878e-15 at index 12
 312 // max(|f - f_GSL| / |f_GSL|): 5.2867763077388426e-16
 313 // mean(f - f_GSL): -8.4588420923821446e-17
 314 // variance(f - f_GSL): 3.7564805010524372e-34
 315 // stddev(f - f_GSL): 1.9381642090009910e-17
 316 const testcase_assoc_legendre<double>
 317 data006[21] =
 318 {
 319   { 0.0000000000000000, 2, 2,
 320           -1.0000000000000000, 0.0 },
 321   { 0.56999999999999984, 2, 2,
 322           -0.90000000000000002, 0.0 },
 323   { 1.0799999999999996, 2, 2,
 324           -0.80000000000000004, 0.0 },
 325   { 1.5300000000000005, 2, 2,
 326           -0.69999999999999996, 0.0 },
 327   { 1.9200000000000004, 2, 2,
 328           -0.59999999999999998, 0.0 },
 329   { 2.2500000000000000, 2, 2,
 330           -0.50000000000000000, 0.0 },
 331   { 2.5200000000000000, 2, 2,
 332           -0.39999999999999991, 0.0 },
 333   { 2.7299999999999995, 2, 2,
 334           -0.29999999999999993, 0.0 },
 335   { 2.8799999999999999, 2, 2,
 336           -0.19999999999999996, 0.0 },
 337   { 2.9700000000000002, 2, 2,
 338           -0.099999999999999978, 0.0 },
 339   { 3.0000000000000000, 2, 2,
 340           0.0000000000000000, 0.0 },
 341   { 2.9700000000000002, 2, 2,
 342           0.10000000000000009, 0.0 },
 343   { 2.8799999999999999, 2, 2,
 344           0.20000000000000018, 0.0 },
 345   { 2.7300000000000004, 2, 2,
 346           0.30000000000000004, 0.0 },
 347   { 2.5199999999999991, 2, 2,
 348           0.40000000000000013, 0.0 },
 349   { 2.2500000000000000, 2, 2,
 350           0.50000000000000000, 0.0 },
 351   { 1.9199999999999997, 2, 2,
 352           0.60000000000000009, 0.0 },
 353   { 1.5299999999999989, 2, 2,
 354           0.70000000000000018, 0.0 },
 355   { 1.0799999999999996, 2, 2,
 356           0.80000000000000004, 0.0 },
 357   { 0.56999999999999929, 2, 2,
 358           0.90000000000000013, 0.0 },
 359   { 0.0000000000000000, 2, 2,
 360           1.0000000000000000, 0.0 },
 361 };
 362 const double toler006 = 2.5000000000000020e-13;
 363
 364 // Test data for l=5, m=0.
 365 // max(|f - f_GSL|): 2.0122792321330962e-16 at index 19
 366 // max(|f - f_GSL| / |f_GSL|): 4.8911475274405560e-15
 367 // mean(f - f_GSL): -2.3129646346357427e-18
 368 // variance(f - f_GSL): 2.8086478355647191e-37
 369 // stddev(f - f_GSL): 5.2996677589870847e-19
 370 const testcase_assoc_legendre<double>
 371 data007[21] =
 372 {
 373   { -1.0000000000000000, 5, 0,
 374           -1.0000000000000000, 0.0 },
 375   { 0.041141250000000087, 5, 0,
 376           -0.90000000000000002, 0.0 },
 377   { 0.39951999999999993, 5, 0,
 378           -0.80000000000000004, 0.0 },
 379   { 0.36519874999999991, 5, 0,
 380           -0.69999999999999996, 0.0 },
 381   { 0.15263999999999994, 5, 0,
 382           -0.59999999999999998, 0.0 },
 383   { -0.089843750000000000, 5, 0,
 384           -0.50000000000000000, 0.0 },
 385   { -0.27064000000000010, 5, 0,
 386           -0.39999999999999991, 0.0 },
 387   { -0.34538624999999995, 5, 0,
 388           -0.29999999999999993, 0.0 },
 389   { -0.30751999999999996, 5, 0,
 390           -0.19999999999999996, 0.0 },
 391   { -0.17882874999999995, 5, 0,
 392           -0.099999999999999978, 0.0 },
 393   { 0.0000000000000000, 5, 0,
 394           0.0000000000000000, 0.0 },
 395   { 0.17882875000000015, 5, 0,
 396           0.10000000000000009, 0.0 },
 397   { 0.30752000000000013, 5, 0,
 398           0.20000000000000018, 0.0 },
 399   { 0.34538625000000001, 5, 0,
 400           0.30000000000000004, 0.0 },
 401   { 0.27063999999999988, 5, 0,
 402           0.40000000000000013, 0.0 },
 403   { 0.089843750000000000, 5, 0,
 404           0.50000000000000000, 0.0 },
 405   { -0.15264000000000016, 5, 0,
 406           0.60000000000000009, 0.0 },
 407   { -0.36519875000000024, 5, 0,
 408           0.70000000000000018, 0.0 },
 409   { -0.39951999999999993, 5, 0,
 410           0.80000000000000004, 0.0 },
 411   { -0.041141249999999151, 5, 0,
 412           0.90000000000000013, 0.0 },
 413   { 1.0000000000000000, 5, 0,
 414           1.0000000000000000, 0.0 },
 415 };
 416 const double toler007 = 2.5000000000000020e-13;
 417
 418 // Test data for l=5, m=1.
 419 // max(|f - f_GSL|): 6.6613381477509392e-16 at index 14
 420 // max(|f - f_GSL| / |f_GSL|): 5.5186908001167120e-16
 421 // mean(f - f_GSL): 1.0573552615477681e-17
 422 // variance(f - f_GSL): 5.8695007828944331e-36
 423 // stddev(f - f_GSL): 2.4227052612512388e-18
 424 const testcase_assoc_legendre<double>
 425 data008[21] =
 426 {
 427   { -0.0000000000000000, 5, 1,
 428           -1.0000000000000000, 0.0 },
 429   { 2.8099369608350981, 5, 1,
 430           -0.90000000000000002, 0.0 },
 431   { 0.72180000000000089, 5, 1,
 432           -0.80000000000000004, 0.0 },
 433   { -1.0951826834447254, 5, 1,
 434           -0.69999999999999996, 0.0 },
 435   { -1.9775999999999998, 5, 1,
 436           -0.59999999999999998, 0.0 },
 437   { -1.9282596881137892, 5, 1,
 438           -0.50000000000000000, 0.0 },
 439   { -1.2070504380513671, 5, 1,
 440           -0.39999999999999991, 0.0 },
 441   { -0.16079837663884300, 5, 1,
 442           -0.29999999999999993, 0.0 },
 443   { 0.87005875663658538, 5, 1,
 444           -0.19999999999999996, 0.0 },
 445   { 1.6083350053680323, 5, 1,
 446           -0.099999999999999978, 0.0 },
 447   { 1.8750000000000000, 5, 1,
 448           0.0000000000000000, 0.0 },
 449   { 1.6083350053680314, 5, 1,
 450           0.10000000000000009, 0.0 },
 451   { 0.87005875663658327, 5, 1,
 452           0.20000000000000018, 0.0 },
 453   { -0.16079837663884422, 5, 1,
 454           0.30000000000000004, 0.0 },
 455   { -1.2070504380513694, 5, 1,
 456           0.40000000000000013, 0.0 },
 457   { -1.9282596881137892, 5, 1,
 458           0.50000000000000000, 0.0 },
 459   { -1.9775999999999998, 5, 1,
 460           0.60000000000000009, 0.0 },
 461   { -1.0951826834447216, 5, 1,
 462           0.70000000000000018, 0.0 },
 463   { 0.72180000000000089, 5, 1,
 464           0.80000000000000004, 0.0 },
 465   { 2.8099369608350999, 5, 1,
 466           0.90000000000000013, 0.0 },
 467   { -0.0000000000000000, 5, 1,
 468           1.0000000000000000, 0.0 },
 469 };
 470 const double toler008 = 2.5000000000000020e-13;
 471
 472 // Test data for l=5, m=2.
 473 // max(|f - f_GSL|): 3.5527136788005009e-15 at index 12
 474 // max(|f - f_GSL| / |f_GSL|): 5.4136589391245733e-16
 475 // mean(f - f_GSL): 0.0000000000000000
 476 // variance(f - f_GSL): 0.0000000000000000
 477 // stddev(f - f_GSL): 0.0000000000000000
 478 const testcase_assoc_legendre<double>
 479 data009[21] =
 480 {
 481   { 0.0000000000000000, 5, 2,
 482           -1.0000000000000000, 0.0 },
 483   { -12.837825000000000, 5, 2,
 484           -0.90000000000000002, 0.0 },
 485   { -13.910400000000001, 5, 2,
 486           -0.80000000000000004, 0.0 },
 487   { -8.8089749999999967, 5, 2,
 488           -0.69999999999999996, 0.0 },
 489   { -1.6128000000000000, 5, 2,
 490           -0.59999999999999998, 0.0 },
 491   { 4.9218750000000000, 5, 2,
 492           -0.50000000000000000, 0.0 },
 493   { 9.1728000000000005, 5, 2,
 494           -0.39999999999999991, 0.0 },
 495   { 10.462724999999997, 5, 2,
 496           -0.29999999999999993, 0.0 },
 497   { 8.8703999999999983, 5, 2,
 498           -0.19999999999999996, 0.0 },
 499   { 5.0415749999999990, 5, 2,
 500           -0.099999999999999978, 0.0 },
 501   { -0.0000000000000000, 5, 2,
 502           0.0000000000000000, 0.0 },
 503   { -5.0415750000000044, 5, 2,
 504           0.10000000000000009, 0.0 },
 505   { -8.8704000000000054, 5, 2,
 506           0.20000000000000018, 0.0 },
 507   { -10.462725000000001, 5, 2,
 508           0.30000000000000004, 0.0 },
 509   { -9.1727999999999970, 5, 2,
 510           0.40000000000000013, 0.0 },
 511   { -4.9218750000000000, 5, 2,
 512           0.50000000000000000, 0.0 },
 513   { 1.6128000000000047, 5, 2,
 514           0.60000000000000009, 0.0 },
 515   { 8.8089750000000109, 5, 2,
 516           0.70000000000000018, 0.0 },
 517   { 13.910400000000001, 5, 2,
 518           0.80000000000000004, 0.0 },
 519   { 12.837824999999990, 5, 2,
 520           0.90000000000000013, 0.0 },
 521   { 0.0000000000000000, 5, 2,
 522           1.0000000000000000, 0.0 },
 523 };
 524 const double toler009 = 2.5000000000000020e-13;
 525
 526 // Test data for l=5, m=5.
 527 // max(|f - f_GSL|): 1.0231815394945443e-12 at index 12
 528 // max(|f - f_GSL| / |f_GSL|): 1.1990652164995755e-15
 529 // mean(f - f_GSL): -3.1128538899966297e-14
 530 // variance(f - f_GSL): 5.0871761537452526e-29
 531 // stddev(f - f_GSL): 7.1324442891236469e-15
 532 const testcase_assoc_legendre<double>
 533 data010[21] =
 534 {
 535   { 0.0000000000000000, 5, 5,
 536           -1.0000000000000000, 0.0 },
 537   { 14.870165800941818, 5, 5,
 538           -0.90000000000000002, 0.0 },
 539   { 73.483199999999925, 5, 5,
 540           -0.80000000000000004, 0.0 },
 541   { 175.53238298794764, 5, 5,
 542           -0.69999999999999996, 0.0 },
 543   { 309.65760000000006, 5, 5,
 544           -0.59999999999999998, 0.0 },
 545   { 460.34662869916559, 5, 5,
 546           -0.50000000000000000, 0.0 },
 547   { 611.12496255819883, 5, 5,
 548           -0.39999999999999991, 0.0 },
 549   { 746.50941479523703, 5, 5,
 550           -0.29999999999999993, 0.0 },
 551   { 853.31600434671316, 5, 5,
 552           -0.19999999999999996, 0.0 },
 553   { 921.55189181724734, 5, 5,
 554           -0.099999999999999978, 0.0 },
 555   { 945.00000000000000, 5, 5,
 556           0.0000000000000000, 0.0 },
 557   { 921.55189181724734, 5, 5,
 558           0.10000000000000009, 0.0 },
 559   { 853.31600434671316, 5, 5,
 560           0.20000000000000018, 0.0 },
 561   { 746.50941479523760, 5, 5,
 562           0.30000000000000004, 0.0 },
 563   { 611.12496255819838, 5, 5,
 564           0.40000000000000013, 0.0 },
 565   { 460.34662869916559, 5, 5,
 566           0.50000000000000000, 0.0 },
 567   { 309.65759999999989, 5, 5,
 568           0.60000000000000009, 0.0 },
 569   { 175.53238298794724, 5, 5,
 570           0.70000000000000018, 0.0 },
 571   { 73.483199999999925, 5, 5,
 572           0.80000000000000004, 0.0 },
 573   { 14.870165800941789, 5, 5,
 574           0.90000000000000013, 0.0 },
 575   { 0.0000000000000000, 5, 5,
 576           1.0000000000000000, 0.0 },
 577 };
 578 const double toler010 = 2.5000000000000020e-13;
 579
 580 // Test data for l=10, m=0.
 581 // max(|f - f_GSL|): 3.8857805861880479e-16 at index 19
 582 // max(|f - f_GSL| / |f_GSL|): 1.4766655123690915e-15
 583 // mean(f - f_GSL): -2.5112187461759493e-17
 584 // variance(f - f_GSL): 3.3107652853513909e-35
 585 // stddev(f - f_GSL): 5.7539249954716919e-18
 586 const testcase_assoc_legendre<double>
 587 data011[21] =
 588 {
 589   { 1.0000000000000000, 10, 0,
 590           -1.0000000000000000, 0.0 },
 591   { -0.26314561785585960, 10, 0,
 592           -0.90000000000000002, 0.0 },
 593   { 0.30052979560000004, 10, 0,
 594           -0.80000000000000004, 0.0 },
 595   { 0.085805795531640333, 10, 0,
 596           -0.69999999999999996, 0.0 },
 597   { -0.24366274560000001, 10, 0,
 598           -0.59999999999999998, 0.0 },
 599   { -0.18822860717773438, 10, 0,
 600           -0.50000000000000000, 0.0 },
 601   { 0.096839064400000258, 10, 0,
 602           -0.39999999999999991, 0.0 },
 603   { 0.25147634951601561, 10, 0,
 604           -0.29999999999999993, 0.0 },
 605   { 0.12907202559999983, 10, 0,
 606           -0.19999999999999996, 0.0 },
 607   { -0.12212499738710943, 10, 0,
 608           -0.099999999999999978, 0.0 },
 609   { -0.24609375000000000, 10, 0,
 610           0.0000000000000000, 0.0 },
 611   { -0.12212499738710922, 10, 0,
 612           0.10000000000000009, 0.0 },
 613   { 0.12907202560000042, 10, 0,
 614           0.20000000000000018, 0.0 },
 615   { 0.25147634951601561, 10, 0,
 616           0.30000000000000004, 0.0 },
 617   { 0.096839064399999633, 10, 0,
 618           0.40000000000000013, 0.0 },
 619   { -0.18822860717773438, 10, 0,
 620           0.50000000000000000, 0.0 },
 621   { -0.24366274559999984, 10, 0,
 622           0.60000000000000009, 0.0 },
 623   { 0.085805795531641277, 10, 0,
 624           0.70000000000000018, 0.0 },
 625   { 0.30052979560000004, 10, 0,
 626           0.80000000000000004, 0.0 },
 627   { -0.26314561785586010, 10, 0,
 628           0.90000000000000013, 0.0 },
 629   { 1.0000000000000000, 10, 0,
 630           1.0000000000000000, 0.0 },
 631 };
 632 const double toler011 = 2.5000000000000020e-13;
 633
 634 // Test data for l=10, m=1.
 635 // max(|f - f_GSL|): 1.3322676295501878e-15 at index 14
 636 // max(|f - f_GSL| / |f_GSL|): 5.3672431731635395e-16
 637 // mean(f - f_GSL): -6.3441315692866085e-17
 638 // variance(f - f_GSL): 2.1130202818419960e-34
 639 // stddev(f - f_GSL): 1.4536231567507432e-17
 640 const testcase_assoc_legendre<double>
 641 data012[21] =
 642 {
 643   { 0.0000000000000000, 10, 1,
 644           -1.0000000000000000, 0.0 },
 645   { 3.0438748781479039, 10, 1,
 646           -0.90000000000000002, 0.0 },
 647   { 0.87614260800000254, 10, 1,
 648           -0.80000000000000004, 0.0 },
 649   { -2.9685359952934527, 10, 1,
 650           -0.69999999999999996, 0.0 },
 651   { -1.2511825919999997, 10, 1,
 652           -0.59999999999999998, 0.0 },
 653   { 2.0066877394361260, 10, 1,
 654           -0.50000000000000000, 0.0 },
 655   { 2.4822196173476647, 10, 1,
 656           -0.39999999999999991, 0.0 },
 657   { 0.12309508907433593, 10, 1,
 658           -0.29999999999999993, 0.0 },
 659   { -2.2468221751958413, 10, 1,
 660           -0.19999999999999996, 0.0 },
 661   { -2.2472659777983512, 10, 1,
 662           -0.099999999999999978, 0.0 },
 663   { 0.0000000000000000, 10, 1,
 664           0.0000000000000000, 0.0 },
 665   { 2.2472659777983535, 10, 1,
 666           0.10000000000000009, 0.0 },
 667   { 2.2468221751958377, 10, 1,
 668           0.20000000000000018, 0.0 },
 669   { -0.12309508907433910, 10, 1,
 670           0.30000000000000004, 0.0 },
 671   { -2.4822196173476669, 10, 1,
 672           0.40000000000000013, 0.0 },
 673   { -2.0066877394361260, 10, 1,
 674           0.50000000000000000, 0.0 },
 675   { 1.2511825920000037, 10, 1,
 676           0.60000000000000009, 0.0 },
 677   { 2.9685359952934505, 10, 1,
 678           0.70000000000000018, 0.0 },
 679   { -0.87614260800000254, 10, 1,
 680           0.80000000000000004, 0.0 },
 681   { -3.0438748781478981, 10, 1,
 682           0.90000000000000013, 0.0 },
 683   { -0.0000000000000000, 10, 1,
 684           1.0000000000000000, 0.0 },
 685 };
 686 const double toler012 = 2.5000000000000020e-13;
 687
 688 // Test data for l=10, m=2.
 689 // max(|f - f_GSL|): 1.2434497875801753e-14 at index 12
 690 // max(|f - f_GSL| / |f_GSL|): 1.9632900580960751e-15
 691 // mean(f - f_GSL): -8.4588420923821446e-17
 692 // variance(f - f_GSL): 3.7564805010524372e-34
 693 // stddev(f - f_GSL): 1.9381642090009910e-17
 694 const testcase_assoc_legendre<double>
 695 data013[21] =
 696 {
 697   { 0.0000000000000000, 10, 2,
 698           -1.0000000000000000, 0.0 },
 699   { 16.376387762496137, 10, 2,
 700           -0.90000000000000002, 0.0 },
 701   { -35.394657804000005, 10, 2,
 702           -0.80000000000000004, 0.0 },
 703   { -3.6191429423788648, 10, 2,
 704           -0.69999999999999996, 0.0 },
 705   { 28.679675904000014, 10, 2,
 706           -0.59999999999999998, 0.0 },
 707   { 18.388023376464844, 10, 2,
 708           -0.50000000000000000, 0.0 },
 709   { -12.818955996000021, 10, 2,
 710           -0.39999999999999991, 0.0 },
 711   { -27.739821675972646, 10, 2,
 712           -0.29999999999999993, 0.0 },
 713   { -13.280661503999987, 10, 2,
 714           -0.19999999999999996, 0.0 },
 715   { 13.885467170308601, 10, 2,
 716           -0.099999999999999978, 0.0 },
 717   { 27.070312500000000, 10, 2,
 718           0.0000000000000000, 0.0 },
 719   { 13.885467170308573, 10, 2,
 720           0.10000000000000009, 0.0 },
 721   { -13.280661504000051, 10, 2,
 722           0.20000000000000018, 0.0 },
 723   { -27.739821675972664, 10, 2,
 724           0.30000000000000004, 0.0 },
 725   { -12.818955995999961, 10, 2,
 726           0.40000000000000013, 0.0 },
 727   { 18.388023376464844, 10, 2,
 728           0.50000000000000000, 0.0 },
 729   { 28.679675904000000, 10, 2,
 730           0.60000000000000009, 0.0 },
 731   { -3.6191429423789856, 10, 2,
 732           0.70000000000000018, 0.0 },
 733   { -35.394657804000005, 10, 2,
 734           0.80000000000000004, 0.0 },
 735   { 16.376387762496201, 10, 2,
 736           0.90000000000000013, 0.0 },
 737   { 0.0000000000000000, 10, 2,
 738           1.0000000000000000, 0.0 },
 739 };
 740 const double toler013 = 2.5000000000000020e-13;
 741
 742 // Test data for l=10, m=5.
 743 // max(|f - f_GSL|): 2.9103830456733704e-11 at index 12
 744 // max(|f - f_GSL| / |f_GSL|): 1.4825078449301893e-15
 745 // mean(f - f_GSL): -2.4253192047278085e-12
 746 // variance(f - f_GSL): 3.0881409535313035e-25
 747 // stddev(f - f_GSL): 5.5571044200476413e-13
 748 const testcase_assoc_legendre<double>
 749 data014[21] =
 750 {
 751   { -0.0000000000000000, 10, 5,
 752           -1.0000000000000000, 0.0 },
 753   { -21343.618518164680, 10, 5,
 754           -0.90000000000000002, 0.0 },
 755   { -40457.016407807983, 10, 5,
 756           -0.80000000000000004, 0.0 },
 757   { -20321.279317331315, 10, 5,
 758           -0.69999999999999996, 0.0 },
 759   { 14410.820616192004, 10, 5,
 760           -0.59999999999999998, 0.0 },
 761   { 30086.169706116176, 10, 5,
 762           -0.50000000000000000, 0.0 },
 763   { 17177.549337582834, 10, 5,
 764           -0.39999999999999991, 0.0 },
 765   { -9272.5119495412546, 10, 5,
 766           -0.29999999999999993, 0.0 },
 767   { -26591.511184414714, 10, 5,
 768           -0.19999999999999996, 0.0 },
 769   { -21961.951238504211, 10, 5,
 770           -0.099999999999999978, 0.0 },
 771   { 0.0000000000000000, 10, 5,
 772           0.0000000000000000, 0.0 },
 773   { 21961.951238504229, 10, 5,
 774           0.10000000000000009, 0.0 },
 775   { 26591.511184414703, 10, 5,
 776           0.20000000000000018, 0.0 },
 777   { 9272.5119495412364, 10, 5,
 778           0.30000000000000004, 0.0 },
 779   { -17177.549337582877, 10, 5,
 780           0.40000000000000013, 0.0 },
 781   { -30086.169706116176, 10, 5,
 782           0.50000000000000000, 0.0 },
 783   { -14410.820616191972, 10, 5,
 784           0.60000000000000009, 0.0 },
 785   { 20321.279317331391, 10, 5,
 786           0.70000000000000018, 0.0 },
 787   { 40457.016407807983, 10, 5,
 788           0.80000000000000004, 0.0 },
 789   { 21343.618518164636, 10, 5,
 790           0.90000000000000013, 0.0 },
 791   { -0.0000000000000000, 10, 5,
 792           1.0000000000000000, 0.0 },
 793 };
 794 const double toler014 = 2.5000000000000020e-13;
 795
 796 // Test data for l=10, m=10.
 797 // max(|f - f_GSL|): 1.1920928955078125e-06 at index 12
 798 // max(|f - f_GSL| / |f_GSL|): 2.3944993944674592e-15
 799 // mean(f - f_GSL): -1.5610740298316593e-08
 800 // variance(f - f_GSL): 1.2793998664727994e-17
 801 // stddev(f - f_GSL): 3.5768699535666649e-09
 802 const testcase_assoc_legendre<double>
 803 data015[21] =
 804 {
 805   { 0.0000000000000000, 10, 10,
 806           -1.0000000000000000, 0.0 },
 807   { 162117.40078784220, 10, 10,
 808           -0.90000000000000002, 0.0 },
 809   { 3958896.3481267113, 10, 10,
 810           -0.80000000000000004, 0.0 },
 811   { 22589806.343887307, 10, 10,
 812           -0.69999999999999996, 0.0 },
 813   { 70300999.121633321, 10, 10,
 814           -0.59999999999999998, 0.0 },
 815   { 155370278.54003900, 10, 10,
 816           -0.50000000000000000, 0.0 },
 817   { 273815518.20150518, 10, 10,
 818           -0.39999999999999991, 0.0 },
 819   { 408571989.13158852, 10, 10,
 820           -0.29999999999999993, 0.0 },
 821   { 533848212.07990247, 10, 10,
 822           -0.19999999999999996, 0.0 },
 823   { 622640835.70523083, 10, 10,
 824           -0.099999999999999978, 0.0 },
 825   { 654729075.00000000, 10, 10,
 826           0.0000000000000000, 0.0 },
 827   { 622640835.70523083, 10, 10,
 828           0.10000000000000009, 0.0 },
 829   { 533848212.07990247, 10, 10,
 830           0.20000000000000018, 0.0 },
 831   { 408571989.13158917, 10, 10,
 832           0.30000000000000004, 0.0 },
 833   { 273815518.20150483, 10, 10,
 834           0.40000000000000013, 0.0 },
 835   { 155370278.54003900, 10, 10,
 836           0.50000000000000000, 0.0 },
 837   { 70300999.121633217, 10, 10,
 838           0.60000000000000009, 0.0 },
 839   { 22589806.343887202, 10, 10,
 840           0.70000000000000018, 0.0 },
 841   { 3958896.3481267113, 10, 10,
 842           0.80000000000000004, 0.0 },
 843   { 162117.40078784159, 10, 10,
 844           0.90000000000000013, 0.0 },
 845   { 0.0000000000000000, 10, 10,
 846           1.0000000000000000, 0.0 },
 847 };
 848 const double toler015 = 2.5000000000000020e-13;
 849
 850 // Test data for l=20, m=0.
 851 // max(|f - f_GSL|): 3.6082248300317588e-16 at index 19
 852 // max(|f - f_GSL| / |f_GSL|): 2.4166281401316513e-15
 853 // mean(f - f_GSL): 0.0000000000000000
 854 // variance(f - f_GSL): 4.9424644697959907e-65
 855 // stddev(f - f_GSL): 7.0302663319365015e-33
 856 const testcase_assoc_legendre<double>
 857 data016[21] =
 858 {
 859   { 1.0000000000000000, 20, 0,
 860           -1.0000000000000000, 0.0 },
 861   { -0.14930823530984835, 20, 0,
 862           -0.90000000000000002, 0.0 },
 863   { 0.22420460541741347, 20, 0,
 864           -0.80000000000000004, 0.0 },
 865   { -0.20457394463834172, 20, 0,
 866           -0.69999999999999996, 0.0 },
 867   { 0.15916752910098109, 20, 0,
 868           -0.59999999999999998, 0.0 },
 869   { -0.048358381067373557, 20, 0,
 870           -0.50000000000000000, 0.0 },
 871   { -0.10159261558628112, 20, 0,
 872           -0.39999999999999991, 0.0 },
 873   { 0.18028715947998047, 20, 0,
 874           -0.29999999999999993, 0.0 },
 875   { -0.098042194344594796, 20, 0,
 876           -0.19999999999999996, 0.0 },
 877   { -0.082077130944527663, 20, 0,
 878           -0.099999999999999978, 0.0 },
 879   { 0.17619705200195312, 20, 0,
 880           0.0000000000000000, 0.0 },
 881   { -0.082077130944528023, 20, 0,
 882           0.10000000000000009, 0.0 },
 883   { -0.098042194344594089, 20, 0,
 884           0.20000000000000018, 0.0 },
 885   { 0.18028715947998042, 20, 0,
 886           0.30000000000000004, 0.0 },
 887   { -0.10159261558628192, 20, 0,
 888           0.40000000000000013, 0.0 },
 889   { -0.048358381067373557, 20, 0,
 890           0.50000000000000000, 0.0 },
 891   { 0.15916752910098075, 20, 0,
 892           0.60000000000000009, 0.0 },
 893   { -0.20457394463834136, 20, 0,
 894           0.70000000000000018, 0.0 },
 895   { 0.22420460541741347, 20, 0,
 896           0.80000000000000004, 0.0 },
 897   { -0.14930823530984758, 20, 0,
 898           0.90000000000000013, 0.0 },
 899   { 1.0000000000000000, 20, 0,
 900           1.0000000000000000, 0.0 },
 901 };
 902 const double toler016 = 2.5000000000000020e-13;
 903
 904 // Test data for l=20, m=1.
 905 // max(|f - f_GSL|): 1.3322676295501878e-15 at index 12
 906 // max(|f - f_GSL| / |f_GSL|): 1.0752784502271902e-15
 907 // mean(f - f_GSL): -1.4274296030894871e-16
 908 // variance(f - f_GSL): 1.0697165176825104e-33
 909 // stddev(f - f_GSL): 3.2706521026891724e-17
 910 const testcase_assoc_legendre<double>
 911 data017[21] =
 912 {
 913   { -0.0000000000000000, 20, 1,
 914           -1.0000000000000000, 0.0 },
 915   { -4.3838334818220499, 20, 1,
 916           -0.90000000000000002, 0.0 },
 917   { 0.63138296146340844, 20, 1,
 918           -0.80000000000000004, 0.0 },
 919   { -0.72274871413391395, 20, 1,
 920           -0.69999999999999996, 0.0 },
 921   { 2.3203528743824910, 20, 1,
 922           -0.59999999999999998, 0.0 },
 923   { -3.7399919228791405, 20, 1,
 924           -0.50000000000000000, 0.0 },
 925   { 3.1692202279270085, 20, 1,
 926           -0.39999999999999991, 0.0 },
 927   { -0.15804468835345031, 20, 1,
 928           -0.29999999999999993, 0.0 },
 929   { -3.0366182393271171, 20, 1,
 930           -0.19999999999999996, 0.0 },
 931   { 3.2115523815580209, 20, 1,
 932           -0.099999999999999978, 0.0 },
 933   { -0.0000000000000000, 20, 1,
 934           0.0000000000000000, 0.0 },
 935   { -3.2115523815580169, 20, 1,
 936           0.10000000000000009, 0.0 },
 937   { 3.0366182393271259, 20, 1,
 938           0.20000000000000018, 0.0 },
 939   { 0.15804468835344135, 20, 1,
 940           0.30000000000000004, 0.0 },
 941   { -3.1692202279269970, 20, 1,
 942           0.40000000000000013, 0.0 },
 943   { 3.7399919228791405, 20, 1,
 944           0.50000000000000000, 0.0 },
 945   { -2.3203528743824995, 20, 1,
 946           0.60000000000000009, 0.0 },
 947   { 0.72274871413393793, 20, 1,
 948           0.70000000000000018, 0.0 },
 949   { -0.63138296146340844, 20, 1,
 950           0.80000000000000004, 0.0 },
 951   { 4.3838334818220686, 20, 1,
 952           0.90000000000000013, 0.0 },
 953   { -0.0000000000000000, 20, 1,
 954           1.0000000000000000, 0.0 },
 955 };
 956 const double toler017 = 2.5000000000000020e-13;
 957
 958 // Test data for l=20, m=2.
 959 // max(|f - f_GSL|): 2.4868995751603507e-14 at index 5
 960 // max(|f - f_GSL| / |f_GSL|): 1.0097407175524082e-15
 961 // mean(f - f_GSL): -2.0301221021717147e-15
 962 // variance(f - f_GSL): 2.1637327686062039e-31
 963 // stddev(f - f_GSL): 4.6515941016023782e-16
 964 const testcase_assoc_legendre<double>
 965 data018[21] =
 966 {
 967   { 0.0000000000000000, 20, 2,
 968           -1.0000000000000000, 0.0 },
 969   { 80.812425587310102, 20, 2,
 970           -0.90000000000000002, 0.0 },
 971   { -95.849622172549374, 20, 2,
 972           -0.80000000000000004, 0.0 },
 973   { 87.337927630325510, 20, 2,
 974           -0.69999999999999996, 0.0 },
 975   { -70.330891533985834, 20, 2,
 976           -0.59999999999999998, 0.0 },
 977   { 24.629090735179489, 20, 2,
 978           -0.50000000000000000, 0.0 },
 979   { 39.902576338912247, 20, 2,
 980           -0.39999999999999991, 0.0 },
 981   { -75.621201471396546, 20, 2,
 982           -0.29999999999999993, 0.0 },
 983   { 42.417415829726494, 20, 2,
 984           -0.19999999999999996, 0.0 },
 985   { 33.826848678871293, 20, 2,
 986           -0.099999999999999978, 0.0 },
 987   { -74.002761840820312, 20, 2,
 988           0.0000000000000000, 0.0 },
 989   { 33.826848678871464, 20, 2,
 990           0.10000000000000009, 0.0 },
 991   { 42.417415829726188, 20, 2,
 992           0.20000000000000018, 0.0 },
 993   { -75.621201471396603, 20, 2,
 994           0.30000000000000004, 0.0 },
 995   { 39.902576338912553, 20, 2,
 996           0.40000000000000013, 0.0 },
 997   { 24.629090735179489, 20, 2,
 998           0.50000000000000000, 0.0 },
 999   { -70.330891533985721, 20, 2,
1000           0.60000000000000009, 0.0 },
1001   { 87.337927630325453, 20, 2,
1002           0.70000000000000018, 0.0 },
1003   { -95.849622172549374, 20, 2,
1004           0.80000000000000004, 0.0 },
1005   { 80.812425587309747, 20, 2,
1006           0.90000000000000013, 0.0 },
1007   { 0.0000000000000000, 20, 2,
1008           1.0000000000000000, 0.0 },
1009 };
1010 const double toler018 = 2.5000000000000020e-13;
1011
1012 // Test data for l=20, m=5.
1013 // max(|f - f_GSL|): 1.2805685400962830e-09 at index 14
1014 // max(|f - f_GSL| / |f_GSL|): 2.0861530799041223e-15
1015 // mean(f - f_GSL): -1.1641532182693481e-10
1016 // variance(f - f_GSL): 7.1150767569361226e-22
1017 // stddev(f - f_GSL): 2.6674101216228678e-11
1018 const testcase_assoc_legendre<double>
1019 data019[21] =
1020 {
1021   { 0.0000000000000000, 20, 5,
1022           -1.0000000000000000, 0.0 },
1023   { 315702.32715134218, 20, 5,
1024           -0.90000000000000002, 0.0 },
1025   { -503060.91484852589, 20, 5,
1026           -0.80000000000000004, 0.0 },
1027   { 298127.28360361955, 20, 5,
1028           -0.69999999999999996, 0.0 },
1029   { 114444.61447464029, 20, 5,
1030           -0.59999999999999998, 0.0 },
1031   { -543428.40914592845, 20, 5,
1032           -0.50000000000000000, 0.0 },
1033   { 613842.07728185481, 20, 5,
1034           -0.39999999999999991, 0.0 },
1035   { -143765.42411271061, 20, 5,
1036           -0.29999999999999993, 0.0 },
1037   { -472600.45321372285, 20, 5,
1038           -0.19999999999999996, 0.0 },
1039   { 563861.76771496492, 20, 5,
1040           -0.099999999999999978, 0.0 },
1041   { -0.0000000000000000, 20, 5,
1042           0.0000000000000000, 0.0 },
1043   { -563861.76771496458, 20, 5,
1044           0.10000000000000009, 0.0 },
1045   { 472600.45321372483, 20, 5,
1046           0.20000000000000018, 0.0 },
1047   { 143765.42411270936, 20, 5,
1048           0.30000000000000004, 0.0 },
1049   { -613842.07728185353, 20, 5,
1050           0.40000000000000013, 0.0 },
1051   { 543428.40914592845, 20, 5,
1052           0.50000000000000000, 0.0 },
1053   { -114444.61447464178, 20, 5,
1054           0.60000000000000009, 0.0 },
1055   { -298127.28360361519, 20, 5,
1056           0.70000000000000018, 0.0 },
1057   { 503060.91484852589, 20, 5,
1058           0.80000000000000004, 0.0 },
1059   { -315702.32715133618, 20, 5,
1060           0.90000000000000013, 0.0 },
1061   { -0.0000000000000000, 20, 5,
1062           1.0000000000000000, 0.0 },
1063 };
1064 const double toler019 = 2.5000000000000020e-13;
1065
1066 // Test data for l=20, m=10.
1067 // max(|f - f_GSL|): 0.0039062500000000000 at index 12
1068 // max(|f - f_GSL| / |f_GSL|): 2.6634565089260680e-15
1069 // mean(f - f_GSL): -0.00043596540178571431
1070 // variance(f - f_GSL): 9.9784562846311215e-09
1071 // stddev(f - f_GSL): 9.9892223344117843e-05
1072 const testcase_assoc_legendre<double>
1073 data020[21] =
1074 {
1075   { -0.0000000000000000, 20, 10,
1076           -1.0000000000000000, 0.0 },
1077   { 990017476694.99084, 20, 10,
1078           -0.90000000000000002, 0.0 },
1079   { 2392757933281.0498, 20, 10,
1080           -0.80000000000000004, 0.0 },
1081   { -1548364524949.5808, 20, 10,
1082           -0.69999999999999996, 0.0 },
1083   { -424471915195.05627, 20, 10,
1084           -0.59999999999999998, 0.0 },
1085   { 1744502295946.2073, 20, 10,
1086           -0.50000000000000000, 0.0 },
1087   { -899973487310.55530, 20, 10,
1088           -0.39999999999999991, 0.0 },
1089   { -1092420454297.7119, 20, 10,
1090           -0.29999999999999993, 0.0 },
1091   { 1466609267659.8816, 20, 10,
1092           -0.19999999999999996, 0.0 },
1093   { 356041756390.71674, 20, 10,
1094           -0.099999999999999978, 0.0 },
1095   { -1612052956674.3164, 20, 10,
1096           0.0000000000000000, 0.0 },
1097   { 356041756390.71985, 20, 10,
1098           0.10000000000000009, 0.0 },
1099   { 1466609267659.8796, 20, 10,
1100           0.20000000000000018, 0.0 },
1101   { -1092420454297.7161, 20, 10,
1102           0.30000000000000004, 0.0 },
1103   { -899973487310.54810, 20, 10,
1104           0.40000000000000013, 0.0 },
1105   { 1744502295946.2073, 20, 10,
1106           0.50000000000000000, 0.0 },
1107   { -424471915195.05896, 20, 10,
1108           0.60000000000000009, 0.0 },
1109   { -1548364524949.5730, 20, 10,
1110           0.70000000000000018, 0.0 },
1111   { 2392757933281.0498, 20, 10,
1112           0.80000000000000004, 0.0 },
1113   { 990017476694.98828, 20, 10,
1114           0.90000000000000013, 0.0 },
1115   { 0.0000000000000000, 20, 10,
1116           1.0000000000000000, 0.0 },
1117 };
1118 const double toler020 = 2.5000000000000020e-13;
1119
1120 // Test data for l=20, m=20.
1121 // max(|f - f_GSL|): 838860800.00000000 at index 12
1122 // max(|f - f_GSL| / |f_GSL|): 4.3488507135833189e-15
1123 // mean(f - f_GSL): -2855497.1428571427
1124 // variance(f - f_GSL): inf
1125 // stddev(f - f_GSL): inf
1126 const testcase_assoc_legendre<double>
1127 data021[21] =
1128 {
1129   { 0.0000000000000000, 20, 20,
1130           -1.0000000000000000, 0.0 },
1131   { 19609049712023808., 20, 20,
1132           -0.90000000000000002, 0.0 },
1133   { 1.1693527616833221e+19, 20, 20,
1134           -0.80000000000000004, 0.0 },
1135   { 3.8073455880620691e+20, 20, 20,
1136           -0.69999999999999996, 0.0 },
1137   { 3.6874002249007927e+21, 20, 20,
1138           -0.59999999999999998, 0.0 },
1139   { 1.8010806978179592e+22, 20, 20,
1140           -0.50000000000000000, 0.0 },
1141   { 5.5938832584012466e+22, 20, 20,
1142           -0.39999999999999991, 0.0 },
1143   { 1.2454734132297759e+23, 20, 20,
1144           -0.29999999999999993, 0.0 },
1145   { 2.1263407800797497e+23, 20, 20,
1146           -0.19999999999999996, 0.0 },
1147   { 2.8924941146976873e+23, 20, 20,
1148           -0.099999999999999978, 0.0 },
1149   { 3.1983098677287775e+23, 20, 20,
1150           0.0000000000000000, 0.0 },
1151   { 2.8924941146976873e+23, 20, 20,
1152           0.10000000000000009, 0.0 },
1153   { 2.1263407800797497e+23, 20, 20,
1154           0.20000000000000018, 0.0 },
1155   { 1.2454734132297811e+23, 20, 20,
1156           0.30000000000000004, 0.0 },
1157   { 5.5938832584012366e+22, 20, 20,
1158           0.40000000000000013, 0.0 },
1159   { 1.8010806978179592e+22, 20, 20,
1160           0.50000000000000000, 0.0 },
1161   { 3.6874002249007807e+21, 20, 20,
1162           0.60000000000000009, 0.0 },
1163   { 3.8073455880620343e+20, 20, 20,
1164           0.70000000000000018, 0.0 },
1165   { 1.1693527616833221e+19, 20, 20,
1166           0.80000000000000004, 0.0 },
1167   { 19609049712023672., 20, 20,
1168           0.90000000000000013, 0.0 },
1169   { 0.0000000000000000, 20, 20,
1170           1.0000000000000000, 0.0 },
1171 };
1172 const double toler021 = 2.5000000000000020e-13;
1173
1174 // Test data for l=50, m=0.
1175 // max(|f - f_GSL|): 1.6653345369377348e-16 at index 2
1176 // max(|f - f_GSL| / |f_GSL|): 1.6665460706897444e-15
1177 // mean(f - f_GSL): -8.0953762212251003e-18
1178 // variance(f - f_GSL): 3.4405935985667807e-36
1179 // stddev(f - f_GSL): 1.8548837156454796e-18
1180 const testcase_assoc_legendre<double>
1181 data022[21] =
1182 {
1183   { 1.0000000000000000, 50, 0,
1184           -1.0000000000000000, 0.0 },
1185   { -0.17003765994383671, 50, 0,
1186           -0.90000000000000002, 0.0 },
1187   { 0.13879737345093113, 50, 0,
1188           -0.80000000000000004, 0.0 },
1189   { -0.014572731645892852, 50, 0,
1190           -0.69999999999999996, 0.0 },
1191   { -0.058860798844002096, 50, 0,
1192           -0.59999999999999998, 0.0 },
1193   { -0.031059099239609811, 50, 0,
1194           -0.50000000000000000, 0.0 },
1195   { 0.041569033381824674, 50, 0,
1196           -0.39999999999999991, 0.0 },
1197   { 0.10911051574714790, 50, 0,
1198           -0.29999999999999993, 0.0 },
1199   { 0.083432272204197494, 50, 0,
1200           -0.19999999999999996, 0.0 },
1201   { -0.038205812661313600, 50, 0,
1202           -0.099999999999999978, 0.0 },
1203   { -0.11227517265921705, 50, 0,
1204           0.0000000000000000, 0.0 },
1205   { -0.038205812661314155, 50, 0,
1206           0.10000000000000009, 0.0 },
1207   { 0.083432272204196564, 50, 0,
1208           0.20000000000000018, 0.0 },
1209   { 0.10911051574714797, 50, 0,
1210           0.30000000000000004, 0.0 },
1211   { 0.041569033381826007, 50, 0,
1212           0.40000000000000013, 0.0 },
1213   { -0.031059099239609811, 50, 0,
1214           0.50000000000000000, 0.0 },
1215   { -0.058860798844001430, 50, 0,
1216           0.60000000000000009, 0.0 },
1217   { -0.014572731645890737, 50, 0,
1218           0.70000000000000018, 0.0 },
1219   { 0.13879737345093113, 50, 0,
1220           0.80000000000000004, 0.0 },
1221   { -0.17003765994383679, 50, 0,
1222           0.90000000000000013, 0.0 },
1223   { 1.0000000000000000, 50, 0,
1224           1.0000000000000000, 0.0 },
1225 };
1226 const double toler022 = 2.5000000000000020e-13;
1227
1228 // Test data for l=50, m=1.
1229 // max(|f - f_GSL|): 4.4408920985006262e-15 at index 3
1230 // max(|f - f_GSL| / |f_GSL|): 6.6657627150738456e-16
1231 // mean(f - f_GSL): -6.3441315692866085e-17
1232 // variance(f - f_GSL): 2.1130202818419960e-34
1233 // stddev(f - f_GSL): 1.4536231567507432e-17
1234 const testcase_assoc_legendre<double>
1235 data023[21] =
1236 {
1237   { -0.0000000000000000, 50, 1,
1238           -1.0000000000000000, 0.0 },
1239   { 0.13424149984449490, 50, 1,
1240           -0.90000000000000002, 0.0 },
1241   { -2.2011219672413018, 50, 1,
1242           -0.80000000000000004, 0.0 },
1243   { -6.6622414993232004, 50, 1,
1244           -0.69999999999999996, 0.0 },
1245   { -5.5772846936919249, 50, 1,
1246           -0.59999999999999998, 0.0 },
1247   { -5.8787148815607608, 50, 1,
1248           -0.50000000000000000, 0.0 },
1249   { -5.5473459458634080, 50, 1,
1250           -0.39999999999999991, 0.0 },
1251   { -1.8444956647620248, 50, 1,
1252           -0.29999999999999993, 0.0 },
1253   { 3.8722014306642127, 50, 1,
1254           -0.19999999999999996, 0.0 },
1255   { 5.3488751322285628, 50, 1,
1256           -0.099999999999999978, 0.0 },
1257   { 0.0000000000000000, 50, 1,
1258           0.0000000000000000, 0.0 },
1259   { -5.3488751322285522, 50, 1,
1260           0.10000000000000009, 0.0 },
1261   { -3.8722014306642620, 50, 1,
1262           0.20000000000000018, 0.0 },
1263   { 1.8444956647619930, 50, 1,
1264           0.30000000000000004, 0.0 },
1265   { 5.5473459458633814, 50, 1,
1266           0.40000000000000013, 0.0 },
1267   { 5.8787148815607608, 50, 1,
1268           0.50000000000000000, 0.0 },
1269   { 5.5772846936919453, 50, 1,
1270           0.60000000000000009, 0.0 },
1271   { 6.6622414993232182, 50, 1,
1272           0.70000000000000018, 0.0 },
1273   { 2.2011219672413018, 50, 1,
1274           0.80000000000000004, 0.0 },
1275   { -0.13424149984438935, 50, 1,
1276           0.90000000000000013, 0.0 },
1277   { -0.0000000000000000, 50, 1,
1278           1.0000000000000000, 0.0 },
1279 };
1280 const double toler023 = 2.5000000000000020e-13;
1281
1282 // Test data for l=50, m=2.
1283 // max(|f - f_GSL|): 3.0553337637684308e-13 at index 3
1284 // max(|f - f_GSL| / |f_GSL|): 6.0837685805857877e-15
1285 // mean(f - f_GSL): -3.3835368369528578e-16
1286 // variance(f - f_GSL): 6.0103688016838995e-33
1287 // stddev(f - f_GSL): 7.7526568360039641e-17
1288 const testcase_assoc_legendre<double>
1289 data024[21] =
1290 {
1291   { 0.0000000000000000, 50, 2,
1292           -1.0000000000000000, 0.0 },
1293   { 433.04168483713511, 50, 2,
1294           -0.90000000000000002, 0.0 },
1295   { -348.06364372056424, 50, 2,
1296           -0.80000000000000004, 0.0 },
1297   { 50.221071418108444, 50, 2,
1298           -0.69999999999999996, 0.0 },
1299   { 158.46096409274352, 50, 2,
1300           -0.59999999999999998, 0.0 },
1301   { 85.988858299721457, 50, 2,
1302           -0.50000000000000000, 0.0 },
1303   { -101.15891460879088, 50, 2,
1304           -0.39999999999999991, 0.0 },
1305   { -277.07168105316526, 50, 2,
1306           -0.29999999999999993, 0.0 },
1307   { -214.33311373510401, 50, 2,
1308           -0.19999999999999996, 0.0 },
1309   { 96.349657930951665, 50, 2,
1310           -0.099999999999999978, 0.0 },
1311   { 286.30169028100346, 50, 2,
1312           0.0000000000000000, 0.0 },
1313   { 96.349657930953242, 50, 2,
1314           0.10000000000000009, 0.0 },
1315   { -214.33311373510165, 50, 2,
1316           0.20000000000000018, 0.0 },
1317   { -277.07168105316617, 50, 2,
1318           0.30000000000000004, 0.0 },
1319   { -101.15891460879435, 50, 2,
1320           0.40000000000000013, 0.0 },
1321   { 85.988858299721457, 50, 2,
1322           0.50000000000000000, 0.0 },
1323   { 158.46096409274153, 50, 2,
1324           0.60000000000000009, 0.0 },
1325   { 50.221071418103143, 50, 2,
1326           0.70000000000000018, 0.0 },
1327   { -348.06364372056424, 50, 2,
1328           0.80000000000000004, 0.0 },
1329   { 433.04168483713596, 50, 2,
1330           0.90000000000000013, 0.0 },
1331   { 0.0000000000000000, 50, 2,
1332           1.0000000000000000, 0.0 },
1333 };
1334 const double toler024 = 5.0000000000000039e-13;
1335
1336 // Test data for l=50, m=5.
1337 // max(|f - f_GSL|): 7.4505805969238281e-08 at index 14
1338 // max(|f - f_GSL| / |f_GSL|): 2.0088060426072767e-15
1339 // mean(f - f_GSL): 6.3862119402204238e-09
1340 // variance(f - f_GSL): 2.1411444046342303e-18
1341 // stddev(f - f_GSL): 1.4632649810045447e-09
1342 const testcase_assoc_legendre<double>
1343 data025[21] =
1344 {
1345   { 0.0000000000000000, 50, 5,
1346           -1.0000000000000000, 0.0 },
1347   { 27340473.952132829, 50, 5,
1348           -0.90000000000000002, 0.0 },
1349   { -27753716.768532373, 50, 5,
1350           -0.80000000000000004, 0.0 },
1351   { -40808153.913493633, 50, 5,
1352           -0.69999999999999996, 0.0 },
1353   { -32071189.035790090, 50, 5,
1354           -0.59999999999999998, 0.0 },
1355   { -36265736.218529105, 50, 5,
1356           -0.50000000000000000, 0.0 },
1357   { -37089596.700204931, 50, 5,
1358           -0.39999999999999991, 0.0 },
1359   { -14562029.629244687, 50, 5,
1360           -0.29999999999999993, 0.0 },
1361   { 23686895.217517190, 50, 5,
1362           -0.19999999999999996, 0.0 },
1363   { 34878992.965676002, 50, 5,
1364           -0.099999999999999978, 0.0 },
1365   { 0.0000000000000000, 50, 5,
1366           0.0000000000000000, 0.0 },
1367   { -34878992.965675958, 50, 5,
1368           0.10000000000000009, 0.0 },
1369   { -23686895.217517529, 50, 5,
1370           0.20000000000000018, 0.0 },
1371   { 14562029.629244499, 50, 5,
1372           0.30000000000000004, 0.0 },
1373   { 37089596.700204782, 50, 5,
1374           0.40000000000000013, 0.0 },
1375   { 36265736.218529105, 50, 5,
1376           0.50000000000000000, 0.0 },
1377   { 32071189.035790242, 50, 5,
1378           0.60000000000000009, 0.0 },
1379   { 40808153.913493834, 50, 5,
1380           0.70000000000000018, 0.0 },
1381   { 27753716.768532373, 50, 5,
1382           0.80000000000000004, 0.0 },
1383   { -27340473.952132136, 50, 5,
1384           0.90000000000000013, 0.0 },
1385   { -0.0000000000000000, 50, 5,
1386           1.0000000000000000, 0.0 },
1387 };
1388 const double toler025 = 2.5000000000000020e-13;
1389
1390 // Test data for l=50, m=10.
1391 // max(|f - f_GSL|): 22.000000000000000 at index 3
1392 // max(|f - f_GSL| / |f_GSL|): 2.2163032493360465e-15
1393 // mean(f - f_GSL): -1.9821428571428572
1394 // variance(f - f_GSL): inf
1395 // stddev(f - f_GSL): inf
1396 const testcase_assoc_legendre<double>
1397 data026[21] =
1398 {
1399   { -0.0000000000000000, 50, 10,
1400           -1.0000000000000000, 0.0 },
1401   { -8994661710093155.0, 50, 10,
1402           -0.90000000000000002, 0.0 },
1403   { 932311375306569.62, 50, 10,
1404           -0.80000000000000004, 0.0 },
1405   { 12153535011507012., 50, 10,
1406           -0.69999999999999996, 0.0 },
1407   { 12176690755542240., 50, 10,
1408           -0.59999999999999998, 0.0 },
1409   { 9180035388465754.0, 50, 10,
1410           -0.50000000000000000, 0.0 },
1411   { 889201701866984.00, 50, 10,
1412           -0.39999999999999991, 0.0 },
1413   { -9451384032851544.0, 50, 10,
1414           -0.29999999999999993, 0.0 },
1415   { -9926439446673564.0, 50, 10,
1416           -0.19999999999999996, 0.0 },
1417   { 2794368162749970.5, 50, 10,
1418           -0.099999999999999978, 0.0 },
1419   { 11452238249246346., 50, 10,
1420           0.0000000000000000, 0.0 },
1421   { 2794368162750031.0, 50, 10,
1422           0.10000000000000009, 0.0 },
1423   { -9926439446673506.0, 50, 10,
1424           0.20000000000000018, 0.0 },
1425   { -9451384032851604.0, 50, 10,
1426           0.30000000000000004, 0.0 },
1427   { 889201701866835.25, 50, 10,
1428           0.40000000000000013, 0.0 },
1429   { 9180035388465754.0, 50, 10,
1430           0.50000000000000000, 0.0 },
1431   { 12176690755542214., 50, 10,
1432           0.60000000000000009, 0.0 },
1433   { 12153535011506908., 50, 10,
1434           0.70000000000000018, 0.0 },
1435   { 932311375306569.62, 50, 10,
1436           0.80000000000000004, 0.0 },
1437   { -8994661710093013.0, 50, 10,
1438           0.90000000000000013, 0.0 },
1439   { 0.0000000000000000, 50, 10,
1440           1.0000000000000000, 0.0 },
1441 };
1442 const double toler026 = 2.5000000000000020e-13;
1443
1444 // Test data for l=50, m=20.
1445 // max(|f - f_GSL|): 3.6028797018963968e+18 at index 14
1446 // max(|f - f_GSL| / |f_GSL|): 5.0227025739283085e-15
1447 // mean(f - f_GSL): -1.8529095609752899e+17
1448 // variance(f - f_GSL): inf
1449 // stddev(f - f_GSL): inf
1450 const testcase_assoc_legendre<double>
1451 data027[21] =
1452 {
1453   { 0.0000000000000000, 50, 20,
1454           -1.0000000000000000, 0.0 },
1455   { 1.6630925158645501e+33, 50, 20,
1456           -0.90000000000000002, 0.0 },
1457   { 1.0622676657892052e+33, 50, 20,
1458           -0.80000000000000004, 0.0 },
1459   { 8.6022521164717112e+32, 50, 20,
1460           -0.69999999999999996, 0.0 },
1461   { 4.0860128756808466e+32, 50, 20,
1462           -0.59999999999999998, 0.0 },
1463   { -4.0169860814274459e+32, 50, 20,
1464           -0.50000000000000000, 0.0 },
1465   { -8.2324325279773994e+32, 50, 20,
1466           -0.39999999999999991, 0.0 },
1467   { -4.0054067236247267e+31, 50, 20,
1468           -0.29999999999999993, 0.0 },
1469   { 7.9309266056434309e+32, 50, 20,
1470           -0.19999999999999996, 0.0 },
1471   { 5.4151358290898977e+31, 50, 20,
1472           -0.099999999999999978, 0.0 },
1473   { -7.8735935697332210e+32, 50, 20,
1474           0.0000000000000000, 0.0 },
1475   { 5.4151358290894924e+31, 50, 20,
1476           0.10000000000000009, 0.0 },
1477   { 7.9309266056434453e+32, 50, 20,
1478           0.20000000000000018, 0.0 },
1479   { -4.0054067236243731e+31, 50, 20,
1480           0.30000000000000004, 0.0 },
1481   { -8.2324325279773893e+32, 50, 20,
1482           0.40000000000000013, 0.0 },
1483   { -4.0169860814274459e+32, 50, 20,
1484           0.50000000000000000, 0.0 },
1485   { 4.0860128756807846e+32, 50, 20,
1486           0.60000000000000009, 0.0 },
1487   { 8.6022521164716291e+32, 50, 20,
1488           0.70000000000000018, 0.0 },
1489   { 1.0622676657892052e+33, 50, 20,
1490           0.80000000000000004, 0.0 },
1491   { 1.6630925158645541e+33, 50, 20,
1492           0.90000000000000013, 0.0 },
1493   { 0.0000000000000000, 50, 20,
1494           1.0000000000000000, 0.0 },
1495 };
1496 const double toler027 = 5.0000000000000039e-13;
1497
1498 // Test data for l=50, m=50.
1499 // max(|f - f_GSL|): 1.0695779622587839e+64 at index 12
1500 // max(|f - f_GSL| / |f_GSL|): 1.1061123055945194e-14
1501 // mean(f - f_GSL): -2.0817504896587826e+62
1502 // variance(f - f_GSL): inf
1503 // stddev(f - f_GSL): inf
1504 const testcase_assoc_legendre<double>
1505 data028[21] =
1506 {
1507   { 0.0000000000000000, 50, 50,
1508           -1.0000000000000000, 0.0 },
1509   { 2.5366994974431341e+60, 50, 50,
1510           -0.90000000000000002, 0.0 },
1511   { 2.2028834403101213e+67, 50, 50,
1512           -0.80000000000000004, 0.0 },
1513   { 1.3325496559566651e+71, 50, 50,
1514           -0.69999999999999996, 0.0 },
1515   { 3.8898096431781969e+73, 50, 50,
1516           -0.59999999999999998, 0.0 },
1517   { 2.0509760257037188e+75, 50, 50,
1518           -0.50000000000000000, 0.0 },
1519   { 3.4866724533443283e+76, 50, 50,
1520           -0.39999999999999991, 0.0 },
1521   { 2.5790740224149893e+77, 50, 50,
1522           -0.29999999999999993, 0.0 },
1523   { 9.8222237931680989e+77, 50, 50,
1524           -0.19999999999999996, 0.0 },
1525   { 2.1198682190366617e+78, 50, 50,
1526           -0.099999999999999978, 0.0 },
1527   { 2.7253921397507295e+78, 50, 50,
1528           0.0000000000000000, 0.0 },
1529   { 2.1198682190366617e+78, 50, 50,
1530           0.10000000000000009, 0.0 },
1531   { 9.8222237931680989e+77, 50, 50,
1532           0.20000000000000018, 0.0 },
1533   { 2.5790740224150207e+77, 50, 50,
1534           0.30000000000000004, 0.0 },
1535   { 3.4866724533443123e+76, 50, 50,
1536           0.40000000000000013, 0.0 },
1537   { 2.0509760257037188e+75, 50, 50,
1538           0.50000000000000000, 0.0 },
1539   { 3.8898096431781724e+73, 50, 50,
1540           0.60000000000000009, 0.0 },
1541   { 1.3325496559566344e+71, 50, 50,
1542           0.70000000000000018, 0.0 },
1543   { 2.2028834403101213e+67, 50, 50,
1544           0.80000000000000004, 0.0 },
1545   { 2.5366994974430855e+60, 50, 50,
1546           0.90000000000000013, 0.0 },
1547   { 0.0000000000000000, 50, 50,
1548           1.0000000000000000, 0.0 },
1549 };
1550 const double toler028 = 1.0000000000000008e-12;
1551
1552 // Test data for l=100, m=0.
1553 // max(|f - f_GSL|): 3.4694469519536142e-16 at index 2
1554 // max(|f - f_GSL| / |f_GSL|): 6.8214063779431592e-15
1555 // mean(f - f_GSL): -4.1385545784018113e-17
1556 // variance(f - f_GSL): 8.9920078491655612e-35
1557 // stddev(f - f_GSL): 9.4826198116161765e-18
1558 const testcase_assoc_legendre<double>
1559 data029[21] =
1560 {
1561   { 1.0000000000000000, 100, 0,
1562           -1.0000000000000000, 0.0 },
1563   { 0.10226582055871893, 100, 0,
1564           -0.90000000000000002, 0.0 },
1565   { 0.050861167913584228, 100, 0,
1566           -0.80000000000000004, 0.0 },
1567   { -0.077132507199778641, 100, 0,
1568           -0.69999999999999996, 0.0 },
1569   { -0.023747023905133141, 100, 0,
1570           -0.59999999999999998, 0.0 },
1571   { -0.060518025961861198, 100, 0,
1572           -0.50000000000000000, 0.0 },
1573   { -0.072258202125685025, 100, 0,
1574           -0.39999999999999991, 0.0 },
1575   { 0.057127392202801046, 100, 0,
1576           -0.29999999999999993, 0.0 },
1577   { 0.014681835355659706, 100, 0,
1578           -0.19999999999999996, 0.0 },
1579   { -0.063895098434750205, 100, 0,
1580           -0.099999999999999978, 0.0 },
1581   { 0.079589237387178727, 100, 0,
1582           0.0000000000000000, 0.0 },
1583   { -0.063895098434749761, 100, 0,
1584           0.10000000000000009, 0.0 },
1585   { 0.014681835355657875, 100, 0,
1586           0.20000000000000018, 0.0 },
1587   { 0.057127392202801566, 100, 0,
1588           0.30000000000000004, 0.0 },
1589   { -0.072258202125684082, 100, 0,
1590           0.40000000000000013, 0.0 },
1591   { -0.060518025961861198, 100, 0,
1592           0.50000000000000000, 0.0 },
1593   { -0.023747023905134217, 100, 0,
1594           0.60000000000000009, 0.0 },
1595   { -0.077132507199780501, 100, 0,
1596           0.70000000000000018, 0.0 },
1597   { 0.050861167913584228, 100, 0,
1598           0.80000000000000004, 0.0 },
1599   { 0.10226582055872063, 100, 0,
1600           0.90000000000000013, 0.0 },
1601   { 1.0000000000000000, 100, 0,
1602           1.0000000000000000, 0.0 },
1603 };
1604 const double toler029 = 5.0000000000000039e-13;
1605
1606 // Test data for l=100, m=1.
1607 // max(|f - f_GSL|): 1.1546319456101628e-14 at index 3
1608 // max(|f - f_GSL| / |f_GSL|): 2.1111954004946762e-15
1609 // mean(f - f_GSL): 8.0358999877630379e-16
1610 // variance(f - f_GSL): 3.3902236521998243e-32
1611 // stddev(f - f_GSL): 1.8412559985509414e-16
1612 const testcase_assoc_legendre<double>
1613 data030[21] =
1614 {
1615   { 0.0000000000000000, 100, 1,
1616           -1.0000000000000000, 0.0 },
1617   { -6.5200167187780345, 100, 1,
1618           -0.90000000000000002, 0.0 },
1619   { -9.0065170007027486, 100, 1,
1620           -0.80000000000000004, 0.0 },
1621   { 5.4690908541180976, 100, 1,
1622           -0.69999999999999996, 0.0 },
1623   { 8.6275439170430790, 100, 1,
1624           -0.59999999999999998, 0.0 },
1625   { 6.0909031663448454, 100, 1,
1626           -0.50000000000000000, 0.0 },
1627   { -4.1160338699560395, 100, 1,
1628           -0.39999999999999991, 0.0 },
1629   { -5.8491043010758634, 100, 1,
1630           -0.29999999999999993, 0.0 },
1631   { 7.9435138723089826, 100, 1,
1632           -0.19999999999999996, 0.0 },
1633   { -4.7996285823989355, 100, 1,
1634           -0.099999999999999978, 0.0 },
1635   { -0.0000000000000000, 100, 1,
1636           0.0000000000000000, 0.0 },
1637   { 4.7996285823990101, 100, 1,
1638           0.10000000000000009, 0.0 },
1639   { -7.9435138723090155, 100, 1,
1640           0.20000000000000018, 0.0 },
1641   { 5.8491043010758013, 100, 1,
1642           0.30000000000000004, 0.0 },
1643   { 4.1160338699562162, 100, 1,
1644           0.40000000000000013, 0.0 },
1645   { -6.0909031663448454, 100, 1,
1646           0.50000000000000000, 0.0 },
1647   { -8.6275439170430470, 100, 1,
1648           0.60000000000000009, 0.0 },
1649   { -5.4690908541178693, 100, 1,
1650           0.70000000000000018, 0.0 },
1651   { 9.0065170007027486, 100, 1,
1652           0.80000000000000004, 0.0 },
1653   { 6.5200167187777787, 100, 1,
1654           0.90000000000000013, 0.0 },
1655   { -0.0000000000000000, 100, 1,
1656           1.0000000000000000, 0.0 },
1657 };
1658 const double toler030 = 2.5000000000000020e-13;
1659
1660 // Test data for l=100, m=2.
1661 // max(|f - f_GSL|): 6.8212102632969618e-13 at index 5
1662 // max(|f - f_GSL| / |f_GSL|): 2.0632212207447263e-15
1663 // mean(f - f_GSL): 8.2558298821649733e-14
1664 // variance(f - f_GSL): 3.5783331697705263e-28
1665 // stddev(f - f_GSL): 1.8916482679849673e-14
1666 const testcase_assoc_legendre<double>
1667 data031[21] =
1668 {
1669   { 0.0000000000000000, 100, 2,
1670           -1.0000000000000000, 0.0 },
1671   { -1005.9604880761002, 100, 2,
1672           -0.90000000000000002, 0.0 },
1673   { -489.68041725865947, 100, 2,
1674           -0.80000000000000004, 0.0 },
1675   { 768.31676011669924, 100, 2,
1676           -0.69999999999999996, 0.0 },
1677   { 226.90362556627937, 100, 2,
1678           -0.59999999999999998, 0.0 },
1679   { 604.19889304940341, 100, 2,
1680           -0.50000000000000000, 0.0 },
1681   { 733.40061037838518, 100, 2,
1682           -0.39999999999999991, 0.0 },
1683   { -573.30774483995629, 100, 2,
1684           -0.29999999999999993, 0.0 },
1685   { -151.52946305080880, 100, 2,
1686           -0.19999999999999996, 0.0 },
1687   { 646.30525583587985, 100, 2,
1688           -0.099999999999999978, 0.0 },
1689   { -803.85129761050518, 100, 2,
1690           0.0000000000000000, 0.0 },
1691   { 646.30525583587439, 100, 2,
1692           0.10000000000000009, 0.0 },
1693   { -151.52946305079013, 100, 2,
1694           0.20000000000000018, 0.0 },
1695   { -573.30774483996390, 100, 2,
1696           0.30000000000000004, 0.0 },
1697   { 733.40061037837552, 100, 2,
1698           0.40000000000000013, 0.0 },
1699   { 604.19889304940341, 100, 2,
1700           0.50000000000000000, 0.0 },
1701   { 226.90362556629168, 100, 2,
1702           0.60000000000000009, 0.0 },
1703   { 768.31676011671766, 100, 2,
1704           0.70000000000000018, 0.0 },
1705   { -489.68041725865947, 100, 2,
1706           0.80000000000000004, 0.0 },
1707   { -1005.9604880761161, 100, 2,
1708           0.90000000000000013, 0.0 },
1709   { 0.0000000000000000, 100, 2,
1710           1.0000000000000000, 0.0 },
1711 };
1712 const double toler031 = 2.5000000000000020e-13;
1713
1714 // Test data for l=100, m=5.
1715 // max(|f - f_GSL|): 1.4305114746093750e-06 at index 14
1716 // max(|f - f_GSL| / |f_GSL|): 3.7628882298853693e-15
1717 // mean(f - f_GSL): 5.1089695521763390e-08
1718 // variance(f - f_GSL): 1.3703324189659077e-16
1719 // stddev(f - f_GSL): 1.1706119848036358e-08
1720 const testcase_assoc_legendre<double>
1721 data032[21] =
1722 {
1723   { -0.0000000000000000, 100, 5,
1724           -1.0000000000000000, 0.0 },
1725   { -900551126.09653807, 100, 5,
1726           -0.90000000000000002, 0.0 },
1727   { -988567431.55756140, 100, 5,
1728           -0.80000000000000004, 0.0 },
1729   { 645646451.90344620, 100, 5,
1730           -0.69999999999999996, 0.0 },
1731   { 897114585.29920685, 100, 5,
1732           -0.59999999999999998, 0.0 },
1733   { 661710744.42483854, 100, 5,
1734           -0.50000000000000000, 0.0 },
1735   { -380163158.51424754, 100, 5,
1736           -0.39999999999999991, 0.0 },
1737   { -617391071.36633193, 100, 5,
1738           -0.29999999999999993, 0.0 },
1739   { 805288801.85509109, 100, 5,
1740           -0.19999999999999996, 0.0 },
1741   { -481041740.16728652, 100, 5,
1742           -0.099999999999999978, 0.0 },
1743   { -0.0000000000000000, 100, 5,
1744           0.0000000000000000, 0.0 },
1745   { 481041740.16729391, 100, 5,
1746           0.10000000000000009, 0.0 },
1747   { -805288801.85509515, 100, 5,
1748           0.20000000000000018, 0.0 },
1749   { 617391071.36632574, 100, 5,
1750           0.30000000000000004, 0.0 },
1751   { 380163158.51426536, 100, 5,
1752           0.40000000000000013, 0.0 },
1753   { -661710744.42483854, 100, 5,
1754           0.50000000000000000, 0.0 },
1755   { -897114585.29920483, 100, 5,
1756           0.60000000000000009, 0.0 },
1757   { -645646451.90342283, 100, 5,
1758           0.70000000000000018, 0.0 },
1759   { 988567431.55756140, 100, 5,
1760           0.80000000000000004, 0.0 },
1761   { 900551126.09651637, 100, 5,
1762           0.90000000000000013, 0.0 },
1763   { -0.0000000000000000, 100, 5,
1764           1.0000000000000000, 0.0 },
1765 };
1766 const double toler032 = 2.5000000000000020e-13;
1767
1768 // Test data for l=100, m=10.
1769 // max(|f - f_GSL|): 14336.000000000000 at index 14
1770 // max(|f - f_GSL| / |f_GSL|): 3.4905902237930355e-15
1771 // mean(f - f_GSL): 341.33333333333331
1772 // variance(f - f_GSL): inf
1773 // stddev(f - f_GSL): inf
1774 const testcase_assoc_legendre<double>
1775 data033[21] =
1776 {
1777   { 0.0000000000000000, 100, 10,
1778           -1.0000000000000000, 0.0 },
1779   { 2.5643395957658602e+17, 100, 10,
1780           -0.90000000000000002, 0.0 },
1781   { 1.5778673545673485e+18, 100, 10,
1782           -0.80000000000000004, 0.0 },
1783   { 4.4355048487496801e+18, 100, 10,
1784           -0.69999999999999996, 0.0 },
1785   { -9.5936111659124288e+17, 100, 10,
1786           -0.59999999999999998, 0.0 },
1787   { 4.2387123021963438e+18, 100, 10,
1788           -0.50000000000000000, 0.0 },
1789   { 8.2370834618426767e+18, 100, 10,
1790           -0.39999999999999991, 0.0 },
1791   { -4.9089358388051978e+18, 100, 10,
1792           -0.29999999999999993, 0.0 },
1793   { -2.3468810358091274e+18, 100, 10,
1794           -0.19999999999999996, 0.0 },
1795   { 6.8627855225034568e+18, 100, 10,
1796           -0.099999999999999978, 0.0 },
1797   { -8.2494597181670380e+18, 100, 10,
1798           0.0000000000000000, 0.0 },
1799   { 6.8627855225034056e+18, 100, 10,
1800           0.10000000000000009, 0.0 },
1801   { -2.3468810358089518e+18, 100, 10,
1802           0.20000000000000018, 0.0 },
1803   { -4.9089358388052941e+18, 100, 10,
1804           0.30000000000000004, 0.0 },
1805   { 8.2370834618426112e+18, 100, 10,
1806           0.40000000000000013, 0.0 },
1807   { 4.2387123021963438e+18, 100, 10,
1808           0.50000000000000000, 0.0 },
1809   { -9.5936111659112640e+17, 100, 10,
1810           0.60000000000000009, 0.0 },
1811   { 4.4355048487499668e+18, 100, 10,
1812           0.70000000000000018, 0.0 },
1813   { 1.5778673545673485e+18, 100, 10,
1814           0.80000000000000004, 0.0 },
1815   { 2.5643395957630058e+17, 100, 10,
1816           0.90000000000000013, 0.0 },
1817   { 0.0000000000000000, 100, 10,
1818           1.0000000000000000, 0.0 },
1819 };
1820 const double toler033 = 2.5000000000000020e-13;
1821
1822 // Test data for l=100, m=20.
1823 // max(|f - f_GSL|): 3.9290089137475448e+24 at index 14
1824 // max(|f - f_GSL| / |f_GSL|): 5.1615589395022552e-15
1825 // mean(f - f_GSL): -5.0371909150609551e+23
1826 // variance(f - f_GSL): inf
1827 // stddev(f - f_GSL): inf
1828 const testcase_assoc_legendre<double>
1829 data034[21] =
1830 {
1831   { 0.0000000000000000, 100, 20,
1832           -1.0000000000000000, 0.0 },
1833   { 7.1604344878780134e+37, 100, 20,
1834           -0.90000000000000002, 0.0 },
1835   { -8.3963895116962231e+38, 100, 20,
1836           -0.80000000000000004, 0.0 },
1837   { 7.9022236853110145e+38, 100, 20,
1838           -0.69999999999999996, 0.0 },
1839   { 8.2680005574121013e+38, 100, 20,
1840           -0.59999999999999998, 0.0 },
1841   { 3.0750497039999552e+38, 100, 20,
1842           -0.50000000000000000, 0.0 },
1843   { -7.6120586043843556e+38, 100, 20,
1844           -0.39999999999999991, 0.0 },
1845   { 1.1474496891900921e+38, 100, 20,
1846           -0.29999999999999993, 0.0 },
1847   { 4.3966251307444241e+38, 100, 20,
1848           -0.19999999999999996, 0.0 },
1849   { -7.0503266451702591e+38, 100, 20,
1850           -0.099999999999999978, 0.0 },
1851   { 7.7727439836159581e+38, 100, 20,
1852           0.0000000000000000, 0.0 },
1853   { -7.0503266451702213e+38, 100, 20,
1854           0.10000000000000009, 0.0 },
1855   { 4.3966251307442783e+38, 100, 20,
1856           0.20000000000000018, 0.0 },
1857   { 1.1474496891901797e+38, 100, 20,
1858           0.30000000000000004, 0.0 },
1859   { -7.6120586043844176e+38, 100, 20,
1860           0.40000000000000013, 0.0 },
1861   { 3.0750497039999552e+38, 100, 20,
1862           0.50000000000000000, 0.0 },
1863   { 8.2680005574120394e+38, 100, 20,
1864           0.60000000000000009, 0.0 },
1865   { 7.9022236853108422e+38, 100, 20,
1866           0.70000000000000018, 0.0 },
1867   { -8.3963895116962231e+38, 100, 20,
1868           0.80000000000000004, 0.0 },
1869   { 7.1604344878751847e+37, 100, 20,
1870           0.90000000000000013, 0.0 },
1871   { 0.0000000000000000, 100, 20,
1872           1.0000000000000000, 0.0 },
1873 };
1874 const double toler034 = 5.0000000000000039e-13;
1875
1876 // Test data for l=100, m=50.
1877 // max(|f - f_GSL|): 6.8296953242310788e+83 at index 14
1878 // max(|f - f_GSL| / |f_GSL|): 2.0345950516284159e-14
1879 // mean(f - f_GSL): 7.0103750947133818e+82
1880 // variance(f - f_GSL): inf
1881 // stddev(f - f_GSL): inf
1882 const testcase_assoc_legendre<double>
1883 data035[21] =
1884 {
1885   { 0.0000000000000000, 100, 50,
1886           -1.0000000000000000, 0.0 },
1887   { 9.3231278516893716e+96, 100, 50,
1888           -0.90000000000000002, 0.0 },
1889   { -1.1029797977454281e+98, 100, 50,
1890           -0.80000000000000004, 0.0 },
1891   { 1.8089333903465606e+97, 100, 50,
1892           -0.69999999999999996, 0.0 },
1893   { 5.9364045925669405e+97, 100, 50,
1894           -0.59999999999999998, 0.0 },
1895   { -8.2252620339727118e+97, 100, 50,
1896           -0.50000000000000000, 0.0 },
1897   { 7.1431385093740728e+97, 100, 50,
1898           -0.39999999999999991, 0.0 },
1899   { -3.3520602067479935e+97, 100, 50,
1900           -0.29999999999999993, 0.0 },
1901   { -2.7791149588121382e+97, 100, 50,
1902           -0.19999999999999996, 0.0 },
1903   { 9.0119338550180417e+97, 100, 50,
1904           -0.099999999999999978, 0.0 },
1905   { -1.1712145031578381e+98, 100, 50,
1906           0.0000000000000000, 0.0 },
1907   { 9.0119338550181207e+97, 100, 50,
1908           0.10000000000000009, 0.0 },
1909   { -2.7791149588123644e+97, 100, 50,
1910           0.20000000000000018, 0.0 },
1911   { -3.3520602067479344e+97, 100, 50,
1912           0.30000000000000004, 0.0 },
1913   { 7.1431385093738816e+97, 100, 50,
1914           0.40000000000000013, 0.0 },
1915   { -8.2252620339727118e+97, 100, 50,
1916           0.50000000000000000, 0.0 },
1917   { 5.9364045925668024e+97, 100, 50,
1918           0.60000000000000009, 0.0 },
1919   { 1.8089333903469005e+97, 100, 50,
1920           0.70000000000000018, 0.0 },
1921   { -1.1029797977454281e+98, 100, 50,
1922           0.80000000000000004, 0.0 },
1923   { 9.3231278516892938e+96, 100, 50,
1924           0.90000000000000013, 0.0 },
1925   { 0.0000000000000000, 100, 50,
1926           1.0000000000000000, 0.0 },
1927 };
1928 const double toler035 = 2.5000000000000015e-12;
1929
1930 // Test data for l=100, m=100.
1931 // max(|f - f_GSL|): 1.9201920817492002e+172 at index 12
1932 // max(|f - f_GSL| / |f_GSL|): 2.3348104188683898e-14
1933 // mean(f - f_GSL): -4.0709851490379341e+170
1934 // variance(f - f_GSL): inf
1935 // stddev(f - f_GSL): inf
1936 const testcase_assoc_legendre<double>
1937 data036[21] =
1938 {
1939   { 0.0000000000000000, 100, 100,
1940           -1.0000000000000000, 0.0 },
1941   { 5.7751792255758316e+150, 100, 100,
1942           -0.90000000000000002, 0.0 },
1943   { 4.3552236041585515e+164, 100, 100,
1944           -0.80000000000000004, 0.0 },
1945   { 1.5936546850595123e+172, 100, 100,
1946           -0.69999999999999996, 0.0 },
1947   { 1.3579510590289176e+177, 100, 100,
1948           -0.59999999999999998, 0.0 },
1949   { 3.7752749682889513e+180, 100, 100,
1950           -0.50000000000000000, 0.0 },
1951   { 1.0910627330458913e+183, 100, 100,
1952           -0.39999999999999991, 0.0 },
1953   { 5.9697347526821064e+184, 100, 100,
1954           -0.29999999999999993, 0.0 },
1955   { 8.6585879147526714e+185, 100, 100,
1956           -0.19999999999999996, 0.0 },
1957   { 4.0331571908057011e+186, 100, 100,
1958           -0.099999999999999978, 0.0 },
1959   { 6.6663086700729543e+186, 100, 100,
1960           0.0000000000000000, 0.0 },
1961   { 4.0331571908057011e+186, 100, 100,
1962           0.10000000000000009, 0.0 },
1963   { 8.6585879147526714e+185, 100, 100,
1964           0.20000000000000018, 0.0 },
1965   { 5.9697347526822483e+184, 100, 100,
1966           0.30000000000000004, 0.0 },
1967   { 1.0910627330458797e+183, 100, 100,
1968           0.40000000000000013, 0.0 },
1969   { 3.7752749682889513e+180, 100, 100,
1970           0.50000000000000000, 0.0 },
1971   { 1.3579510590289000e+177, 100, 100,
1972           0.60000000000000009, 0.0 },
1973   { 1.5936546850594382e+172, 100, 100,
1974           0.70000000000000018, 0.0 },
1975   { 4.3552236041585515e+164, 100, 100,
1976           0.80000000000000004, 0.0 },
1977   { 5.7751792255756128e+150, 100, 100,
1978           0.90000000000000013, 0.0 },
1979   { 0.0000000000000000, 100, 100,
1980           1.0000000000000000, 0.0 },
1981 };
1982 const double toler036 = 2.5000000000000015e-12;
1983
1984 template<typename Ret, unsigned int Num>
1985   void
1986   test(const testcase_assoc_legendre<Ret> (&data)[Num], Ret toler)
1987   {
1988     bool test __attribute__((unused)) = true;
1989     const Ret eps = std::numeric_limits<Ret>::epsilon();
1990     Ret max_abs_diff = -Ret(1);
1991     Ret max_abs_frac = -Ret(1);
1992     unsigned int num_datum = Num;
1993     for (unsigned int i = 0; i < num_datum; ++i)
1994          {
1995         const Ret f = std::tr1::assoc_legendre(data[i].l, data[i].m,
1996                      data[i].x);
1997         const Ret f0 = data[i].f0;
1998         const Ret diff = f - f0;
1999         if (std::abs(diff) > max_abs_diff)
2000           max_abs_diff = std::abs(diff);
2001         if (std::abs(f0) > Ret(10) * eps
2002          && std::abs(f) > Ret(10) * eps)
2003           {
2004             const Ret frac = diff / f0;
2005             if (std::abs(frac) > max_abs_frac)
2006               max_abs_frac = std::abs(frac);
2007           }
2008       }
2009     VERIFY(max_abs_frac < toler);
2010   }
2011
2012 int
2013 main()
2014 {
2015   test(data001, toler001);
2016   test(data002, toler002);
2017   test(data003, toler003);
2018   test(data004, toler004);
2019   test(data005, toler005);
2020   test(data006, toler006);
2021   test(data007, toler007);
2022   test(data008, toler008);
2023   test(data009, toler009);
2024   test(data010, toler010);
2025   test(data011, toler011);
2026   test(data012, toler012);
2027   test(data013, toler013);
2028   test(data014, toler014);
2029   test(data015, toler015);
2030   test(data016, toler016);
2031   test(data017, toler017);
2032   test(data018, toler018);
2033   test(data019, toler019);
2034   test(data020, toler020);
2035   test(data021, toler021);
2036   test(data022, toler022);
2037   test(data023, toler023);
2038   test(data024, toler024);
2039   test(data025, toler025);
2040   test(data026, toler026);
2041   test(data027, toler027);
2042   test(data028, toler028);
2043   test(data029, toler029);
2044   test(data030, toler030);
2045   test(data031, toler031);
2046   test(data032, toler032);
2047   test(data033, toler033);
2048   test(data034, toler034);
2049   test(data035, toler035);
2050   test(data036, toler036);
2051   return 0;
2052 }