libstdc++-v3/testsuite/special_functions/20_sph_legendre/check_value.cc

   1 // { dg-do run { target c++11 } }
   2 // { dg-options "-D__STDCPP_WANT_MATH_SPEC_FUNCS__" }
   3 //
   4 // Copyright (C) 2016-2020 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 //  sph_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 <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         << std::endl; \
  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_sph_legendre<double>
  47 data001[21] =
  48 {
  49   { 0.28209479177387814, 0, 0,
  50           0.0000000000000000, 0.0 },
  51   { 0.28209479177387814, 0, 0,
  52           0.15707963267948966, 0.0 },
  53   { 0.28209479177387814, 0, 0,
  54           0.31415926535897931, 0.0 },
  55   { 0.28209479177387814, 0, 0,
  56           0.47123889803846897, 0.0 },
  57   { 0.28209479177387814, 0, 0,
  58           0.62831853071795862, 0.0 },
  59   { 0.28209479177387814, 0, 0,
  60           0.78539816339744828, 0.0 },
  61   { 0.28209479177387814, 0, 0,
  62           0.94247779607693793, 0.0 },
  63   { 0.28209479177387814, 0, 0,
  64           1.0995574287564276, 0.0 },
  65   { 0.28209479177387814, 0, 0,
  66           1.2566370614359172, 0.0 },
  67   { 0.28209479177387814, 0, 0,
  68           1.4137166941154069, 0.0 },
  69   { 0.28209479177387814, 0, 0,
  70           1.5707963267948966, 0.0 },
  71   { 0.28209479177387814, 0, 0,
  72           1.7278759594743862, 0.0 },
  73   { 0.28209479177387814, 0, 0,
  74           1.8849555921538759, 0.0 },
  75   { 0.28209479177387814, 0, 0,
  76           2.0420352248333655, 0.0 },
  77   { 0.28209479177387814, 0, 0,
  78           2.1991148575128552, 0.0 },
  79   { 0.28209479177387814, 0, 0,
  80           2.3561944901923448, 0.0 },
  81   { 0.28209479177387814, 0, 0,
  82           2.5132741228718345, 0.0 },
  83   { 0.28209479177387814, 0, 0,
  84           2.6703537555513241, 0.0 },
  85   { 0.28209479177387814, 0, 0,
  86           2.8274333882308138, 0.0 },
  87   { 0.28209479177387814, 0, 0,
  88           2.9845130209103035, 0.0 },
  89   { 0.28209479177387814, 0, 0,
  90           3.1415926535897931, 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_sph_legendre<double>
 101 data002[21] =
 102 {
 103   { 0.48860251190291992, 1, 0,
 104           0.0000000000000000, 0.0 },
 105   { 0.48258700419201095, 1, 0,
 106           0.15707963267948966, 0.0 },
 107   { 0.46468860282345237, 1, 0,
 108           0.31415926535897931, 0.0 },
 109   { 0.43534802584032634, 1, 0,
 110           0.47123889803846897, 0.0 },
 111   { 0.39528773562374969, 1, 0,
 112           0.62831853071795862, 0.0 },
 113   { 0.34549414947133550, 1, 0,
 114           0.78539816339744828, 0.0 },
 115   { 0.28719335072959395, 1, 0,
 116           0.94247779607693793, 0.0 },
 117   { 0.22182089855280449, 1, 0,
 118           1.0995574287564276, 0.0 },
 119   { 0.15098647967228981, 1, 0,
 120           1.2566370614359172, 0.0 },
 121   { 0.076434272566846345, 1, 0,
 122           1.4137166941154069, 0.0 },
 123   { 2.9918275112863332e-17, 1, 0,
 124           1.5707963267948966, 0.0 },
 125   { -0.076434272566846290, 1, 0,
 126           1.7278759594743862, 0.0 },
 127   { -0.15098647967228976, 1, 0,
 128           1.8849555921538759, 0.0 },
 129   { -0.22182089855280446, 1, 0,
 130           2.0420352248333655, 0.0 },
 131   { -0.28719335072959384, 1, 0,
 132           2.1991148575128552, 0.0 },
 133   { -0.34549414947133544, 1, 0,
 134           2.3561944901923448, 0.0 },
 135   { -0.39528773562374975, 1, 0,
 136           2.5132741228718345, 0.0 },
 137   { -0.43534802584032628, 1, 0,
 138           2.6703537555513241, 0.0 },
 139   { -0.46468860282345231, 1, 0,
 140           2.8274333882308138, 0.0 },
 141   { -0.48258700419201095, 1, 0,
 142           2.9845130209103035, 0.0 },
 143   { -0.48860251190291992, 1, 0,
 144           3.1415926535897931, 0.0 },
 145 };
 146 const double toler002 = 2.5000000000000020e-13;
 147
 148 // Test data for l=1, m=1.
 149 // max(|f - f_GSL|): 1.9984014443252818e-15 at index 9
 150 // max(|f - f_GSL| / |f_GSL|): 5.9507621559271917e-15
 151 // mean(f - f_GSL): 1.1981156807413147e-15
 152 // variance(f - f_GSL): 7.5362762183006766e-32
 153 // stddev(f - f_GSL): 2.7452278991553101e-16
 154 const testcase_sph_legendre<double>
 155 data003[21] =
 156 {
 157   { 0.0000000000000000, 1, 1,
 158           0.0000000000000000, 0.0 },
 159   { -0.054047192447078167, 1, 1,
 160           0.15707963267948966, 0.0 },
 161   { -0.10676356364376091, 1, 1,
 162           0.31415926535897931, 0.0 },
 163   { -0.15685106157558129, 1, 1,
 164           0.47123889803846897, 0.0 },
 165   { -0.20307636581258248, 1, 1,
 166           0.62831853071795862, 0.0 },
 167   { -0.24430125595146007, 1, 1,
 168           0.78539816339744828, 0.0 },
 169   { -0.27951063837942880, 1, 1,
 170           0.94247779607693793, 0.0 },
 171   { -0.30783754124787122, 1, 1,
 172           1.0995574287564276, 0.0 },
 173   { -0.32858446219656556, 1, 1,
 174           1.2566370614359172, 0.0 },
 175   { -0.34124054317667202, 1, 1,
 176           1.4137166941154069, 0.0 },
 177   { -0.34549414947133567, 1, 1,
 178           1.5707963267948966, 0.0 },
 179   { -0.34124054317667202, 1, 1,
 180           1.7278759594743862, 0.0 },
 181   { -0.32858446219656556, 1, 1,
 182           1.8849555921538759, 0.0 },
 183   { -0.30783754124787127, 1, 1,
 184           2.0420352248333655, 0.0 },
 185   { -0.27951063837942880, 1, 1,
 186           2.1991148575128552, 0.0 },
 187   { -0.24430125595146013, 1, 1,
 188           2.3561944901923448, 0.0 },
 189   { -0.20307636581258243, 1, 1,
 190           2.5132741228718345, 0.0 },
 191   { -0.15685106157558140, 1, 1,
 192           2.6703537555513241, 0.0 },
 193   { -0.10676356364376104, 1, 1,
 194           2.8274333882308138, 0.0 },
 195   { -0.054047192447078167, 1, 1,
 196           2.9845130209103035, 0.0 },
 197   { 0.0000000000000000, 1, 1,
 198           3.1415926535897931, 0.0 },
 199 };
 200 const double toler003 = 5.0000000000000039e-13;
 201
 202 // Test data for l=2, m=0.
 203 // max(|f - f_GSL|): 1.1102230246251565e-16 at index 0
 204 // max(|f - f_GSL| / |f_GSL|): 3.1667226501700652e-15
 205 // mean(f - f_GSL): -3.0233752009881492e-17
 206 // variance(f - f_GSL): 3.4265646892248527e-34
 207 // stddev(f - f_GSL): 1.8510982386747748e-17
 208 const testcase_sph_legendre<double>
 209 data004[21] =
 210 {
 211   { 0.63078313050504009, 2, 0,
 212           0.0000000000000000, 0.0 },
 213   { 0.60762858760316585, 2, 0,
 214           0.15707963267948966, 0.0 },
 215   { 0.54043148688396603, 2, 0,
 216           0.31415926535897931, 0.0 },
 217   { 0.43576954875556589, 2, 0,
 218           0.47123889803846897, 0.0 },
 219   { 0.30388781294457567, 2, 0,
 220           0.62831853071795862, 0.0 },
 221   { 0.15769578262626011, 2, 0,
 222           0.78539816339744828, 0.0 },
 223   { 0.011503752307944374, 2, 0,
 224           0.94247779607693793, 0.0 },
 225   { -0.12037798350304570, 2, 0,
 226           1.0995574287564276, 0.0 },
 227   { -0.22503992163144576, 2, 0,
 228           1.2566370614359172, 0.0 },
 229   { -0.29223702235064597, 2, 0,
 230           1.4137166941154069, 0.0 },
 231   { -0.31539156525252005, 2, 0,
 232           1.5707963267948966, 0.0 },
 233   { -0.29223702235064597, 2, 0,
 234           1.7278759594743862, 0.0 },
 235   { -0.22503992163144584, 2, 0,
 236           1.8849555921538759, 0.0 },
 237   { -0.12037798350304577, 2, 0,
 238           2.0420352248333655, 0.0 },
 239   { 0.011503752307944164, 2, 0,
 240           2.1991148575128552, 0.0 },
 241   { 0.15769578262625994, 2, 0,
 242           2.3561944901923448, 0.0 },
 243   { 0.30388781294457579, 2, 0,
 244           2.5132741228718345, 0.0 },
 245   { 0.43576954875556562, 2, 0,
 246           2.6703537555513241, 0.0 },
 247   { 0.54043148688396569, 2, 0,
 248           2.8274333882308138, 0.0 },
 249   { 0.60762858760316585, 2, 0,
 250           2.9845130209103035, 0.0 },
 251   { 0.63078313050504009, 2, 0,
 252           3.1415926535897931, 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-15 at index 5
 258 // max(|f - f_GSL| / |f_GSL|): 5.9294218689659490e-15
 259 // mean(f - f_GSL): 6.6084703846736834e-19
 260 // variance(f - f_GSL): 2.2927737433182295e-38
 261 // stddev(f - f_GSL): 1.5141907882820546e-19
 262 const testcase_sph_legendre<double>
 263 data005[21] =
 264 {
 265   { 0.0000000000000000, 2, 1,
 266           0.0000000000000000, 0.0 },
 267   { -0.11936529291378781, 2, 1,
 268           0.15707963267948966, 0.0 },
 269   { -0.22704627929027421, 2, 1,
 270           0.31415926535897931, 0.0 },
 271   { -0.31250239392538215, 2, 1,
 272           0.47123889803846897, 0.0 },
 273   { -0.36736859691086526, 2, 1,
 274           0.62831853071795862, 0.0 },
 275   { -0.38627420202318979, 2, 1,
 276           0.78539816339744828, 0.0 },
 277   { -0.36736859691086532, 2, 1,
 278           0.94247779607693793, 0.0 },
 279   { -0.31250239392538226, 2, 1,
 280           1.0995574287564276, 0.0 },
 281   { -0.22704627929027438, 2, 1,
 282           1.2566370614359172, 0.0 },
 283   { -0.11936529291378740, 2, 1,
 284           1.4137166941154069, 0.0 },
 285   { -4.7304946510089680e-17, 2, 1,
 286           1.5707963267948966, 0.0 },
 287   { 0.11936529291378731, 2, 1,
 288           1.7278759594743862, 0.0 },
 289   { 0.22704627929027429, 2, 1,
 290           1.8849555921538759, 0.0 },
 291   { 0.31250239392538226, 2, 1,
 292           2.0420352248333655, 0.0 },
 293   { 0.36736859691086521, 2, 1,
 294           2.1991148575128552, 0.0 },
 295   { 0.38627420202318979, 2, 1,
 296           2.3561944901923448, 0.0 },
 297   { 0.36736859691086526, 2, 1,
 298           2.5132741228718345, 0.0 },
 299   { 0.31250239392538232, 2, 1,
 300           2.6703537555513241, 0.0 },
 301   { 0.22704627929027449, 2, 1,
 302           2.8274333882308138, 0.0 },
 303   { 0.11936529291378781, 2, 1,
 304           2.9845130209103035, 0.0 },
 305   { 0.0000000000000000, 2, 1,
 306           3.1415926535897931, 0.0 },
 307 };
 308 const double toler005 = 5.0000000000000039e-13;
 309
 310 // Test data for l=2, m=2.
 311 // max(|f - f_GSL|): 2.2759572004815709e-15 at index 10
 312 // max(|f - f_GSL| / |f_GSL|): 6.0197685443914339e-15
 313 // mean(f - f_GSL): -1.0489294618073093e-15
 314 // variance(f - f_GSL): 5.7763283331987012e-32
 315 // stddev(f - f_GSL): 2.4033993287006430e-16
 316 const testcase_sph_legendre<double>
 317 data006[21] =
 318 {
 319   { 0.0000000000000000, 2, 2,
 320           0.0000000000000000, 0.0 },
 321   { 0.0094528025561623485, 2, 2,
 322           0.15707963267948966, 0.0 },
 323   { 0.036885904048903732, 2, 2,
 324           0.31415926535897931, 0.0 },
 325   { 0.079613961366457681, 2, 2,
 326           0.47123889803846897, 0.0 },
 327   { 0.13345445455470126, 2, 2,
 328           0.62831853071795862, 0.0 },
 329   { 0.19313710101159490, 2, 2,
 330           0.78539816339744828, 0.0 },
 331   { 0.25281974746848851, 2, 2,
 332           0.94247779607693793, 0.0 },
 333   { 0.30666024065673209, 2, 2,
 334           1.0995574287564276, 0.0 },
 335   { 0.34938829797428600, 2, 2,
 336           1.2566370614359172, 0.0 },
 337   { 0.37682139946702753, 2, 2,
 338           1.4137166941154069, 0.0 },
 339   { 0.38627420202318985, 2, 2,
 340           1.5707963267948966, 0.0 },
 341   { 0.37682139946702753, 2, 2,
 342           1.7278759594743862, 0.0 },
 343   { 0.34938829797428606, 2, 2,
 344           1.8849555921538759, 0.0 },
 345   { 0.30666024065673214, 2, 2,
 346           2.0420352248333655, 0.0 },
 347   { 0.25281974746848856, 2, 2,
 348           2.1991148575128552, 0.0 },
 349   { 0.19313710101159498, 2, 2,
 350           2.3561944901923448, 0.0 },
 351   { 0.13345445455470123, 2, 2,
 352           2.5132741228718345, 0.0 },
 353   { 0.079613961366457764, 2, 2,
 354           2.6703537555513241, 0.0 },
 355   { 0.036885904048903809, 2, 2,
 356           2.8274333882308138, 0.0 },
 357   { 0.0094528025561623485, 2, 2,
 358           2.9845130209103035, 0.0 },
 359   { 0.0000000000000000, 2, 2,
 360           3.1415926535897931, 0.0 },
 361 };
 362 const double toler006 = 5.0000000000000039e-13;
 363
 364 // Test data for l=5, m=0.
 365 // max(|f - f_GSL|): 1.0547118733938987e-15 at index 18
 366 // max(|f - f_GSL| / |f_GSL|): 2.9385557676213648e-15
 367 // mean(f - f_GSL): -4.2955057500378082e-17
 368 // variance(f - f_GSL): 9.6869690655191325e-35
 369 // stddev(f - f_GSL): 9.8422401238331574e-18
 370 const testcase_sph_legendre<double>
 371 data007[21] =
 372 {
 373   { 0.93560257962738880, 5, 0,
 374           0.0000000000000000, 0.0 },
 375   { 0.77014422942079852, 5, 0,
 376           0.15707963267948966, 0.0 },
 377   { 0.35892185032365348, 5, 0,
 378           0.31415926535897931, 0.0 },
 379   { -0.090214932090594294, 5, 0,
 380           0.47123889803846897, 0.0 },
 381   { -0.36214460396518905, 5, 0,
 382           0.62831853071795862, 0.0 },
 383   { -0.35145955579226906, 5, 0,
 384           0.78539816339744828, 0.0 },
 385   { -0.11441703594725193, 5, 0,
 386           0.94247779607693793, 0.0 },
 387   { 0.17248966720808107, 5, 0,
 388           1.0995574287564276, 0.0 },
 389   { 0.32128384287200523, 5, 0,
 390           1.2566370614359172, 0.0 },
 391   { 0.24377632246714948, 5, 0,
 392           1.4137166941154069, 0.0 },
 393   { 1.0741712853887687e-16, 5, 0,
 394           1.5707963267948966, 0.0 },
 395   { -0.24377632246714936, 5, 0,
 396           1.7278759594743862, 0.0 },
 397   { -0.32128384287200534, 5, 0,
 398           1.8849555921538759, 0.0 },
 399   { -0.17248966720808118, 5, 0,
 400           2.0420352248333655, 0.0 },
 401   { 0.11441703594725151, 5, 0,
 402           2.1991148575128552, 0.0 },
 403   { 0.35145955579226895, 5, 0,
 404           2.3561944901923448, 0.0 },
 405   { 0.36214460396518883, 5, 0,
 406           2.5132741228718345, 0.0 },
 407   { 0.090214932090594752, 5, 0,
 408           2.6703537555513241, 0.0 },
 409   { -0.35892185032365165, 5, 0,
 410           2.8274333882308138, 0.0 },
 411   { -0.77014422942079852, 5, 0,
 412           2.9845130209103035, 0.0 },
 413   { -0.93560257962738880, 5, 0,
 414           3.1415926535897931, 0.0 },
 415 };
 416 const double toler007 = 2.5000000000000020e-13;
 417
 418 // Test data for l=5, m=1.
 419 // max(|f - f_GSL|): 3.3306690738754696e-15 at index 2
 420 // max(|f - f_GSL| / |f_GSL|): 6.7265986294921411e-15
 421 // mean(f - f_GSL): 6.3573485100559560e-16
 422 // variance(f - f_GSL): 2.1218337041113107e-32
 423 // stddev(f - f_GSL): 1.4566515383273072e-16
 424 const testcase_sph_legendre<double>
 425 data008[21] =
 426 {
 427   { 0.0000000000000000, 5, 1,
 428           0.0000000000000000, 0.0 },
 429   { -0.36712373713318402, 5, 1,
 430           0.15707963267948966, 0.0 },
 431   { -0.54610329010534708, 5, 1,
 432           0.31415926535897931, 0.0 },
 433   { -0.45381991493631763, 5, 1,
 434           0.47123889803846897, 0.0 },
 435   { -0.15679720635769906, 5, 1,
 436           0.62831853071795862, 0.0 },
 437   { 0.16985499419838601, 5, 1,
 438           0.78539816339744828, 0.0 },
 439   { 0.34468004499725169, 5, 1,
 440           0.94247779607693793, 0.0 },
 441   { 0.28349471119605985, 5, 1,
 442           1.0995574287564276, 0.0 },
 443   { 0.044286619339675815, 5, 1,
 444           1.2566370614359172, 0.0 },
 445   { -0.21193784177193470, 5, 1,
 446           1.4137166941154069, 0.0 },
 447   { -0.32028164857621527, 5, 1,
 448           1.5707963267948966, 0.0 },
 449   { -0.21193784177193487, 5, 1,
 450           1.7278759594743862, 0.0 },
 451   { 0.044286619339675592, 5, 1,
 452           1.8849555921538759, 0.0 },
 453   { 0.28349471119605973, 5, 1,
 454           2.0420352248333655, 0.0 },
 455   { 0.34468004499725174, 5, 1,
 456           2.1991148575128552, 0.0 },
 457   { 0.16985499419838640, 5, 1,
 458           2.3561944901923448, 0.0 },
 459   { -0.15679720635769961, 5, 1,
 460           2.5132741228718345, 0.0 },
 461   { -0.45381991493631768, 5, 1,
 462           2.6703537555513241, 0.0 },
 463   { -0.54610329010534753, 5, 1,
 464           2.8274333882308138, 0.0 },
 465   { -0.36712373713318402, 5, 1,
 466           2.9845130209103035, 0.0 },
 467   { 0.0000000000000000, 5, 1,
 468           3.1415926535897931, 0.0 },
 469 };
 470 const double toler008 = 5.0000000000000039e-13;
 471
 472 // Test data for l=5, m=2.
 473 // max(|f - f_GSL|): 2.6645352591003757e-15 at index 4
 474 // max(|f - f_GSL| / |f_GSL|): 6.1144597746666128e-15
 475 // mean(f - f_GSL): -4.3615904538845404e-17
 476 // variance(f - f_GSL): 9.9873224258937960e-35
 477 // stddev(f - f_GSL): 9.9936592026613535e-18
 478 const testcase_sph_legendre<double>
 479 data009[21] =
 480 {
 481   { 0.0000000000000000, 5, 2,
 482           0.0000000000000000, 0.0 },
 483   { 0.078919441745546909, 5, 2,
 484           0.15707963267948966, 0.0 },
 485   { 0.26373799140437948, 5, 2,
 486           0.31415926535897931, 0.0 },
 487   { 0.43002359842080096, 5, 2,
 488           0.47123889803846897, 0.0 },
 489   { 0.45642486439050983, 5, 2,
 490           0.62831853071795862, 0.0 },
 491   { 0.29959604906083298, 5, 2,
 492           0.78539816339744828, 0.0 },
 493   { 0.023781239849532509, 5, 2,
 494           0.94247779607693793, 0.0 },
 495   { -0.23313989334673826, 5, 2,
 496           1.0995574287564276, 0.0 },
 497   { -0.33799912776303714, 5, 2,
 498           1.2566370614359172, 0.0 },
 499   { -0.23964508489529746, 5, 2,
 500           1.4137166941154069, 0.0 },
 501   { -1.0377480524338157e-16, 5, 2,
 502           1.5707963267948966, 0.0 },
 503   { 0.23964508489529732, 5, 2,
 504           1.7278759594743862, 0.0 },
 505   { 0.33799912776303714, 5, 2,
 506           1.8849555921538759, 0.0 },
 507   { 0.23313989334673843, 5, 2,
 508           2.0420352248333655, 0.0 },
 509   { -0.023781239849531916, 5, 2,
 510           2.1991148575128552, 0.0 },
 511   { -0.29959604906083270, 5, 2,
 512           2.3561944901923448, 0.0 },
 513   { -0.45642486439050983, 5, 2,
 514           2.5132741228718345, 0.0 },
 515   { -0.43002359842080118, 5, 2,
 516           2.6703537555513241, 0.0 },
 517   { -0.26373799140437981, 5, 2,
 518           2.8274333882308138, 0.0 },
 519   { -0.078919441745546909, 5, 2,
 520           2.9845130209103035, 0.0 },
 521   { 0.0000000000000000, 5, 2,
 522           3.1415926535897931, 0.0 },
 523 };
 524 const double toler009 = 5.0000000000000039e-13;
 525
 526 // Test data for l=5, m=5.
 527 // max(|f - f_GSL|): 2.3869795029440866e-15 at index 10
 528 // max(|f - f_GSL| / |f_GSL|): 5.4714279627676651e-15
 529 // mean(f - f_GSL): 7.7966720691209837e-16
 530 // variance(f - f_GSL): 3.1913750060540917e-32
 531 // stddev(f - f_GSL): 1.7864419962747438e-16
 532 const testcase_sph_legendre<double>
 533 data010[21] =
 534 {
 535   { 0.0000000000000000, 5, 5,
 536           0.0000000000000000, 0.0 },
 537   { -4.3481439097910151e-05, 5, 5,
 538           0.15707963267948966, 0.0 },
 539   { -0.0013078367086431740, 5, 5,
 540           0.31415926535897931, 0.0 },
 541   { -0.0089510818191922761, 5, 5,
 542           0.47123889803846897, 0.0 },
 543   { -0.032563803777573910, 5, 5,
 544           0.62831853071795862, 0.0 },
 545   { -0.082047757105021241, 5, 5,
 546           0.78539816339744828, 0.0 },
 547   { -0.16085328164143806, 5, 5,
 548           0.94247779607693793, 0.0 },
 549   { -0.26064303436645381, 5, 5,
 550           1.0995574287564276, 0.0 },
 551   { -0.36113811790820571, 5, 5,
 552           1.2566370614359172, 0.0 },
 553   { -0.43625592459446139, 5, 5,
 554           1.4137166941154069, 0.0 },
 555   { -0.46413220344085809, 5, 5,
 556           1.5707963267948966, 0.0 },
 557   { -0.43625592459446144, 5, 5,
 558           1.7278759594743862, 0.0 },
 559   { -0.36113811790820577, 5, 5,
 560           1.8849555921538759, 0.0 },
 561   { -0.26064303436645381, 5, 5,
 562           2.0420352248333655, 0.0 },
 563   { -0.16085328164143822, 5, 5,
 564           2.1991148575128552, 0.0 },
 565   { -0.082047757105021310, 5, 5,
 566           2.3561944901923448, 0.0 },
 567   { -0.032563803777573896, 5, 5,
 568           2.5132741228718345, 0.0 },
 569   { -0.0089510818191923004, 5, 5,
 570           2.6703537555513241, 0.0 },
 571   { -0.0013078367086431812, 5, 5,
 572           2.8274333882308138, 0.0 },
 573   { -4.3481439097910151e-05, 5, 5,
 574           2.9845130209103035, 0.0 },
 575   { 0.0000000000000000, 5, 5,
 576           3.1415926535897931, 0.0 },
 577 };
 578 const double toler010 = 5.0000000000000039e-13;
 579
 580 // Test data for l=10, m=0.
 581 // max(|f - f_GSL|): 9.9920072216264089e-16 at index 18
 582 // max(|f - f_GSL| / |f_GSL|): 4.3176588435352323e-15
 583 // mean(f - f_GSL): 5.6171998269725183e-18
 584 // variance(f - f_GSL): 1.6565290295473546e-36
 585 // stddev(f - f_GSL): 1.2870621700397205e-18
 586 const testcase_sph_legendre<double>
 587 data011[21] =
 588 {
 589   { 1.2927207364566027, 10, 0,
 590           0.0000000000000000, 0.0 },
 591   { 0.55288895150522011, 10, 0,
 592           0.15707963267948966, 0.0 },
 593   { -0.44874428379711329, 10, 0,
 594           0.31415926535897931, 0.0 },
 595   { -0.25532095827149687, 10, 0,
 596           0.47123889803846897, 0.0 },
 597   { 0.36625249688013994, 10, 0,
 598           0.62831853071795862, 0.0 },
 599   { 0.14880806329084206, 10, 0,
 600           0.78539816339744828, 0.0 },
 601   { -0.33533356797848740, 10, 0,
 602           0.94247779607693793, 0.0 },
 603   { -0.080639967662335665, 10, 0,
 604           1.0995574287564276, 0.0 },
 605   { 0.32197986450174521, 10, 0,
 606           1.2566370614359172, 0.0 },
 607   { 0.025713542103667848, 10, 0,
 608           1.4137166941154069, 0.0 },
 609   { -0.31813049373736707, 10, 0,
 610           1.5707963267948966, 0.0 },
 611   { 0.025713542103667528, 10, 0,
 612           1.7278759594743862, 0.0 },
 613   { 0.32197986450174543, 10, 0,
 614           1.8849555921538759, 0.0 },
 615   { -0.080639967662335416, 10, 0,
 616           2.0420352248333655, 0.0 },
 617   { -0.33533356797848757, 10, 0,
 618           2.1991148575128552, 0.0 },
 619   { 0.14880806329084156, 10, 0,
 620           2.3561944901923448, 0.0 },
 621   { 0.36625249688013961, 10, 0,
 622           2.5132741228718345, 0.0 },
 623   { -0.25532095827149576, 10, 0,
 624           2.6703537555513241, 0.0 },
 625   { -0.44874428379711545, 10, 0,
 626           2.8274333882308138, 0.0 },
 627   { 0.55288895150522011, 10, 0,
 628           2.9845130209103035, 0.0 },
 629   { 1.2927207364566027, 10, 0,
 630           3.1415926535897931, 0.0 },
 631 };
 632 const double toler011 = 2.5000000000000020e-13;
 633
 634 // Test data for l=10, m=1.
 635 // max(|f - f_GSL|): 6.2172489379008766e-15 at index 1
 636 // max(|f - f_GSL| / |f_GSL|): 1.0515246092567303e-14
 637 // mean(f - f_GSL): -3.3042351923367698e-17
 638 // variance(f - f_GSL): 5.7319343582953255e-35
 639 // stddev(f - f_GSL): 7.5709539414101082e-18
 640 const testcase_sph_legendre<double>
 641 data012[21] =
 642 {
 643   { 0.0000000000000000, 10, 1,
 644           0.0000000000000000, 0.0 },
 645   { -0.74373723919064050, 10, 1,
 646           0.15707963267948966, 0.0 },
 647   { -0.29035110456209651, 10, 1,
 648           0.31415926535897931, 0.0 },
 649   { 0.42219282075271530, 10, 1,
 650           0.47123889803846897, 0.0 },
 651   { 0.17109256898931161, 10, 1,
 652           0.62831853071795862, 0.0 },
 653   { -0.35583574648544281, 10, 1,
 654           0.78539816339744828, 0.0 },
 655   { -0.10089212303544023, 10, 1,
 656           0.94247779607693793, 0.0 },
 657   { 0.32997652649321085, 10, 1,
 658           1.0995574287564276, 0.0 },
 659   { 0.047416376890032939, 10, 1,
 660           1.2566370614359172, 0.0 },
 661   { -0.31999356750295660, 10, 1,
 662           1.4137166941154069, 0.0 },
 663   { -2.0430664782290742e-16, 10, 1,
 664           1.5707963267948966, 0.0 },
 665   { 0.31999356750295660, 10, 1,
 666           1.7278759594743862, 0.0 },
 667   { -0.047416376890032544, 10, 1,
 668           1.8849555921538759, 0.0 },
 669   { -0.32997652649321096, 10, 1,
 670           2.0420352248333655, 0.0 },
 671   { 0.10089212303543935, 10, 1,
 672           2.1991148575128552, 0.0 },
 673   { 0.35583574648544292, 10, 1,
 674           2.3561944901923448, 0.0 },
 675   { -0.17109256898931269, 10, 1,
 676           2.5132741228718345, 0.0 },
 677   { -0.42219282075271569, 10, 1,
 678           2.6703537555513241, 0.0 },
 679   { 0.29035110456209601, 10, 1,
 680           2.8274333882308138, 0.0 },
 681   { 0.74373723919064050, 10, 1,
 682           2.9845130209103035, 0.0 },
 683   { 0.0000000000000000, 10, 1,
 684           3.1415926535897931, 0.0 },
 685 };
 686 const double toler012 = 1.0000000000000008e-12;
 687
 688 // Test data for l=10, m=2.
 689 // max(|f - f_GSL|): 3.3306690738754696e-15 at index 2
 690 // max(|f - f_GSL| / |f_GSL|): 7.9132043938587384e-15
 691 // mean(f - f_GSL): -3.3504944850294902e-16
 692 // variance(f - f_GSL): 5.8935519794618406e-33
 693 // stddev(f - f_GSL): 7.6769472965898634e-17
 694 const testcase_sph_legendre<double>
 695 data013[21] =
 696 {
 697   { 0.0000000000000000, 10, 2,
 698           0.0000000000000000, 0.0 },
 699   { 0.34571695599980556, 10, 2,
 700           0.15707963267948966, 0.0 },
 701   { 0.62485535978198148, 10, 2,
 702           0.31415926535897931, 0.0 },
 703   { 0.098210039644716252, 10, 2,
 704           0.47123889803846897, 0.0 },
 705   { -0.41494799233049695, 10, 2,
 706           0.62831853071795862, 0.0 },
 707   { -0.081698973831472732, 10, 2,
 708           0.78539816339744828, 0.0 },
 709   { 0.35253132222271277, 10, 2,
 710           0.94247779607693793, 0.0 },
 711   { 0.049026298555980979, 10, 2,
 712           1.0995574287564276, 0.0 },
 713   { -0.32791246874130792, 10, 2,
 714           1.2566370614359172, 0.0 },
 715   { -0.016196782433946871, 10, 2,
 716           1.4137166941154069, 0.0 },
 717   { 0.32106263400438334, 10, 2,
 718           1.5707963267948966, 0.0 },
 719   { -0.016196782433946497, 10, 2,
 720           1.7278759594743862, 0.0 },
 721   { -0.32791246874130803, 10, 2,
 722           1.8849555921538759, 0.0 },
 723   { 0.049026298555980730, 10, 2,
 724           2.0420352248333655, 0.0 },
 725   { 0.35253132222271266, 10, 2,
 726           2.1991148575128552, 0.0 },
 727   { -0.081698973831472121, 10, 2,
 728           2.3561944901923448, 0.0 },
 729   { -0.41494799233049684, 10, 2,
 730           2.5132741228718345, 0.0 },
 731   { 0.098210039644715197, 10, 2,
 732           2.6703537555513241, 0.0 },
 733   { 0.62485535978198081, 10, 2,
 734           2.8274333882308138, 0.0 },
 735   { 0.34571695599980556, 10, 2,
 736           2.9845130209103035, 0.0 },
 737   { 0.0000000000000000, 10, 2,
 738           3.1415926535897931, 0.0 },
 739 };
 740 const double toler013 = 5.0000000000000039e-13;
 741
 742 // Test data for l=10, m=5.
 743 // max(|f - f_GSL|): 2.8310687127941492e-15 at index 4
 744 // max(|f - f_GSL| / |f_GSL|): 6.7700052018114004e-15
 745 // mean(f - f_GSL): 3.8329128231106644e-17
 746 // variance(f - f_GSL): 7.7128908725222343e-35
 747 // stddev(f - f_GSL): 8.7823065720357509e-18
 748 const testcase_sph_legendre<double>
 749 data014[21] =
 750 {
 751   { 0.0000000000000000, 10, 5,
 752           0.0000000000000000, 0.0 },
 753   { -0.0030300124052750873, 10, 5,
 754           0.15707963267948966, 0.0 },
 755   { -0.070348585248056511, 10, 5,
 756           0.31415926535897931, 0.0 },
 757   { -0.30055029290703639, 10, 5,
 758           0.47123889803846897, 0.0 },
 759   { -0.49987818144009138, 10, 5,
 760           0.62831853071795862, 0.0 },
 761   { -0.28108771757150108, 10, 5,
 762           0.78539816339744828, 0.0 },
 763   { 0.22068081187249255, 10, 5,
 764           0.94247779607693793, 0.0 },
 765   { 0.33689502212592121, 10, 5,
 766           1.0995574287564276, 0.0 },
 767   { -0.086095515520764110, 10, 5,
 768           1.2566370614359172, 0.0 },
 769   { -0.33935827318511558, 10, 5,
 770           1.4137166941154069, 0.0 },
 771   { -1.9213014340664546e-16, 10, 5,
 772           1.5707963267948966, 0.0 },
 773   { 0.33935827318511558, 10, 5,
 774           1.7278759594743862, 0.0 },
 775   { 0.086095515520764512, 10, 5,
 776           1.8849555921538759, 0.0 },
 777   { -0.33689502212592104, 10, 5,
 778           2.0420352248333655, 0.0 },
 779   { -0.22068081187249344, 10, 5,
 780           2.1991148575128552, 0.0 },
 781   { 0.28108771757150064, 10, 5,
 782           2.3561944901923448, 0.0 },
 783   { 0.49987818144009155, 10, 5,
 784           2.5132741228718345, 0.0 },
 785   { 0.30055029290703672, 10, 5,
 786           2.6703537555513241, 0.0 },
 787   { 0.070348585248056802, 10, 5,
 788           2.8274333882308138, 0.0 },
 789   { 0.0030300124052750873, 10, 5,
 790           2.9845130209103035, 0.0 },
 791   { 0.0000000000000000, 10, 5,
 792           3.1415926535897931, 0.0 },
 793 };
 794 const double toler014 = 5.0000000000000039e-13;
 795
 796 // Test data for l=10, m=10.
 797 // max(|f - f_GSL|): 2.8865798640254070e-15 at index 10
 798 // max(|f - f_GSL| / |f_GSL|): 5.5362170773729674e-15
 799 // mean(f - f_GSL): -6.6245801611758948e-16
 800 // variance(f - f_GSL): 2.3039657713718755e-32
 801 // stddev(f - f_GSL): 1.5178820017945649e-16
 802 const testcase_sph_legendre<double>
 803 data015[21] =
 804 {
 805   { 0.0000000000000000, 10, 10,
 806           0.0000000000000000, 0.0 },
 807   { 4.7624282733345673e-09, 10, 10,
 808           0.15707963267948966, 0.0 },
 809   { 4.3085156534549315e-06, 10, 10,
 810           0.31415926535897931, 0.0 },
 811   { 0.00020182347649472387, 10, 10,
 812           0.47123889803846897, 0.0 },
 813   { 0.0026711045506511706, 10, 10,
 814           0.62831853071795862, 0.0 },
 815   { 0.016957196623256909, 10, 10,
 816           0.78539816339744828, 0.0 },
 817   { 0.065174916004990271, 10, 10,
 818           0.94247779607693793, 0.0 },
 819   { 0.17112476903017845, 10, 10,
 820           1.0995574287564276, 0.0 },
 821   { 0.32852414199733554, 10, 10,
 822           1.2566370614359172, 0.0 },
 823   { 0.47940582314838287, 10, 10,
 824           1.4137166941154069, 0.0 },
 825   { 0.54263029194422152, 10, 10,
 826           1.5707963267948966, 0.0 },
 827   { 0.47940582314838293, 10, 10,
 828           1.7278759594743862, 0.0 },
 829   { 0.32852414199733571, 10, 10,
 830           1.8849555921538759, 0.0 },
 831   { 0.17112476903017851, 10, 10,
 832           2.0420352248333655, 0.0 },
 833   { 0.065174916004990410, 10, 10,
 834           2.1991148575128552, 0.0 },
 835   { 0.016957196623256943, 10, 10,
 836           2.3561944901923448, 0.0 },
 837   { 0.0026711045506511684, 10, 10,
 838           2.5132741228718345, 0.0 },
 839   { 0.00020182347649472493, 10, 10,
 840           2.6703537555513241, 0.0 },
 841   { 4.3085156534549772e-06, 10, 10,
 842           2.8274333882308138, 0.0 },
 843   { 4.7624282733345673e-09, 10, 10,
 844           2.9845130209103035, 0.0 },
 845   { 0.0000000000000000, 10, 10,
 846           3.1415926535897931, 0.0 },
 847 };
 848 const double toler015 = 5.0000000000000039e-13;
 849
 850 // Test data for l=20, m=0.
 851 // max(|f - f_GSL|): 1.2212453270876722e-15 at index 1
 852 // max(|f - f_GSL| / |f_GSL|): 2.0731910827878347e-15
 853 // mean(f - f_GSL): 1.1102230246251565e-16
 854 // variance(f - f_GSL): 6.4711246131411122e-34
 855 // stddev(f - f_GSL): 2.5438405243138007e-17
 856 const testcase_sph_legendre<double>
 857 data016[21] =
 858 {
 859   { 1.8062879984608917, 20, 0,
 860           0.0000000000000000, 0.0 },
 861   { -0.58906549291416732, 20, 0,
 862           0.15707963267948966, 0.0 },
 863   { 0.45624611402342063, 20, 0,
 864           0.31415926535897931, 0.0 },
 865   { -0.39955402700466724, 20, 0,
 866           0.47123889803846897, 0.0 },
 867   { 0.36818552901640805, 20, 0,
 868           0.62831853071795862, 0.0 },
 869   { -0.34873131330857743, 20, 0,
 870           0.78539816339744828, 0.0 },
 871   { 0.33600882829186468, 20, 0,
 872           0.94247779607693793, 0.0 },
 873   { -0.32759286308122931, 20, 0,
 874           1.0995574287564276, 0.0 },
 875   { 0.32222458068091325, 20, 0,
 876           1.2566370614359172, 0.0 },
 877   { -0.31922731037135965, 20, 0,
 878           1.4137166941154069, 0.0 },
 879   { 0.31826262039531755, 20, 0,
 880           1.5707963267948966, 0.0 },
 881   { -0.31922731037135971, 20, 0,
 882           1.7278759594743862, 0.0 },
 883   { 0.32222458068091342, 20, 0,
 884           1.8849555921538759, 0.0 },
 885   { -0.32759286308122926, 20, 0,
 886           2.0420352248333655, 0.0 },
 887   { 0.33600882829186518, 20, 0,
 888           2.1991148575128552, 0.0 },
 889   { -0.34873131330857782, 20, 0,
 890           2.3561944901923448, 0.0 },
 891   { 0.36818552901640772, 20, 0,
 892           2.5132741228718345, 0.0 },
 893   { -0.39955402700466824, 20, 0,
 894           2.6703537555513241, 0.0 },
 895   { 0.45624611402342408, 20, 0,
 896           2.8274333882308138, 0.0 },
 897   { -0.58906549291416732, 20, 0,
 898           2.9845130209103035, 0.0 },
 899   { 1.8062879984608917, 20, 0,
 900           3.1415926535897931, 0.0 },
 901 };
 902 const double toler016 = 2.5000000000000020e-13;
 903
 904 // Test data for l=20, m=1.
 905 // max(|f - f_GSL|): 6.2727600891321345e-15 at index 1
 906 // max(|f - f_GSL| / |f_GSL|): 1.3664592043119372e-14
 907 // mean(f - f_GSL): 2.8416422654096149e-17
 908 // variance(f - f_GSL): 4.2393386513952014e-35
 909 // stddev(f - f_GSL): 6.5110203896126770e-18
 910 const testcase_sph_legendre<double>
 911 data017[21] =
 912 {
 913   { 0.0000000000000000, 20, 1,
 914           0.0000000000000000, 0.0 },
 915   { -0.45905213045059046, 20, 1,
 916           0.15707963267948966, 0.0 },
 917   { 0.31166370423309414, 20, 1,
 918           0.31415926535897931, 0.0 },
 919   { -0.23278757741246778, 20, 1,
 920           0.47123889803846897, 0.0 },
 921   { 0.17937240823503983, 20, 1,
 922           0.62831853071795862, 0.0 },
 923   { -0.13857299972299839, 20, 1,
 924           0.78539816339744828, 0.0 },
 925   { 0.10495324841927815, 20, 1,
 926           0.94247779607693793, 0.0 },
 927   { -0.075707774352163665, 20, 1,
 928           1.0995574287564276, 0.0 },
 929   { 0.049168697683476224, 20, 1,
 930           1.2566370614359172, 0.0 },
 931   { -0.024216050551253303, 20, 1,
 932           1.4137166941154069, 0.0 },
 933   { 3.9938443510694310e-16, 20, 1,
 934           1.5707963267948966, 0.0 },
 935   { 0.024216050551252380, 20, 1,
 936           1.7278759594743862, 0.0 },
 937   { -0.049168697683475482, 20, 1,
 938           1.8849555921538759, 0.0 },
 939   { 0.075707774352163346, 20, 1,
 940           2.0420352248333655, 0.0 },
 941   { -0.10495324841927638, 20, 1,
 942           2.1991148575128552, 0.0 },
 943   { 0.13857299972299741, 20, 1,
 944           2.3561944901923448, 0.0 },
 945   { -0.17937240823504172, 20, 1,
 946           2.5132741228718345, 0.0 },
 947   { 0.23278757741246703, 20, 1,
 948           2.6703537555513241, 0.0 },
 949   { -0.31166370423309253, 20, 1,
 950           2.8274333882308138, 0.0 },
 951   { 0.45905213045059046, 20, 1,
 952           2.9845130209103035, 0.0 },
 953   { 0.0000000000000000, 20, 1,
 954           3.1415926535897931, 0.0 },
 955 };
 956 const double toler017 = 1.0000000000000008e-12;
 957
 958 // Test data for l=20, m=2.
 959 // max(|f - f_GSL|): 3.2196467714129540e-15 at index 2
 960 // max(|f - f_GSL| / |f_GSL|): 7.1072009465107655e-15
 961 // mean(f - f_GSL): 9.2518585385429716e-17
 962 // variance(f - f_GSL): 4.4938365369035506e-34
 963 // stddev(f - f_GSL): 2.1198671035948340e-17
 964 const testcase_sph_legendre<double>
 965 data018[21] =
 966 {
 967   { 0.0000000000000000, 20, 2,
 968           0.0000000000000000, 0.0 },
 969   { 0.87399805141574205, 20, 2,
 970           0.15707963267948966, 0.0 },
 971   { -0.55116854080894906, 20, 2,
 972           0.31415926535897931, 0.0 },
 973   { 0.44520137308557572, 20, 2,
 974           0.47123889803846897, 0.0 },
 975   { -0.39321637877908278, 20, 2,
 976           0.62831853071795862, 0.0 },
 977   { 0.36312025711350954, 20, 2,
 978           0.78539816339744828, 0.0 },
 979   { -0.34427103004873089, 20, 2,
 980           0.94247779607693793, 0.0 },
 981   { 0.33214917638387642, 20, 2,
 982           1.0995574287564276, 0.0 },
 983   { -0.32455734448839091, 20, 2,
 984           1.2566370614359172, 0.0 },
 985   { 0.32036529628513255, 20, 2,
 986           1.4137166941154069, 0.0 },
 987   { -0.31902310563819986, 20, 2,
 988           1.5707963267948966, 0.0 },
 989   { 0.32036529628513261, 20, 2,
 990           1.7278759594743862, 0.0 },
 991   { -0.32455734448839102, 20, 2,
 992           1.8849555921538759, 0.0 },
 993   { 0.33214917638387659, 20, 2,
 994           2.0420352248333655, 0.0 },
 995   { -0.34427103004873105, 20, 2,
 996           2.1991148575128552, 0.0 },
 997   { 0.36312025711350981, 20, 2,
 998           2.3561944901923448, 0.0 },
 999   { -0.39321637877908228, 20, 2,
1000           2.5132741228718345, 0.0 },
1001   { 0.44520137308557650, 20, 2,
1002           2.6703537555513241, 0.0 },
1003   { -0.55116854080895039, 20, 2,
1004           2.8274333882308138, 0.0 },
1005   { 0.87399805141574205, 20, 2,
1006           2.9845130209103035, 0.0 },
1007   { 0.0000000000000000, 20, 2,
1008           3.1415926535897931, 0.0 },
1009 };
1010 const double toler018 = 5.0000000000000039e-13;
1011
1012 // Test data for l=20, m=5.
1013 // max(|f - f_GSL|): 3.7747582837255322e-15 at index 2
1014 // max(|f - f_GSL| / |f_GSL|): 2.9846361325102863e-14
1015 // mean(f - f_GSL): -1.9825411154020758e-17
1016 // variance(f - f_GSL): 2.0634963689863462e-35
1017 // stddev(f - f_GSL): 4.5425723648460969e-18
1018 const testcase_sph_legendre<double>
1019 data019[21] =
1020 {
1021   { 0.0000000000000000, 20, 5,
1022           0.0000000000000000, 0.0 },
1023   { -0.10024848623505046, 20, 5,
1024           0.15707963267948966, 0.0 },
1025   { -0.68115361075940650, 20, 5,
1026           0.31415926535897931, 0.0 },
1027   { 0.31774532551156298, 20, 5,
1028           0.47123889803846897, 0.0 },
1029   { -0.16011868165390658, 20, 5,
1030           0.62831853071795862, 0.0 },
1031   { 0.085844143304115578, 20, 5,
1032           0.78539816339744828, 0.0 },
1033   { -0.047467540840863798, 20, 5,
1034           0.94247779607693793, 0.0 },
1035   { 0.026283575189471796, 20, 5,
1036           1.0995574287564276, 0.0 },
1037   { -0.013891104052597688, 20, 5,
1038           1.2566370614359172, 0.0 },
1039   { 0.0059873308239496957, 20, 5,
1040           1.4137166941154069, 0.0 },
1041   { 3.9355286582083051e-16, 20, 5,
1042           1.5707963267948966, 0.0 },
1043   { -0.0059873308239503324, 20, 5,
1044           1.7278759594743862, 0.0 },
1045   { 0.013891104052598547, 20, 5,
1046           1.8849555921538759, 0.0 },
1047   { -0.026283575189472153, 20, 5,
1048           2.0420352248333655, 0.0 },
1049   { 0.047467540840865928, 20, 5,
1050           2.1991148575128552, 0.0 },
1051   { -0.085844143304117007, 20, 5,
1052           2.3561944901923448, 0.0 },
1053   { 0.16011868165390544, 20, 5,
1054           2.5132741228718345, 0.0 },
1055   { -0.31774532551156381, 20, 5,
1056           2.6703537555513241, 0.0 },
1057   { 0.68115361075940484, 20, 5,
1058           2.8274333882308138, 0.0 },
1059   { 0.10024848623505046, 20, 5,
1060           2.9845130209103035, 0.0 },
1061   { 0.0000000000000000, 20, 5,
1062           3.1415926535897931, 0.0 },
1063 };
1064 const double toler019 = 2.5000000000000015e-12;
1065
1066 // Test data for l=20, m=10.
1067 // max(|f - f_GSL|): 2.3314683517128287e-15 at index 4
1068 // max(|f - f_GSL| / |f_GSL|): 1.0575432165608034e-14
1069 // mean(f - f_GSL): -1.4002922808555493e-16
1070 // variance(f - f_GSL): 1.0294296977074090e-33
1071 // stddev(f - f_GSL): 3.2084726860414586e-17
1072 const testcase_sph_legendre<double>
1073 data020[21] =
1074 {
1075   { 0.0000000000000000, 20, 10,
1076           0.0000000000000000, 0.0 },
1077   { 3.0595797603707888e-05, 20, 10,
1078           0.15707963267948966, 0.0 },
1079   { 0.015924453916396863, 20, 10,
1080           0.31415926535897931, 0.0 },
1081   { 0.26588079118745744, 20, 10,
1082           0.47123889803846897, 0.0 },
1083   { 0.54045081420686736, 20, 10,
1084           0.62831853071795862, 0.0 },
1085   { -0.28215279394285531, 20, 10,
1086           0.78539816339744828, 0.0 },
1087   { 0.0085297337582239223, 20, 10,
1088           0.94247779607693793, 0.0 },
1089   { 0.16930127953533738, 20, 10,
1090           1.0995574287564276, 0.0 },
1091   { -0.27215134048018325, 20, 10,
1092           1.2566370614359172, 0.0 },
1093   { 0.32456597088029526, 20, 10,
1094           1.4137166941154069, 0.0 },
1095   { -0.34057893241353715, 20, 10,
1096           1.5707963267948966, 0.0 },
1097   { 0.32456597088029493, 20, 10,
1098           1.7278759594743862, 0.0 },
1099   { -0.27215134048018291, 20, 10,
1100           1.8849555921538759, 0.0 },
1101   { 0.16930127953533702, 20, 10,
1102           2.0420352248333655, 0.0 },
1103   { 0.0085297337582257438, 20, 10,
1104           2.1991148575128552, 0.0 },
1105   { -0.28215279394285619, 20, 10,
1106           2.3561944901923448, 0.0 },
1107   { 0.54045081420686869, 20, 10,
1108           2.5132741228718345, 0.0 },
1109   { 0.26588079118745828, 20, 10,
1110           2.6703537555513241, 0.0 },
1111   { 0.015924453916397002, 20, 10,
1112           2.8274333882308138, 0.0 },
1113   { 3.0595797603707888e-05, 20, 10,
1114           2.9845130209103035, 0.0 },
1115   { 0.0000000000000000, 20, 10,
1116           3.1415926535897931, 0.0 },
1117 };
1118 const double toler020 = 1.0000000000000008e-12;
1119
1120 // Test data for l=20, m=20.
1121 // max(|f - f_GSL|): 2.6645352591003757e-15 at index 10
1122 // max(|f - f_GSL| / |f_GSL|): 4.5133454703584717e-15
1123 // mean(f - f_GSL): -4.3857545834694365e-16
1124 // variance(f - f_GSL): 1.0098292714872165e-32
1125 // stddev(f - f_GSL): 1.0049026179124106e-16
1126 const testcase_sph_legendre<double>
1127 data021[21] =
1128 {
1129   { 0.0000000000000000, 20, 20,
1130           0.0000000000000000, 0.0 },
1131   { 4.9264471419250786e-17, 20, 20,
1132           0.15707963267948966, 0.0 },
1133   { 4.0321091681530921e-11, 20, 20,
1134           0.31415926535897931, 0.0 },
1135   { 8.8474944184471664e-08, 20, 20,
1136           0.47123889803846897, 0.0 },
1137   { 1.5497395129387791e-05, 20, 20,
1138           0.62831853071795862, 0.0 },
1139   { 0.00062457564282984495, 20, 20,
1140           0.78539816339744828, 0.0 },
1141   { 0.0092265192458967308, 20, 20,
1142           0.94247779607693793, 0.0 },
1143   { 0.063606673236323297, 20, 20,
1144           1.0995574287564276, 0.0 },
1145   { 0.23442909509776316, 20, 20,
1146           1.2566370614359172, 0.0 },
1147   { 0.49921030481087009, 20, 20,
1148           1.4137166941154069, 0.0 },
1149   { 0.63956545825776223, 20, 20,
1150           1.5707963267948966, 0.0 },
1151   { 0.49921030481087025, 20, 20,
1152           1.7278759594743862, 0.0 },
1153   { 0.23442909509776336, 20, 20,
1154           1.8849555921538759, 0.0 },
1155   { 0.063606673236323324, 20, 20,
1156           2.0420352248333655, 0.0 },
1157   { 0.0092265192458967742, 20, 20,
1158           2.1991148575128552, 0.0 },
1159   { 0.00062457564282984766, 20, 20,
1160           2.3561944901923448, 0.0 },
1161   { 1.5497395129387764e-05, 20, 20,
1162           2.5132741228718345, 0.0 },
1163   { 8.8474944184472617e-08, 20, 20,
1164           2.6703537555513241, 0.0 },
1165   { 4.0321091681531780e-11, 20, 20,
1166           2.8274333882308138, 0.0 },
1167   { 4.9264471419250786e-17, 20, 20,
1168           2.9845130209103035, 0.0 },
1169   { 0.0000000000000000, 20, 20,
1170           3.1415926535897931, 0.0 },
1171 };
1172 const double toler021 = 2.5000000000000020e-13;
1173
1174 // Test data for l=50, m=0.
1175 // max(|f - f_GSL|): 4.8849813083506888e-15 at index 1
1176 // max(|f - f_GSL| / |f_GSL|): 9.1896305557688689e-15
1177 // mean(f - f_GSL): -4.0592529337857286e-16
1178 // variance(f - f_GSL): 8.6507055497352237e-33
1179 // stddev(f - f_GSL): 9.3009169170223341e-17
1180 const testcase_sph_legendre<double>
1181 data022[21] =
1182 {
1183   { 2.8350175706934717, 50, 0,
1184           0.0000000000000000, 0.0 },
1185   { 0.53157537495172758, 50, 0,
1186           0.15707963267948966, 0.0 },
1187   { -0.46056183476300561, 50, 0,
1188           0.31415926535897931, 0.0 },
1189   { -0.24876032079677909, 50, 0,
1190           0.47123889803846897, 0.0 },
1191   { 0.36926172901532667, 50, 0,
1192           0.62831853071795862, 0.0 },
1193   { 0.14571730283563575, 50, 0,
1194           0.78539816339744828, 0.0 },
1195   { -0.33636199170850750, 50, 0,
1196           0.94247779607693793, 0.0 },
1197   { -0.079132716267091507, 50, 0,
1198           1.0995574287564276, 0.0 },
1199   { 0.32232921941301451, 50, 0,
1200           1.2566370614359172, 0.0 },
1201   { 0.025253991969481544, 50, 0,
1202           1.4137166941154069, 0.0 },
1203   { -0.31830208724152359, 50, 0,
1204           1.5707963267948966, 0.0 },
1205   { 0.025253991969479882, 50, 0,
1206           1.7278759594743862, 0.0 },
1207   { 0.32232921941301479, 50, 0,
1208           1.8849555921538759, 0.0 },
1209   { -0.079132716267090064, 50, 0,
1210           2.0420352248333655, 0.0 },
1211   { -0.33636199170850883, 50, 0,
1212           2.1991148575128552, 0.0 },
1213   { 0.14571730283563347, 50, 0,
1214           2.3561944901923448, 0.0 },
1215   { 0.36926172901532522, 50, 0,
1216           2.5132741228718345, 0.0 },
1217   { -0.24876032079677354, 50, 0,
1218           2.6703537555513241, 0.0 },
1219   { -0.46056183476301255, 50, 0,
1220           2.8274333882308138, 0.0 },
1221   { 0.53157537495172758, 50, 0,
1222           2.9845130209103035, 0.0 },
1223   { 2.8350175706934717, 50, 0,
1224           3.1415926535897931, 0.0 },
1225 };
1226 const double toler022 = 5.0000000000000039e-13;
1227
1228 // Test data for l=50, m=1.
1229 // max(|f - f_GSL|): 7.8825834748386114e-15 at index 1
1230 // max(|f - f_GSL| / |f_GSL|): 1.6168319946399610e-14
1231 // mean(f - f_GSL): -8.8223079635391627e-17
1232 // variance(f - f_GSL): 4.0862386846851427e-34
1233 // stddev(f - f_GSL): 2.0214447023564960e-17
1234 const testcase_sph_legendre<double>
1235 data023[21] =
1236 {
1237   { 0.0000000000000000, 50, 1,
1238           0.0000000000000000, 0.0 },
1239   { -0.63751752155228247, 50, 1,
1240           0.15707963267948966, 0.0 },
1241   { -0.32616619317605133, 50, 1,
1242           0.31415926535897931, 0.0 },
1243   { 0.40649930826162850, 50, 1,
1244           0.47123889803846897, 0.0 },
1245   { 0.18473991408343635, 50, 1,
1246           0.62831853071795862, 0.0 },
1247   { -0.35083930302013117, 50, 1,
1248           0.78539816339744828, 0.0 },
1249   { -0.10755382110947342, 50, 1,
1250           0.94247779607693793, 0.0 },
1251   { 0.32822568316499900, 50, 1,
1252           1.0995574287564276, 0.0 },
1253   { 0.050286056609797389, 50, 1,
1254           1.2566370614359172, 0.0 },
1255   { -0.31935368562159644, 50, 1,
1256           1.4137166941154069, 0.0 },
1257   { -9.8421602686195843e-16, 50, 1,
1258           1.5707963267948966, 0.0 },
1259   { 0.31935368562159649, 50, 1,
1260           1.7278759594743862, 0.0 },
1261   { -0.050286056609795446, 50, 1,
1262           1.8849555921538759, 0.0 },
1263   { -0.32822568316499923, 50, 1,
1264           2.0420352248333655, 0.0 },
1265   { 0.10755382110946902, 50, 1,
1266           2.1991148575128552, 0.0 },
1267   { 0.35083930302013205, 50, 1,
1268           2.3561944901923448, 0.0 },
1269   { -0.18473991408344057, 50, 1,
1270           2.5132741228718345, 0.0 },
1271   { -0.40649930826163011, 50, 1,
1272           2.6703537555513241, 0.0 },
1273   { 0.32616619317604606, 50, 1,
1274           2.8274333882308138, 0.0 },
1275   { 0.63751752155228247, 50, 1,
1276           2.9845130209103035, 0.0 },
1277   { 0.0000000000000000, 50, 1,
1278           3.1415926535897931, 0.0 },
1279 };
1280 const double toler023 = 1.0000000000000008e-12;
1281
1282 // Test data for l=50, m=2.
1283 // max(|f - f_GSL|): 3.7747582837255322e-15 at index 1
1284 // max(|f - f_GSL| / |f_GSL|): 1.3049193367556654e-14
1285 // mean(f - f_GSL): 2.4038311024250043e-16
1286 // variance(f - f_GSL): 3.0336620837175507e-33
1287 // stddev(f - f_GSL): 5.5078689923758629e-17
1288 const testcase_sph_legendre<double>
1289 data024[21] =
1290 {
1291   { 0.0000000000000000, 50, 2,
1292           0.0000000000000000, 0.0 },
1293   { -0.37230261163836259, 50, 2,
1294           0.15707963267948966, 0.0 },
1295   { 0.50051599680315517, 50, 2,
1296           0.31415926535897931, 0.0 },
1297   { 0.21724795180329495, 50, 2,
1298           0.47123889803846897, 0.0 },
1299   { -0.37948127307611107, 50, 2,
1300           0.62831853071795862, 0.0 },
1301   { -0.13187372121003413, 50, 2,
1302           0.78539816339744828, 0.0 },
1303   { 0.33959009162400128, 50, 2,
1304           0.94247779607693793, 0.0 },
1305   { 0.072537503112489563, 50, 2,
1306           1.0995574287564276, 0.0 },
1307   { -0.32310306941855266, 50, 2,
1308           1.2566370614359172, 0.0 },
1309   { -0.023259822816436636, 50, 2,
1310           1.4137166941154069, 0.0 },
1311   { 0.31842698506357275, 50, 2,
1312           1.5707963267948966, 0.0 },
1313   { -0.023259822816434638, 50, 2,
1314           1.7278759594743862, 0.0 },
1315   { -0.32310306941855316, 50, 2,
1316           1.8849555921538759, 0.0 },
1317   { 0.072537503112488411, 50, 2,
1318           2.0420352248333655, 0.0 },
1319   { 0.33959009162400267, 50, 2,
1320           2.1991148575128552, 0.0 },
1321   { -0.13187372121003124, 50, 2,
1322           2.3561944901923448, 0.0 },
1323   { -0.37948127307610924, 50, 2,
1324           2.5132741228718345, 0.0 },
1325   { 0.21724795180329090, 50, 2,
1326           2.6703537555513241, 0.0 },
1327   { 0.50051599680316095, 50, 2,
1328           2.8274333882308138, 0.0 },
1329   { -0.37230261163836259, 50, 2,
1330           2.9845130209103035, 0.0 },
1331   { 0.0000000000000000, 50, 2,
1332           3.1415926535897931, 0.0 },
1333 };
1334 const double toler024 = 1.0000000000000008e-12;
1335
1336 // Test data for l=50, m=5.
1337 // max(|f - f_GSL|): 6.4254157550180935e-15 at index 18
1338 // max(|f - f_GSL| / |f_GSL|): 8.3058098108785765e-14
1339 // mean(f - f_GSL): 3.9419525844577751e-16
1340 // variance(f - f_GSL): 8.1579698435094999e-33
1341 // stddev(f - f_GSL): 9.0321480521022794e-17
1342 const testcase_sph_legendre<double>
1343 data025[21] =
1344 {
1345   { 0.0000000000000000, 50, 5,
1346           0.0000000000000000, 0.0 },
1347   { -0.57750385903191603, 50, 5,
1348           0.15707963267948966, 0.0 },
1349   { 0.077360497065570286, 50, 5,
1350           0.31415926535897931, 0.0 },
1351   { 0.47707267400540226, 50, 5,
1352           0.47123889803846897, 0.0 },
1353   { 0.055370615126626811, 50, 5,
1354           0.62831853071795862, 0.0 },
1355   { -0.37629451847202833, 50, 5,
1356           0.78539816339744828, 0.0 },
1357   { -0.048042277801963115, 50, 5,
1358           0.94247779607693793, 0.0 },
1359   { 0.33619379362228718, 50, 5,
1360           1.0995574287564276, 0.0 },
1361   { 0.025265227185718764, 50, 5,
1362           1.2566370614359172, 0.0 },
1363   { -0.32083679430964535, 50, 5,
1364           1.4137166941154069, 0.0 },
1365   { -9.8189201019751726e-16, 50, 5,
1366           1.5707963267948966, 0.0 },
1367   { 0.32083679430964546, 50, 5,
1368           1.7278759594743862, 0.0 },
1369   { -0.025265227185716790, 50, 5,
1370           1.8849555921538759, 0.0 },
1371   { -0.33619379362228730, 50, 5,
1372           2.0420352248333655, 0.0 },
1373   { 0.048042277801958064, 50, 5,
1374           2.1991148575128552, 0.0 },
1375   { 0.37629451847202872, 50, 5,
1376           2.3561944901923448, 0.0 },
1377   { -0.055370615126630517, 50, 5,
1378           2.5132741228718345, 0.0 },
1379   { -0.47707267400540176, 50, 5,
1380           2.6703537555513241, 0.0 },
1381   { -0.077360497065588632, 50, 5,
1382           2.8274333882308138, 0.0 },
1383   { 0.57750385903191603, 50, 5,
1384           2.9845130209103035, 0.0 },
1385   { 0.0000000000000000, 50, 5,
1386           3.1415926535897931, 0.0 },
1387 };
1388 const double toler025 = 5.0000000000000029e-12;
1389
1390 // Test data for l=50, m=10.
1391 // max(|f - f_GSL|): 4.2188474935755949e-15 at index 17
1392 // max(|f - f_GSL| / |f_GSL|): 8.5566462271658482e-15
1393 // mean(f - f_GSL): 2.1378401694418937e-16
1394 // variance(f - f_GSL): 2.3994393097916561e-33
1395 // stddev(f - f_GSL): 4.8984072000923484e-17
1396 const testcase_sph_legendre<double>
1397 data026[21] =
1398 {
1399   { 0.0000000000000000, 50, 10,
1400           0.0000000000000000, 0.0 },
1401   { 0.15606941844801256, 50, 10,
1402           0.15707963267948966, 0.0 },
1403   { -0.53748868836814168, 50, 10,
1404           0.31415926535897931, 0.0 },
1405   { -0.49304919025183969, 50, 10,
1406           0.47123889803846897, 0.0 },
1407   { -0.26267582750427920, 50, 10,
1408           0.62831853071795862, 0.0 },
1409   { 0.22058983666314153, 50, 10,
1410           0.78539816339744828, 0.0 },
1411   { 0.32936725160671843, 50, 10,
1412           0.94247779607693793, 0.0 },
1413   { -0.092053311559447959, 50, 10,
1414           1.0995574287564276, 0.0 },
1415   { -0.32542913495935522, 50, 10,
1416           1.2566370614359172, 0.0 },
1417   { 0.025673223789103351, 50, 10,
1418           1.4137166941154069, 0.0 },
1419   { 0.32150019350255743, 50, 10,
1420           1.5707963267948966, 0.0 },
1421   { 0.025673223789105259, 50, 10,
1422           1.7278759594743862, 0.0 },
1423   { -0.32542913495935510, 50, 10,
1424           1.8849555921538759, 0.0 },
1425   { -0.092053311559449028, 50, 10,
1426           2.0420352248333655, 0.0 },
1427   { 0.32936725160671687, 50, 10,
1428           2.1991148575128552, 0.0 },
1429   { 0.22058983666314380, 50, 10,
1430           2.3561944901923448, 0.0 },
1431   { -0.26267582750428364, 50, 10,
1432           2.5132741228718345, 0.0 },
1433   { -0.49304919025184135, 50, 10,
1434           2.6703537555513241, 0.0 },
1435   { -0.53748868836814501, 50, 10,
1436           2.8274333882308138, 0.0 },
1437   { 0.15606941844801256, 50, 10,
1438           2.9845130209103035, 0.0 },
1439   { 0.0000000000000000, 50, 10,
1440           3.1415926535897931, 0.0 },
1441 };
1442 const double toler026 = 5.0000000000000039e-13;
1443
1444 // Test data for l=50, m=20.
1445 // max(|f - f_GSL|): 5.3290705182007514e-15 at index 17
1446 // max(|f - f_GSL| / |f_GSL|): 8.3091988651062939e-15
1447 // mean(f - f_GSL): -6.3705667112909982e-16
1448 // variance(f - f_GSL): 2.1306663117079729e-32
1449 // stddev(f - f_GSL): 1.4596802087128444e-16
1450 const testcase_sph_legendre<double>
1451 data027[21] =
1452 {
1453   { 0.0000000000000000, 50, 20,
1454           0.0000000000000000, 0.0 },
1455   { 3.0409598712835887e-07, 50, 20,
1456           0.15707963267948966, 0.0 },
1457   { 0.030940518122881844, 50, 20,
1458           0.31415926535897931, 0.0 },
1459   { 0.64134588721659802, 50, 20,
1460           0.47123889803846897, 0.0 },
1461   { 0.29895244392136594, 50, 20,
1462           0.62831853071795862, 0.0 },
1463   { 0.25309324781873871, 50, 20,
1464           0.78539816339744828, 0.0 },
1465   { 0.34368634714931567, 50, 20,
1466           0.94247779607693793, 0.0 },
1467   { 0.33996764360663945, 50, 20,
1468           1.0995574287564276, 0.0 },
1469   { 0.12866267745104024, 50, 20,
1470           1.2566370614359172, 0.0 },
1471   { -0.18201114398922874, 50, 20,
1472           1.4137166941154069, 0.0 },
1473   { -0.33216683431510857, 50, 20,
1474           1.5707963267948966, 0.0 },
1475   { -0.18201114398923010, 50, 20,
1476           1.7278759594743862, 0.0 },
1477   { 0.12866267745103857, 50, 20,
1478           1.8849555921538759, 0.0 },
1479   { 0.33996764360663895, 50, 20,
1480           2.0420352248333655, 0.0 },
1481   { 0.34368634714931812, 50, 20,
1482           2.1991148575128552, 0.0 },
1483   { 0.25309324781874126, 50, 20,
1484           2.3561944901923448, 0.0 },
1485   { 0.29895244392136405, 50, 20,
1486           2.5132741228718345, 0.0 },
1487   { 0.64134588721659869, 50, 20,
1488           2.6703537555513241, 0.0 },
1489   { 0.030940518122882274, 50, 20,
1490           2.8274333882308138, 0.0 },
1491   { 3.0409598712835887e-07, 50, 20,
1492           2.9845130209103035, 0.0 },
1493   { 0.0000000000000000, 50, 20,
1494           3.1415926535897931, 0.0 },
1495 };
1496 const double toler027 = 5.0000000000000039e-13;
1497
1498 // Test data for l=50, m=50.
1499 // max(|f - f_GSL|): 8.8817841970012523e-16 at index 10
1500 // max(|f - f_GSL| / |f_GSL|): 1.6947090122367552e-15
1501 // mean(f - f_GSL): 8.9876810627650302e-17
1502 // variance(f - f_GSL): 4.2408665715142198e-34
1503 // stddev(f - f_GSL): 2.0593364396120950e-17
1504 const testcase_sph_legendre<double>
1505 data028[21] =
1506 {
1507   { 0.0000000000000000, 50, 50,
1508           0.0000000000000000, 0.0 },
1509   { 4.1649039898161316e-41, 50, 50,
1510           0.15707963267948966, 0.0 },
1511   { 2.5240684647722935e-26, 50, 50,
1512           0.31415926535897931, 0.0 },
1513   { 5.6927376423967334e-18, 50, 50,
1514           0.47123889803846897, 0.0 },
1515   { 2.3116239814797222e-12, 50, 50,
1516           0.62831853071795862, 0.0 },
1517   { 2.3835981241325056e-08, 50, 50,
1518           0.78539816339744828, 0.0 },
1519   { 1.9992410287270217e-05, 50, 50,
1520           0.94247779607693793, 0.0 },
1521   { 0.0024947505670829830, 50, 50,
1522           1.0995574287564276, 0.0 },
1523   { 0.065057774647971231, 50, 50,
1524           1.2566370614359172, 0.0 },
1525   { 0.43050607056732243, 50, 50,
1526           1.4137166941154069, 0.0 },
1527   { 0.79980281171531975, 50, 50,
1528           1.5707963267948966, 0.0 },
1529   { 0.43050607056732287, 50, 50,
1530           1.7278759594743862, 0.0 },
1531   { 0.065057774647971398, 50, 50,
1532           1.8849555921538759, 0.0 },
1533   { 0.0024947505670829878, 50, 50,
1534           2.0420352248333655, 0.0 },
1535   { 1.9992410287270430e-05, 50, 50,
1536           2.1991148575128552, 0.0 },
1537   { 2.3835981241325311e-08, 50, 50,
1538           2.3561944901923448, 0.0 },
1539   { 2.3116239814797057e-12, 50, 50,
1540           2.5132741228718345, 0.0 },
1541   { 5.6927376423968952e-18, 50, 50,
1542           2.6703537555513241, 0.0 },
1543   { 2.5240684647724192e-26, 50, 50,
1544           2.8274333882308138, 0.0 },
1545   { 4.1649039898161316e-41, 50, 50,
1546           2.9845130209103035, 0.0 },
1547   { 0.0000000000000000, 50, 50,
1548           3.1415926535897931, 0.0 },
1549 };
1550 const double toler028 = 2.5000000000000020e-13;
1551
1552 // Test data for l=100, m=0.
1553 // max(|f - f_GSL|): 2.2759572004815709e-15 at index 18
1554 // max(|f - f_GSL| / |f_GSL|): 4.9270578369513465e-15
1555 // mean(f - f_GSL): -2.0618427600181479e-16
1556 // variance(f - f_GSL): 2.2318776726956083e-33
1557 // stddev(f - f_GSL): 4.7242752594399153e-17
1558 const testcase_sph_legendre<double>
1559 data029[21] =
1560 {
1561   { 3.9993839251484076, 100, 0,
1562           0.0000000000000000, 0.0 },
1563   { -0.60770160285939456, 100, 0,
1564           0.15707963267948966, 0.0 },
1565   { 0.46193027883954441, 100, 0,
1566           0.31415926535897931, 0.0 },
1567   { -0.40218718869815234, 100, 0,
1568           0.47123889803846897, 0.0 },
1569   { 0.36960201406911097, 100, 0,
1570           0.62831853071795862, 0.0 },
1571   { -0.34953726547378389, 100, 0,
1572           0.78539816339744828, 0.0 },
1573   { 0.33646959352497730, 100, 0,
1574           0.94247779607693793, 0.0 },
1575   { -0.32784733067663224, 100, 0,
1576           1.0995574287564276, 0.0 },
1577   { 0.32235624474047969, 100, 0,
1578           1.2566370614359172, 0.0 },
1579   { -0.31929330706601350, 100, 0,
1580           1.4137166941154069, 0.0 },
1581   { 0.31830791662110325, 100, 0,
1582           1.5707963267948966, 0.0 },
1583   { -0.31929330706601333, 100, 0,
1584           1.7278759594743862, 0.0 },
1585   { 0.32235624474048036, 100, 0,
1586           1.8849555921538759, 0.0 },
1587   { -0.32784733067663291, 100, 0,
1588           2.0420352248333655, 0.0 },
1589   { 0.33646959352498013, 100, 0,
1590           2.1991148575128552, 0.0 },
1591   { -0.34953726547378589, 100, 0,
1592           2.3561944901923448, 0.0 },
1593   { 0.36960201406910725, 100, 0,
1594           2.5132741228718345, 0.0 },
1595   { -0.40218718869815723, 100, 0,
1596           2.6703537555513241, 0.0 },
1597   { 0.46193027883955923, 100, 0,
1598           2.8274333882308138, 0.0 },
1599   { -0.60770160285939456, 100, 0,
1600           2.9845130209103035, 0.0 },
1601   { 3.9993839251484076, 100, 0,
1602           3.1415926535897931, 0.0 },
1603 };
1604 const double toler029 = 2.5000000000000020e-13;
1605
1606 // Test data for l=100, m=1.
1607 // max(|f - f_GSL|): 4.7184478546569153e-15 at index 18
1608 // max(|f - f_GSL| / |f_GSL|): 1.6136152688198031e-14
1609 // mean(f - f_GSL): 1.7347234759767562e-17
1610 // variance(f - f_GSL): 1.5798644075050619e-35
1611 // stddev(f - f_GSL): 3.9747508192401971e-18
1612 const testcase_sph_legendre<double>
1613 data030[21] =
1614 {
1615   { 0.0000000000000000, 100, 1,
1616           0.0000000000000000, 0.0 },
1617   { -0.50851949013714159, 100, 1,
1618           0.15707963267948966, 0.0 },
1619   { 0.33129641402223092, 100, 1,
1620           0.31415926535897931, 0.0 },
1621   { -0.24390405750942512, 100, 1,
1622           0.47123889803846897, 0.0 },
1623   { 0.18659755088413388, 100, 1,
1624           0.62831853071795862, 0.0 },
1625   { -0.14355908970517178, 100, 1,
1626           0.78539816339744828, 0.0 },
1627   { 0.10844906813251505, 100, 1,
1628           0.94247779607693793, 0.0 },
1629   { -0.078100088690857675, 100, 1,
1630           1.0995574287564276, 0.0 },
1631   { 0.050670002998302717, 100, 1,
1632           1.2566370614359172, 0.0 },
1633   { -0.024941251747138900, 100, 1,
1634           1.4137166941154069, 0.0 },
1635   { 1.9587949830851608e-15, 100, 1,
1636           1.5707963267948966, 0.0 },
1637   { 0.024941251747135025, 100, 1,
1638           1.7278759594743862, 0.0 },
1639   { -0.050670002998298824, 100, 1,
1640           1.8849555921538759, 0.0 },
1641   { 0.078100088690855773, 100, 1,
1642           2.0420352248333655, 0.0 },
1643   { -0.10844906813250622, 100, 1,
1644           2.1991148575128552, 0.0 },
1645   { 0.14355908970516626, 100, 1,
1646           2.3561944901923448, 0.0 },
1647   { -0.18659755088414104, 100, 1,
1648           2.5132741228718345, 0.0 },
1649   { 0.24390405750941679, 100, 1,
1650           2.6703537555513241, 0.0 },
1651   { -0.33129641402221749, 100, 1,
1652           2.8274333882308138, 0.0 },
1653   { 0.50851949013714159, 100, 1,
1654           2.9845130209103035, 0.0 },
1655   { 0.0000000000000000, 100, 1,
1656           3.1415926535897931, 0.0 },
1657 };
1658 const double toler030 = 1.0000000000000008e-12;
1659
1660 // Test data for l=100, m=2.
1661 // max(|f - f_GSL|): 9.3258734068513149e-15 at index 1
1662 // max(|f - f_GSL| / |f_GSL|): 1.3884756158405121e-14
1663 // mean(f - f_GSL): 1.1049362483174177e-15
1664 // variance(f - f_GSL): 6.4096415924402929e-32
1665 // stddev(f - f_GSL): 2.5317269980075443e-16
1666 const testcase_sph_legendre<double>
1667 data031[21] =
1668 {
1669   { 0.0000000000000000, 100, 2,
1670           0.0000000000000000, 0.0 },
1671   { 0.67166274297196848, 100, 2,
1672           0.15707963267948966, 0.0 },
1673   { -0.48226933687994433, 100, 2,
1674           0.31415926535897931, 0.0 },
1675   { 0.41175421895715525, 100, 2,
1676           0.47123889803846897, 0.0 },
1677   { -0.37475021787822776, 100, 2,
1678           0.62831853071795862, 0.0 },
1679   { 0.35242909383605286, 100, 2,
1680           0.78539816339744828, 0.0 },
1681   { -0.33807110409159863, 100, 2,
1682           0.94247779607693793, 0.0 },
1683   { 0.32867180390710077, 100, 2,
1684           1.0995574287564276, 0.0 },
1685   { -0.32271583790278502, 100, 2,
1686           1.2566370614359172, 0.0 },
1687   { 0.31940354677687466, 100, 2,
1688           1.4137166941154069, 0.0 },
1689   { -0.31833943693772526, 100, 2,
1690           1.5707963267948966, 0.0 },
1691   { 0.31940354677687455, 100, 2,
1692           1.7278759594743862, 0.0 },
1693   { -0.32271583790278552, 100, 2,
1694           1.8849555921538759, 0.0 },
1695   { 0.32867180390710116, 100, 2,
1696           2.0420352248333655, 0.0 },
1697   { -0.33807110409160157, 100, 2,
1698           2.1991148575128552, 0.0 },
1699   { 0.35242909383605503, 100, 2,
1700           2.3561944901923448, 0.0 },
1701   { -0.37475021787822438, 100, 2,
1702           2.5132741228718345, 0.0 },
1703   { 0.41175421895716052, 100, 2,
1704           2.6703537555513241, 0.0 },
1705   { -0.48226933687995360, 100, 2,
1706           2.8274333882308138, 0.0 },
1707   { 0.67166274297196848, 100, 2,
1708           2.9845130209103035, 0.0 },
1709   { 0.0000000000000000, 100, 2,
1710           3.1415926535897931, 0.0 },
1711 };
1712 const double toler031 = 1.0000000000000008e-12;
1713
1714 // Test data for l=100, m=5.
1715 // max(|f - f_GSL|): 1.5432100042289676e-14 at index 1
1716 // max(|f - f_GSL| / |f_GSL|): 2.4665959612621893e-13
1717 // mean(f - f_GSL): -8.4885802091131797e-16
1718 // variance(f - f_GSL): 3.7829396832437676e-32
1719 // stddev(f - f_GSL): 1.9449780675482609e-16
1720 const testcase_sph_legendre<double>
1721 data032[21] =
1722 {
1723   { 0.0000000000000000, 100, 5,
1724           0.0000000000000000, 0.0 },
1725   { 0.062564361105954577, 100, 5,
1726           0.15707963267948966, 0.0 },
1727   { 0.14179554455882706, 100, 5,
1728           0.31415926535897931, 0.0 },
1729   { -0.14356866942905960, 100, 5,
1730           0.47123889803846897, 0.0 },
1731   { 0.12355483388447824, 100, 5,
1732           0.62831853071795862, 0.0 },
1733   { -0.10090029999681642, 100, 5,
1734           0.78539816339744828, 0.0 },
1735   { 0.078905134460235241, 100, 5,
1736           0.94247779607693793, 0.0 },
1737   { -0.058040182398185071, 100, 5,
1738           1.0995574287564276, 0.0 },
1739   { 0.038142759389482424, 100, 5,
1740           1.2566370614359172, 0.0 },
1741   { -0.018906264170660478, 100, 5,
1742           1.4137166941154069, 0.0 },
1743   { 1.9576303042914504e-15, 100, 5,
1744           1.5707963267948966, 0.0 },
1745   { 0.018906264170657019, 100, 5,
1746           1.7278759594743862, 0.0 },
1747   { -0.038142759389478365, 100, 5,
1748           1.8849555921538759, 0.0 },
1749   { 0.058040182398183086, 100, 5,
1750           2.0420352248333655, 0.0 },
1751   { -0.078905134460225707, 100, 5,
1752           2.1991148575128552, 0.0 },
1753   { 0.10090029999681013, 100, 5,
1754           2.3561944901923448, 0.0 },
1755   { -0.12355483388448550, 100, 5,
1756           2.5132741228718345, 0.0 },
1757   { 0.14356866942904906, 100, 5,
1758           2.6703537555513241, 0.0 },
1759   { -0.14179554455879767, 100, 5,
1760           2.8274333882308138, 0.0 },
1761   { -0.062564361105954577, 100, 5,
1762           2.9845130209103035, 0.0 },
1763   { 0.0000000000000000, 100, 5,
1764           3.1415926535897931, 0.0 },
1765 };
1766 const double toler032 = 2.5000000000000014e-11;
1767
1768 // Test data for l=100, m=10.
1769 // max(|f - f_GSL|): 6.9944050551384862e-15 at index 1
1770 // max(|f - f_GSL| / |f_GSL|): 9.2805170220968737e-15
1771 // mean(f - f_GSL): -9.1725568939268895e-16
1772 // variance(f - f_GSL): 4.4171294985470989e-32
1773 // stddev(f - f_GSL): 2.1016968141354497e-16
1774 const testcase_sph_legendre<double>
1775 data033[21] =
1776 {
1777   { 0.0000000000000000, 100, 10,
1778           0.0000000000000000, 0.0 },
1779   { -0.75366545187997991, 100, 10,
1780           0.15707963267948966, 0.0 },
1781   { -0.35914570017278186, 100, 10,
1782           0.31415926535897931, 0.0 },
1783   { 0.43480692911578295, 100, 10,
1784           0.47123889803846897, 0.0 },
1785   { -0.40862111080315500, 100, 10,
1786           0.62831853071795862, 0.0 },
1787   { 0.37832688692909411, 100, 10,
1788           0.78539816339744828, 0.0 },
1789   { -0.35484056194773494, 100, 10,
1790           0.94247779607693793, 0.0 },
1791   { 0.33821981171196341, 100, 10,
1792           1.0995574287564276, 0.0 },
1793   { -0.32729120767830594, 100, 10,
1794           1.2566370614359172, 0.0 },
1795   { 0.32110336937091438, 100, 10,
1796           1.4137166941154069, 0.0 },
1797   { -0.31910064020036194, 100, 10,
1798           1.5707963267948966, 0.0 },
1799   { 0.32110336937091460, 100, 10,
1800           1.7278759594743862, 0.0 },
1801   { -0.32729120767830605, 100, 10,
1802           1.8849555921538759, 0.0 },
1803   { 0.33821981171196364, 100, 10,
1804           2.0420352248333655, 0.0 },
1805   { -0.35484056194773461, 100, 10,
1806           2.1991148575128552, 0.0 },
1807   { 0.37832688692909372, 100, 10,
1808           2.3561944901923448, 0.0 },
1809   { -0.40862111080315755, 100, 10,
1810           2.5132741228718345, 0.0 },
1811   { 0.43480692911577751, 100, 10,
1812           2.6703537555513241, 0.0 },
1813   { -0.35914570017276798, 100, 10,
1814           2.8274333882308138, 0.0 },
1815   { -0.75366545187997991, 100, 10,
1816           2.9845130209103035, 0.0 },
1817   { 0.0000000000000000, 100, 10,
1818           3.1415926535897931, 0.0 },
1819 };
1820 const double toler033 = 5.0000000000000039e-13;
1821
1822 // Test data for l=100, m=20.
1823 // max(|f - f_GSL|): 5.2180482157382357e-15 at index 2
1824 // max(|f - f_GSL| / |f_GSL|): 3.3114331221145530e-14
1825 // mean(f - f_GSL): -1.7066374768419445e-16
1826 // variance(f - f_GSL): 1.5291210256147554e-33
1827 // stddev(f - f_GSL): 3.9103977107383276e-17
1828 const testcase_sph_legendre<double>
1829 data034[21] =
1830 {
1831   { 0.0000000000000000, 100, 20,
1832           0.0000000000000000, 0.0 },
1833   { 0.053569660841557079, 100, 20,
1834           0.15707963267948966, 0.0 },
1835   { 0.57154926874731338, 100, 20,
1836           0.31415926535897931, 0.0 },
1837   { 0.47536909969585633, 100, 20,
1838           0.47123889803846897, 0.0 },
1839   { -0.28882554564109575, 100, 20,
1840           0.62831853071795862, 0.0 },
1841   { 0.020116179014043743, 100, 20,
1842           0.78539816339744828, 0.0 },
1843   { 0.14752195931707024, 100, 20,
1844           0.94247779607693793, 0.0 },
1845   { -0.24069428588868366, 100, 20,
1846           1.0995574287564276, 0.0 },
1847   { 0.29031796025014306, 100, 20,
1848           1.2566370614359172, 0.0 },
1849   { -0.31437256851143458, 100, 20,
1850           1.4137166941154069, 0.0 },
1851   { 0.32153954851141792, 100, 20,
1852           1.5707963267948966, 0.0 },
1853   { -0.31437256851143375, 100, 20,
1854           1.7278759594743862, 0.0 },
1855   { 0.29031796025014139, 100, 20,
1856           1.8849555921538759, 0.0 },
1857   { -0.24069428588868241, 100, 20,
1858           2.0420352248333655, 0.0 },
1859   { 0.14752195931706186, 100, 20,
1860           2.1991148575128552, 0.0 },
1861   { 0.020116179014049562, 100, 20,
1862           2.3561944901923448, 0.0 },
1863   { -0.28882554564109075, 100, 20,
1864           2.5132741228718345, 0.0 },
1865   { 0.47536909969585545, 100, 20,
1866           2.6703537555513241, 0.0 },
1867   { 0.57154926874732348, 100, 20,
1868           2.8274333882308138, 0.0 },
1869   { 0.053569660841557079, 100, 20,
1870           2.9845130209103035, 0.0 },
1871   { 0.0000000000000000, 100, 20,
1872           3.1415926535897931, 0.0 },
1873 };
1874 const double toler034 = 2.5000000000000015e-12;
1875
1876 // Test data for l=100, m=50.
1877 // max(|f - f_GSL|): 2.4980018054066022e-15 at index 15
1878 // max(|f - f_GSL| / |f_GSL|): 7.4927901205617944e-15
1879 // mean(f - f_GSL): 1.5662074559583175e-16
1880 // variance(f - f_GSL): 1.2878280424271984e-33
1881 // stddev(f - f_GSL): 3.5886321104666028e-17
1882 const testcase_sph_legendre<double>
1883 data035[21] =
1884 {
1885   { 0.0000000000000000, 100, 50,
1886           0.0000000000000000, 0.0 },
1887   { 3.3047910392597822e-21, 100, 50,
1888           0.15707963267948966, 0.0 },
1889   { 1.0592655372554564e-07, 100, 50,
1890           0.31415926535897931, 0.0 },
1891   { 0.080418744223952773, 100, 50,
1892           0.47123889803846897, 0.0 },
1893   { -0.56450600580392973, 100, 50,
1894           0.62831853071795862, 0.0 },
1895   { 0.33338739844742110, 100, 50,
1896           0.78539816339744828, 0.0 },
1897   { 0.39741714816514678, 100, 50,
1898           0.94247779607693793, 0.0 },
1899   { 0.35223993750972243, 100, 50,
1900           1.0995574287564276, 0.0 },
1901   { 0.17885891940721577, 100, 50,
1902           1.2566370614359172, 0.0 },
1903   { -0.15341660126461967, 100, 50,
1904           1.4137166941154069, 0.0 },
1905   { -0.34175924303503102, 100, 50,
1906           1.5707963267948966, 0.0 },
1907   { -0.15341660126462270, 100, 50,
1908           1.7278759594743862, 0.0 },
1909   { 0.17885891940721302, 100, 50,
1910           1.8849555921538759, 0.0 },
1911   { 0.35223993750972227, 100, 50,
1912           2.0420352248333655, 0.0 },
1913   { 0.39741714816514595, 100, 50,
1914           2.1991148575128552, 0.0 },
1915   { 0.33338739844741666, 100, 50,
1916           2.3561944901923448, 0.0 },
1917   { -0.56450600580393095, 100, 50,
1918           2.5132741228718345, 0.0 },
1919   { 0.080418744223953911, 100, 50,
1920           2.6703537555513241, 0.0 },
1921   { 1.0592655372554981e-07, 100, 50,
1922           2.8274333882308138, 0.0 },
1923   { 3.3047910392597822e-21, 100, 50,
1924           2.9845130209103035, 0.0 },
1925   { 0.0000000000000000, 100, 50,
1926           3.1415926535897931, 0.0 },
1927 };
1928 const double toler035 = 5.0000000000000039e-13;
1929
1930 // Test data for l=100, m=100.
1931 // max(|f - f_GSL|): 3.6637359812630166e-15 at index 10
1932 // max(|f - f_GSL| / |f_GSL|): 3.8591350247394187e-15
1933 // mean(f - f_GSL): 2.7412914496899967e-16
1934 // variance(f - f_GSL): 3.9452063763753292e-33
1935 // stddev(f - f_GSL): 6.2810877850698192e-17
1936 const testcase_sph_legendre<double>
1937 data036[21] =
1938 {
1939   { 0.0000000000000000, 100, 100,
1940           0.0000000000000000, 0.0 },
1941   { 2.5744136608873899e-81, 100, 100,
1942           0.15707963267948966, 0.0 },
1943   { 9.4551974868947092e-52, 100, 100,
1944           0.31415926535897931, 0.0 },
1945   { 4.8096190703397596e-35, 100, 100,
1946           0.47123889803846897, 0.0 },
1947   { 7.9305393636344023e-24, 100, 100,
1948           0.62831853071795862, 0.0 },
1949   { 8.4320740610944858e-16, 100, 100,
1950           0.78539816339744828, 0.0 },
1951   { 5.9319660146026674e-10, 100, 100,
1952           0.94247779607693793, 0.0 },
1953   { 9.2368225946797243e-06, 100, 100,
1954           1.0995574287564276, 0.0 },
1955   { 0.0062815489742044095, 100, 100,
1956           1.2566370614359172, 0.0 },
1957   { 0.27505966018176986, 100, 100,
1958           1.4137166941154069, 0.0 },
1959   { 0.94936713998764621, 100, 100,
1960           1.5707963267948966, 0.0 },
1961   { 0.27505966018177036, 100, 100,
1962           1.7278759594743862, 0.0 },
1963   { 0.0062815489742044433, 100, 100,
1964           1.8849555921538759, 0.0 },
1965   { 9.2368225946797582e-06, 100, 100,
1966           2.0420352248333655, 0.0 },
1967   { 5.9319660146027946e-10, 100, 100,
1968           2.1991148575128552, 0.0 },
1969   { 8.4320740610946652e-16, 100, 100,
1970           2.3561944901923448, 0.0 },
1971   { 7.9305393636342891e-24, 100, 100,
1972           2.5132741228718345, 0.0 },
1973   { 4.8096190703400333e-35, 100, 100,
1974           2.6703537555513241, 0.0 },
1975   { 9.4551974868956498e-52, 100, 100,
1976           2.8274333882308138, 0.0 },
1977   { 2.5744136608873899e-81, 100, 100,
1978           2.9845130209103035, 0.0 },
1979   { 0.0000000000000000, 100, 100,
1980           3.1415926535897931, 0.0 },
1981 };
1982 const double toler036 = 2.5000000000000020e-13;
1983
1984 template<typename Ret, unsigned int Num>
1985   void
1986   test(const testcase_sph_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::sph_legendre(data[i].l, data[i].m,
1996                      data[i].theta);
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 }