ceph/src/boost/libs/math/test/tanh_sinh_quadrature_test.cpp

   1 // Copyright Nick Thompson, 2017
   2 // Use, modification and distribution are subject to the
   3 // Boost Software License, Version 1.0.
   4 // (See accompanying file LICENSE_1_0.txt
   5 // or copy at http://www.boost.org/LICENSE_1_0.txt)
   6
   7 #define BOOST_TEST_MODULE tanh_sinh_quadrature_test
   8
   9 #include <boost/config.hpp>
  10 #include <boost/detail/workaround.hpp>
  11
  12 #if !defined(BOOST_NO_CXX11_DECLTYPE) && !defined(BOOST_NO_CXX11_TRAILING_RESULT_TYPES) && !defined(BOOST_NO_SFINAE_EXPR)
  13
  14 #include <boost/math/concepts/real_concept.hpp>
  15 #include <boost/test/included/unit_test.hpp>
  16 #include <boost/test/floating_point_comparison.hpp>
  17 #include <boost/math/quadrature/tanh_sinh.hpp>
  18 #include <boost/math/special_functions/sinc.hpp>
  19 #include <boost/multiprecision/cpp_bin_float.hpp>
  20 #include <boost/multiprecision/cpp_dec_float.hpp>
  21 #include <boost/math/special_functions/next.hpp>
  22 #include <boost/math/special_functions/gamma.hpp>
  23 #include <boost/math/special_functions/beta.hpp>
  24 #include <boost/math/special_functions/ellint_rc.hpp>
  25 #include <boost/math/special_functions/ellint_rj.hpp>
  26
  27 #ifdef BOOST_HAS_FLOAT128
  28 #include <boost/multiprecision/float128.hpp>
  29 #endif
  30
  31 #ifdef _MSC_VER
  32 #pragma warning(disable:4127)  // Conditional expression is constant
  33 #endif
  34
  35 #if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4) && !defined(TEST5) && !defined(TEST6) && !defined(TEST7) && !defined(TEST8)\
  36     && !defined(TEST1A) && !defined(TEST2A) && !defined(TEST3A) && !defined(TEST6A)
  37 #  define TEST1
  38 #  define TEST2
  39 #  define TEST3
  40 #  define TEST4
  41 #  define TEST5
  42 #  define TEST6
  43 #  define TEST7
  44 #  define TEST8
  45 #  define TEST1A
  46 #  define TEST2A
  47 #  define TEST3A
  48 #  define TEST6A
  49 #endif
  50
  51 using std::expm1;
  52 using std::atan;
  53 using std::tan;
  54 using std::log;
  55 using std::log1p;
  56 using std::asinh;
  57 using std::atanh;
  58 using std::sqrt;
  59 using std::isnormal;
  60 using std::abs;
  61 using std::sinh;
  62 using std::tanh;
  63 using std::cosh;
  64 using std::pow;
  65 using std::exp;
  66 using std::sin;
  67 using std::cos;
  68 using std::string;
  69 using boost::multiprecision::cpp_bin_float_50;
  70 using boost::multiprecision::cpp_bin_float_100;
  71 using boost::multiprecision::cpp_dec_float_50;
  72 using boost::multiprecision::cpp_dec_float_100;
  73 using boost::multiprecision::cpp_bin_float_quad;
  74 using boost::math::sinc_pi;
  75 using boost::math::quadrature::tanh_sinh;
  76 using boost::math::quadrature::detail::tanh_sinh_detail;
  77 using boost::math::constants::pi;
  78 using boost::math::constants::half_pi;
  79 using boost::math::constants::two_div_pi;
  80 using boost::math::constants::two_pi;
  81 using boost::math::constants::half;
  82 using boost::math::constants::third;
  83 using boost::math::constants::half;
  84 using boost::math::constants::third;
  85 using boost::math::constants::catalan;
  86 using boost::math::constants::ln_two;
  87 using boost::math::constants::root_two;
  88 using boost::math::constants::root_two_pi;
  89 using boost::math::constants::root_pi;
  90
  91 template <class T>
  92 void print_levels(const T& v, const char* suffix)
  93 {
  94    std::cout << "{\n";
  95    for (unsigned i = 0; i < v.size(); ++i)
  96    {
  97       std::cout << "      { ";
  98       for (unsigned j = 0; j < v[i].size(); ++j)
  99       {
 100          std::cout << v[i][j] << suffix << ", ";
 101       }
 102       std::cout << "},\n";
 103    }
 104    std::cout << "   };\n";
 105 }
 106
 107 template <class T>
 108 void print_complement_indexes(const T& v)
 109 {
 110    std::cout << "\n   {";
 111    for (unsigned i = 0; i < v.size(); ++i)
 112    {
 113       unsigned index = 0;
 114       while (v[i][index] >= 0)
 115          ++index;
 116       std::cout << index << ", ";
 117    }
 118    std::cout << "};\n";
 119 }
 120
 121 template <class T>
 122 void print_levels(const std::pair<T, T>& p, const char* suffix = "")
 123 {
 124    std::cout << "   static const std::vector<std::vector<Real> > abscissa = ";
 125    print_levels(p.first, suffix);
 126    std::cout << "   static const std::vector<std::vector<Real> > weights = ";
 127    print_levels(p.second, suffix);
 128    std::cout << "   static const std::vector<unsigned > indexes = ";
 129    print_complement_indexes(p.first);
 130 }
 131
 132 template <class Real>
 133 std::pair<std::vector<std::vector<Real>>, std::vector<std::vector<Real>> > generate_constants(unsigned max_index, unsigned max_rows)
 134 {
 135    using boost::math::constants::half_pi;
 136    using boost::math::constants::two_div_pi;
 137    using boost::math::constants::pi;
 138    auto g = [](Real t) { return tanh(half_pi<Real>()*sinh(t)); };
 139    auto w = [](Real t) { Real cs = cosh(half_pi<Real>() * sinh(t));  return half_pi<Real>() * cosh(t) / (cs * cs); };
 140    auto gc = [](Real t) { Real u2 = half_pi<Real>() * sinh(t);  return 1 / (exp(u2) *cosh(u2)); };
 141    auto g_inv = [](float x)->float { return asinh(two_div_pi<float>()*atanh(x)); };
 142    auto gc_inv = [](float x)
 143    {
 144       float l = log(sqrt((2 - x) / x));
 145       return log((sqrt(4 * l * l + pi<float>() * pi<float>()) + 2 * l) / pi<float>());
 146    };
 147
 148    std::vector<std::vector<Real>> abscissa, weights;
 149
 150    std::vector<Real> temp;
 151
 152    float t_crossover = gc_inv(0.5f);
 153
 154    Real h = 1;
 155    for (unsigned i = 0; i < max_index; ++i)
 156    {
 157       temp.push_back((i < t_crossover) ? g(i * h) : -gc(i * h));
 158    }
 159    abscissa.push_back(temp);
 160    temp.clear();
 161
 162    for (unsigned i = 0; i < max_index; ++i)
 163    {
 164       temp.push_back(w(i * h));
 165    }
 166    weights.push_back(temp);
 167    temp.clear();
 168
 169    for (unsigned row = 1; row < max_rows; ++row)
 170    {
 171       h /= 2;
 172       for (Real t = h; t < max_index - 1; t += 2 * h)
 173          temp.push_back((t < t_crossover) ? g(t) : -gc(t));
 174       abscissa.push_back(temp);
 175       temp.clear();
 176    }
 177    h = 1;
 178    for (unsigned row = 1; row < max_rows; ++row)
 179    {
 180       h /= 2;
 181       for (Real t = h; t < max_index - 1; t += 2 * h)
 182          temp.push_back(w(t));
 183       weights.push_back(temp);
 184       temp.clear();
 185    }
 186
 187    return std::make_pair(abscissa, weights);
 188 }
 189
 190 template <class Real>
 191 const tanh_sinh<Real>& get_integrator()
 192 {
 193    static const tanh_sinh<Real> integrator(15);
 194    return integrator;
 195 }
 196
 197 template <class Real>
 198 Real get_convergence_tolerance()
 199 {
 200    return boost::math::tools::root_epsilon<Real>();
 201 }
 202
 203
 204 template<class Real>
 205 void test_linear()
 206 {
 207     std::cout << "Testing linear functions are integrated properly by tanh_sinh on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 208     Real tol = 10*boost::math::tools::epsilon<Real>();
 209     auto integrator = get_integrator<Real>();
 210     auto f = [](const Real& x)
 211     {
 212        return 5*x + 7;
 213     };
 214     Real error;
 215     Real L1;
 216     Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 217     BOOST_CHECK_CLOSE_FRACTION(Q, 9.5, tol);
 218     BOOST_CHECK_CLOSE_FRACTION(L1, 9.5, tol);
 219 }
 220
 221
 222 template<class Real>
 223 void test_quadratic()
 224 {
 225     std::cout << "Testing quadratic functions are integrated properly by tanh_sinh on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 226     Real tol = 10*boost::math::tools::epsilon<Real>();
 227     auto integrator = get_integrator<Real>();
 228     auto f = [](const Real& x) { return 5*x*x + 7*x + 12; };
 229     Real error;
 230     Real L1;
 231     Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 232     BOOST_CHECK_CLOSE_FRACTION(Q, (Real) 17 + half<Real>()*third<Real>(), tol);
 233     BOOST_CHECK_CLOSE_FRACTION(L1, (Real) 17 + half<Real>()*third<Real>(), tol);
 234 }
 235
 236
 237 template<class Real>
 238 void test_singular()
 239 {
 240     std::cout << "Testing integration of endpoint singularities on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 241     Real tol = 10*boost::math::tools::epsilon<Real>();
 242     Real error;
 243     Real L1;
 244     auto integrator = get_integrator<Real>();
 245     auto f = [](const Real& x) { return log(x)*log(1-x); };
 246     Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 247     Real Q_expected = 2 - pi<Real>()*pi<Real>()*half<Real>()*third<Real>();
 248
 249     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 250     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 251 }
 252
 253
 254 // Examples taken from
 255 //http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/quadrature.pdf
 256 template<class Real>
 257 void test_ca()
 258 {
 259     std::cout << "Testing integration of C(a) on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 260     Real tol = 10 * boost::math::tools::epsilon<Real>();
 261     Real error;
 262     Real L1;
 263
 264     auto integrator = get_integrator<Real>();
 265     auto f1 = [](const Real& x) { return atan(x)/(x*(x*x + 1)) ; };
 266     Real Q = integrator.integrate(f1, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 267     Real Q_expected = pi<Real>()*ln_two<Real>()/8 + catalan<Real>()*half<Real>();
 268     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 269     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 270
 271     auto f2 = [](Real x)->Real { Real t0 = x*x + 1; Real t1 = sqrt(t0); return atan(t1)/(t0*t1); };
 272     Q = integrator.integrate(f2, (Real) 0 , (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 273     Q_expected = pi<Real>()/4 - pi<Real>()/root_two<Real>() + 3*atan(root_two<Real>())/root_two<Real>();
 274     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 275     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 276
 277     auto f5 = [](Real t)->Real { return t*t*log(t)/((t*t - 1)*(t*t*t*t + 1)); };
 278     Q = integrator.integrate(f5, (Real) 0 , (Real) 1);
 279     Q_expected = pi<Real>()*pi<Real>()*(2 - root_two<Real>())/32;
 280     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 281
 282
 283     // Oh it suffers on this one.
 284     auto f6 = [](Real t)->Real { Real x = log(t); return x*x; };
 285     Q = integrator.integrate(f6, (Real) 0 , (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 286     Q_expected = 2;
 287
 288     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 50*tol);
 289     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, 50*tol);
 290
 291
 292     // Although it doesn't get to the requested tolerance on this integral, the error bounds can be queried and are reasonable:
 293     tol = sqrt(boost::math::tools::epsilon<Real>());
 294     auto f7 = [](const Real& t) { return sqrt(tan(t)); };
 295     Q = integrator.integrate(f7, (Real) 0 , (Real) half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 296     Q_expected = pi<Real>()/root_two<Real>();
 297     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 298     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 299
 300     auto f8 = [](const Real& t) { return log(cos(t)); };
 301     Q = integrator.integrate(f8, (Real) 0 , half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 302     Q_expected = -pi<Real>()*ln_two<Real>()*half<Real>();
 303     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 304     BOOST_CHECK_CLOSE_FRACTION(L1, -Q_expected, tol);
 305 }
 306
 307
 308 template<class Real>
 309 void test_three_quadrature_schemes_examples()
 310 {
 311     std::cout << "Testing integral in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 312     Real tol = 10 * boost::math::tools::epsilon<Real>();
 313     Real Q;
 314     Real Q_expected;
 315
 316     auto integrator = get_integrator<Real>();
 317     // Example 1:
 318     auto f1 = [](const Real& t) { return t*boost::math::log1p(t); };
 319     Q = integrator.integrate(f1, (Real) 0 , (Real) 1);
 320     Q_expected = half<Real>()*half<Real>();
 321     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 322
 323
 324     // Example 2:
 325     auto f2 = [](const Real& t) { return t*t*atan(t); };
 326     Q = integrator.integrate(f2, (Real) 0 , (Real) 1);
 327     Q_expected = (pi<Real>() -2 + 2*ln_two<Real>())/12;
 328     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 2 * tol);
 329
 330     // Example 3:
 331     auto f3 = [](const Real& t) { return exp(t)*cos(t); };
 332     Q = integrator.integrate(f3, (Real) 0, half_pi<Real>());
 333     Q_expected = boost::math::expm1(half_pi<Real>())*half<Real>();
 334     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 335
 336     // Example 4:
 337     auto f4 = [](Real x)->Real { Real t0 = sqrt(x*x + 2); return atan(t0)/(t0*(x*x+1)); };
 338     Q = integrator.integrate(f4, (Real) 0 , (Real) 1);
 339     Q_expected = 5*pi<Real>()*pi<Real>()/96;
 340     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 341
 342     // Example 5:
 343     auto f5 = [](const Real& t) { return sqrt(t)*log(t); };
 344     Q = integrator.integrate(f5, (Real) 0 , (Real) 1);
 345     Q_expected = -4/ (Real) 9;
 346     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 347
 348     // Example 6:
 349     auto f6 = [](const Real& t) { return sqrt(1 - t*t); };
 350     Q = integrator.integrate(f6, (Real) 0 , (Real) 1);
 351     Q_expected = pi<Real>()/4;
 352     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 353 }
 354
 355
 356 template<class Real>
 357 void test_integration_over_real_line()
 358 {
 359     std::cout << "Testing integrals over entire real line in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 360     Real tol = 10 * boost::math::tools::epsilon<Real>();
 361     Real Q;
 362     Real Q_expected;
 363     Real error;
 364     Real L1;
 365     auto integrator = get_integrator<Real>();
 366
 367     auto f1 = [](const Real& t) { return 1/(1+t*t);};
 368     Q = integrator.integrate(f1, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 369     Q_expected = pi<Real>();
 370     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 371     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 372
 373     auto f2 = [](const Real& t) { return exp(-t*t*half<Real>()); };
 374     Q = integrator.integrate(f2, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 375     Q_expected = root_two_pi<Real>();
 376     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 2);
 377     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol * 2);
 378
 379     // This test shows how oscillatory integrals are approximated very poorly by this method:
 380     //std::cout << "Testing sinc integral: \n";
 381     //Q = integrator.integrate(boost::math::sinc_pi<Real>, -std::numeric_limits<Real>::infinity(), std::numeric_limits<Real>::infinity(), &error, &L1);
 382     //std::cout << "Error estimate of sinc integral: " << error << std::endl;
 383     //std::cout << "L1 norm of sinc integral " << L1 << std::endl;
 384     //Q_expected = pi<Real>();
 385     //BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
 386
 387     auto f4 = [](const Real& t) { return 1/cosh(t);};
 388     Q = integrator.integrate(f4, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 389     Q_expected = pi<Real>();
 390     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 391     BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 392
 393 }
 394
 395 template<class Real>
 396 void test_right_limit_infinite()
 397 {
 398     std::cout << "Testing right limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 399     Real tol = 10 * boost::math::tools::epsilon<Real>();
 400     Real Q;
 401     Real Q_expected;
 402     Real error;
 403     Real L1;
 404     auto integrator = get_integrator<Real>();
 405
 406     // Example 11:
 407     auto f1 = [](const Real& t) { return 1/(1+t*t);};
 408     Q = integrator.integrate(f1, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 409     Q_expected = half_pi<Real>();
 410     BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
 411
 412     // Example 12
 413     auto f2 = [](const Real& t) { return exp(-t)/sqrt(t); };
 414     Q = integrator.integrate(f2, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 415     Q_expected = root_pi<Real>();
 416     BOOST_CHECK_CLOSE(Q, Q_expected, 1000*tol);
 417
 418     auto f3 = [](const Real& t) { return exp(-t)*cos(t); };
 419     Q = integrator.integrate(f3, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 420     Q_expected = half<Real>();
 421     BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
 422
 423     auto f4 = [](const Real& t) { return 1/(1+t*t); };
 424     Q = integrator.integrate(f4, 1, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 425     Q_expected = pi<Real>()/4;
 426     BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
 427 }
 428
 429 template<class Real>
 430 void test_left_limit_infinite()
 431 {
 432     std::cout << "Testing left limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 433     Real tol = 10 * boost::math::tools::epsilon<Real>();
 434     Real Q;
 435     Real Q_expected;
 436     auto integrator = get_integrator<Real>();
 437
 438     // Example 11:
 439     auto f1 = [](const Real& t) { return 1/(1+t*t);};
 440     Q = integrator.integrate(f1, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), Real(0));
 441     Q_expected = half_pi<Real>();
 442     BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol);
 443 }
 444
 445
 446 // A horrible function taken from
 447 // http://www.chebfun.org/examples/quad/GaussClenCurt.html
 448 template<class Real>
 449 void test_horrible()
 450 {
 451     std::cout << "Testing Trefenthen's horrible integral on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 452     // We only know the integral to double precision, so requesting a higher tolerance doesn't make sense.
 453     Real tol = 10 * std::numeric_limits<float>::epsilon();
 454     Real Q;
 455     Real Q_expected;
 456     Real error;
 457     Real L1;
 458     auto integrator = get_integrator<Real>();
 459
 460     auto f = [](Real x)->Real { return x*sin(2*exp(2*sin(2*exp(2*x) ) ) ); };
 461     Q = integrator.integrate(f, (Real) -1, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 462     // NIntegrate[x*Sin[2*Exp[2*Sin[2*Exp[2*x]]]], {x, -1, 1}, WorkingPrecision -> 130, MaxRecursion -> 100]
 463     Q_expected = boost::lexical_cast<Real>("0.33673283478172753598559003181355241139806404130031017259552729882281");
 464     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 465     // Over again without specifying the bounds:
 466     Q = integrator.integrate(f, get_convergence_tolerance<Real>(), &error, &L1);
 467     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 468 }
 469
 470 // Some examples of tough integrals from NR, section 4.5.4:
 471 template<class Real>
 472 void test_nr_examples()
 473 {
 474     std::cout << "Testing singular integrals from NR 4.5.4 on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 475     Real tol = 10 * boost::math::tools::epsilon<Real>();
 476     Real Q;
 477     Real Q_expected;
 478     Real error;
 479     Real L1;
 480     auto integrator = get_integrator<Real>();
 481
 482     auto f1 = [](Real x)->Real
 483     {
 484        return (sin(x * half<Real>()) * exp(-x) / x) / sqrt(x);
 485     };
 486     Q = integrator.integrate(f1, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 487     Q_expected = sqrt(pi<Real>()*(sqrt((Real) 5) - 2));
 488     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 25*tol);
 489
 490     auto f2 = [](Real x)->Real { return pow(x, -(Real) 2/(Real) 7)*exp(-x*x); };
 491     Q = integrator.integrate(f2, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>());
 492     Q_expected = half<Real>()*boost::math::tgamma((Real) 5/ (Real) 14);
 493     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 3);
 494
 495 }
 496
 497 // Test integrand known to fool some termination schemes:
 498 template<class Real>
 499 void test_early_termination()
 500 {
 501     std::cout << "Testing Clenshaw & Curtis's example of integrand which fools termination schemes on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 502     Real tol = 10 * boost::math::tools::epsilon<Real>();
 503     Real Q;
 504     Real Q_expected;
 505     Real error;
 506     Real L1;
 507     auto integrator = get_integrator<Real>();
 508
 509     auto f1 = [](Real x)->Real { return 23*cosh(x)/ (Real) 25 - cos(x) ; };
 510     Q = integrator.integrate(f1, (Real) -1, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 511     Q_expected = 46*sinh((Real) 1)/(Real) 25 - 2*sin((Real) 1);
 512     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 513     // Over again with no bounds:
 514     Q = integrator.integrate(f1);
 515     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 516 }
 517
 518 // Test some definite integrals from the CRC handbook, 32nd edition:
 519 template<class Real>
 520 void test_crc()
 521 {
 522     std::cout << "Testing CRC formulas on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 523     Real tol = 10 * boost::math::tools::epsilon<Real>();
 524     Real Q;
 525     Real Q_expected;
 526     Real error;
 527     Real L1;
 528     auto integrator = get_integrator<Real>();
 529
 530     // CRC Definite integral 585
 531     auto f1 = [](Real x)->Real { Real t = log(1/x); return x*x*t*t*t; };
 532     Q = integrator.integrate(f1, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 533     Q_expected = (Real) 2/ (Real) 27;
 534     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 535
 536     // CRC 636:
 537     auto f2 = [](Real x)->Real { return sqrt(cos(x)); };
 538     Q = integrator.integrate(f2, (Real) 0, (Real) half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 539     //Q_expected = pow(two_pi<Real>(), 3*half<Real>())/pow(boost::math::tgamma((Real) 1/ (Real) 4), 2);
 540     Q_expected = boost::lexical_cast<Real>("1.198140234735592207439922492280323878227212663215651558263674952946405214143915670835885556489793389375907225");
 541     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 542
 543     // CRC Section 5.5, integral 585:
 544     for (int n = 0; n < 3; ++n) {
 545         for (int m = 0; m < 3; ++m) {
 546             auto f = [&](Real x)->Real { return pow(x, Real(m))*pow(log(1/x), Real(n)); };
 547             Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 548             // Calculation of the tgamma function is not exact, giving spurious failures.
 549             // Casting to cpp_bin_float_100 beforehand fixes most of them.
 550             cpp_bin_float_100 np1 = n + 1;
 551             cpp_bin_float_100 mp1 = m + 1;
 552             Q_expected = boost::lexical_cast<Real>((tgamma(np1)/pow(mp1, np1)).str());
 553             BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 554         }
 555     }
 556
 557     // CRC Section 5.5, integral 591
 558     // The parameter p allows us to control the strength of the singularity.
 559     // Rapid convergence is not guaranteed for this function, as the branch cut makes it non-analytic on a disk.
 560     // This converges only when our test type has an extended exponent range as all the area of the integral
 561     // occurs so close to 0 (or 1) that we need abscissa values exceptionally small to find it.
 562     // "There's a lot of room at the bottom".
 563     // We also use a 2 argument functor so that 1-x is evaluated accurately:
 564     if (std::numeric_limits<Real>::max_exponent > std::numeric_limits<double>::max_exponent)
 565     {
 566        for (Real p = -0.99; p < 1; p += 0.1) {
 567           auto f = [&](Real x, Real cx)->Real
 568           {
 569              //return pow(x, p) / pow(1 - x, p);
 570              return cx < 0 ? exp(p * (log(x) - boost::math::log1p(-x))) : pow(x, p) / pow(cx, p);
 571           };
 572           Q = integrator.integrate(f, (Real)0, (Real)1, get_convergence_tolerance<Real>(), &error, &L1);
 573           Q_expected = 1 / sinc_pi(p*pi<Real>());
 574           BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol);
 575        }
 576     }
 577     // There is an alternative way to evaluate the above integral: by noticing that all the area of the integral
 578     // is near zero for p < 0 and near 1 for p > 0 we can substitute exp(-x) for x and remap the integral to the
 579     // domain (0, INF).  Internally we need to expand out the terms and evaluate using logs to avoid spurous overflow,
 580     // this gives us
 581     // for p > 0:
 582     for (Real p = 0.99; p > 0; p -= 0.1) {
 583        auto f = [&](Real x)->Real
 584        {
 585           return exp(-x * (1 - p) + p * log(-boost::math::expm1(-x)));
 586        };
 587        Q = integrator.integrate(f, 0, boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 588        Q_expected = 1 / sinc_pi(p*pi<Real>());
 589        BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol);
 590     }
 591     // and for p < 1:
 592     for (Real p = -0.99; p < 0; p += 0.1) {
 593        auto f = [&](Real x)->Real
 594        {
 595           return exp(-p * log(-boost::math::expm1(-x)) - (1 + p) * x);
 596        };
 597        Q = integrator.integrate(f, 0, boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 598        Q_expected = 1 / sinc_pi(p*pi<Real>());
 599        BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol);
 600     }
 601
 602     // CRC Section 5.5, integral 635
 603     for (int m = 0; m < 10; ++m) {
 604         auto f = [&](Real x)->Real { return 1/(1 + pow(tan(x), m)); };
 605         Q = integrator.integrate(f, (Real) 0, half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 606         Q_expected = half_pi<Real>()/2;
 607         BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 608     }
 609
 610     // CRC Section 5.5, integral 637:
 611     //
 612     // When h gets very close to 1, the strength of the singularity gradually increases until we
 613     // no longer have enough exponent range to evaluate the integral....
 614     // We also have to use the 2-argument functor version of the integrator to avoid
 615     // cancellation error, since the singularity is near PI/2.
 616     //
 617     Real limit = std::numeric_limits<Real>::max_exponent > std::numeric_limits<double>::max_exponent
 618        ? .95f : .85f;
 619     for (Real h = 0.01; h < limit; h += 0.1) {
 620         auto f = [&](Real x, Real xc)->Real { return xc > 0 ? pow(1/tan(xc), h) : pow(tan(x), h); };
 621         Q = integrator.integrate(f, (Real) 0, half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 622         Q_expected = half_pi<Real>()/cos(h*half_pi<Real>());
 623         BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 624     }
 625     // CRC Section 5.5, integral 637:
 626     //
 627     // Over again, but with argument transformation, we want:
 628     //
 629     // Integral of tan(x)^h over (0, PI/2)
 630     //
 631     // Note that the bulk of the area is next to the singularity at PI/2,
 632     // so we'll start by replacing x by PI/2 - x, and that tan(PI/2 - x) == 1/tan(x)
 633     // so we now have:
 634     //
 635     // Integral of 1/tan(x)^h over (0, PI/2)
 636     //
 637     // Which is almost the ideal form, except that when h is very close to 1
 638     // we run out of exponent range in evaluating the integral arbitrarily close to 0.
 639     // So now we substitute exp(-x) for x: this stretches out the range of the
 640     // integral to (-log(PI/2), +INF) with the singularity at +INF giving:
 641     //
 642     // Integral of exp(-x)/tan(exp(-x))^h over (-log(PI/2), +INF)
 643     //
 644     // We just need a way to evaluate the function without spurious under/overflow
 645     // in the exp terms.  Note that for small x: tan(x) ~= x, so making this
 646     // substitution and evaluating by logs we have:
 647     //
 648     // exp(-x)/tan(exp(-x))^h ~= exp((h - 1) * x)  for x > -log(epsion);
 649     //
 650     // Here's how that looks in code:
 651     //
 652     for (Real i = 80; i < 100; ++i) {
 653        Real h = i / 100;
 654        auto f = [&](Real x)->Real
 655        {
 656           if (x > -log(boost::math::tools::epsilon<Real>()))
 657              return exp((h - 1) * x);
 658           else
 659           {
 660              Real et = exp(-x);
 661              // Need to deal with numeric instability causing et to be greater than PI/2:
 662              return et >= boost::math::constants::half_pi<Real>() ? 0 : et * pow(1 / tan(et), h);
 663           }
 664        };
 665        Q = integrator.integrate(f, -log(half_pi<Real>()), boost::math::tools::max_value<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 666        Q_expected = half_pi<Real>() / cos(h*half_pi<Real>());
 667        BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 5 * tol);
 668     }
 669
 670     // CRC Section 5.5, integral 670:
 671
 672     auto f3 = [](Real x)->Real { return sqrt(log(1/x)); };
 673     Q = integrator.integrate(f3, (Real) 0, (Real) 1, get_convergence_tolerance<Real>(), &error, &L1);
 674     Q_expected = root_pi<Real>()/2;
 675     BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 676
 677 }
 678
 679 template <class Real>
 680 void test_sf()
 681 {
 682    using std::sqrt;
 683    // Test some special functions that we already know how to evaluate:
 684    std::cout << "Testing special functions on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 685    Real tol = 10 * boost::math::tools::epsilon<Real>();
 686    auto integrator = get_integrator<Real>();
 687
 688    // incomplete beta:
 689    if (std::numeric_limits<Real>::digits10 < 37) // Otherwise too slow
 690    {
 691       Real a(100), b(15);
 692       auto f = [&](Real x)->Real { return boost::math::ibeta_derivative(a, b, x); };
 693       BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, Real(0.25)), boost::math::ibeta(100, 15, Real(0.25)), tol * 10);
 694       // Check some really extreme versions:
 695       a = 1000;
 696       b = 500;
 697       BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, 1), Real(1), tol * 10);
 698       //
 699       // This is as extreme as we can get in this domain: otherwise the function has all it's
 700       // area so close to zero we never get in there no matter how many levels we go down:
 701       //
 702       a = Real(1) / 15;
 703       b = 30;
 704       BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, 1), Real(1), tol * 25);
 705    }
 706
 707    Real x = 2, y = 3, z = 0.5, p = 0.25;
 708    // Elliptic integral RC:
 709    Real Q = integrator.integrate([&](const Real& t) { return 1 / (sqrt(t + x) * (t + y)); }, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>()) / 2;
 710    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::ellint_rc(x, y), tol);
 711    // Elliptic Integral RJ:
 712    BOOST_CHECK_CLOSE_FRACTION(Real((Real(3) / 2) * integrator.integrate([&](Real t)->Real { return 1 / (sqrt((t + x) * (t + y) * (t + z)) * (t + p)); }, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>())), boost::math::ellint_rj(x, y, z, p), tol);
 713
 714    z = 5.5;
 715    if (std::numeric_limits<Real>::digits10 > 40)
 716       tol *= 200;
 717    else if (!std::numeric_limits<Real>::is_specialized)
 718       tol *= 3;
 719    // tgamma expressed as an integral:
 720    BOOST_CHECK_CLOSE_FRACTION(integrator.integrate([&](Real t)->Real { using std::pow; using std::exp; return t > 10000 ? Real(0) : Real(pow(t, z - 1) * exp(-t)); }, 0, std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>()),
 721       boost::math::tgamma(z), tol);
 722    BOOST_CHECK_CLOSE_FRACTION(integrator.integrate([](const Real& t) {  using std::exp; return exp(-t*t); }, std::numeric_limits<Real>::has_infinity ? -std::numeric_limits<Real>::infinity() : -boost::math::tools::max_value<Real>(), std::numeric_limits<Real>::has_infinity ? std::numeric_limits<Real>::infinity() : boost::math::tools::max_value<Real>()),
 723       boost::math::constants::root_pi<Real>(), tol);
 724 }
 725
 726 template <class Real>
 727 void test_2_arg()
 728 {
 729    BOOST_MATH_STD_USING
 730    std::cout << "Testing 2 argument functors on type " << boost::typeindex::type_id<Real>().pretty_name() << "\n";
 731    Real tol = 10 * boost::math::tools::epsilon<Real>();
 732
 733    auto integrator = get_integrator<Real>();
 734
 735    //
 736    // There are a whole family of integrals of the general form
 737    // x(1-x)^-N ; N < 1
 738    // which have all the interesting behaviour near the 2 singularities
 739    // and all converge, see: http://www.wolframalpha.com/input/?i=integrate+(x+*+(1-x))%5E-1%2FN+from+0+to+1
 740    //
 741    Real Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
 742    {
 743       return tc < 0 ? 1 / sqrt(t * (1-t)) : 1 / sqrt(t * tc);
 744    }, 0, 1);
 745    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi<Real>(), tol);
 746    Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
 747    {
 748       return tc < 0 ? 1 / boost::math::cbrt(t * (1-t)) : 1 / boost::math::cbrt(t * tc);
 749    }, 0, 1);
 750    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::pow<2>(boost::math::tgamma(Real(2) / 3)) / boost::math::tgamma(Real(4) / 3), tol * 3);
 751    //
 752    // We can do the same thing with ((1+x)(1-x))^-N ; N < 1
 753    //
 754    Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
 755    {
 756       return t < 0 ? 1 / sqrt(-tc * (1-t)) : 1 / sqrt((t + 1) * tc);
 757    }, -1, 1);
 758    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi<Real>(), tol);
 759    Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
 760    {
 761       return t < 0 ? 1 / sqrt(-tc * (1-t)) : 1 / sqrt((t + 1) * tc);
 762    });
 763    BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi<Real>(), tol);
 764    Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
 765    {
 766       return t < 0 ? 1 / boost::math::cbrt(-tc * (1-t)) : 1 / boost::math::cbrt((t + 1) * tc);
 767    }, -1, 1);
 768    BOOST_CHECK_CLOSE_FRACTION(Q, sqrt(boost::math::constants::pi<Real>()) * boost::math::tgamma(Real(2) / 3) / boost::math::tgamma(Real(7) / 6), tol * 10);
 769    Q = integrator.integrate([&](const Real& t, const Real & tc)->Real
 770    {
 771       return t < 0 ? 1 / boost::math::cbrt(-tc * (1-t)) : 1 / boost::math::cbrt((t + 1) * tc);
 772    });
 773    BOOST_CHECK_CLOSE_FRACTION(Q, sqrt(boost::math::constants::pi<Real>()) * boost::math::tgamma(Real(2) / 3) / boost::math::tgamma(Real(7) / 6), tol * 10);
 774    //
 775    // These are taken from above, and do not get to full precision via the single arg functor:
 776    //
 777    auto f7 = [](const Real& t, const Real& tc) { return t < 1 ? sqrt(tan(t)) : sqrt(1/tan(tc)); };
 778    Real error, L1;
 779    Q = integrator.integrate(f7, (Real)0, (Real)half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 780    Real Q_expected = pi<Real>() / root_two<Real>();
 781    BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 782    BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol);
 783
 784    auto f8 = [](const Real& t, const Real& tc) { return t < 1 ? log(cos(t)) : log(sin(tc)); };
 785    Q = integrator.integrate(f8, (Real)0, half_pi<Real>(), get_convergence_tolerance<Real>(), &error, &L1);
 786    Q_expected = -pi<Real>()*ln_two<Real>()*half<Real>();
 787    BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol);
 788    BOOST_CHECK_CLOSE_FRACTION(L1, -Q_expected, tol);
 789 }
 790
 791
 792 BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
 793 {
 794 #ifdef GENERATE_CONSTANTS
 795    //
 796    // Generate pre-computed coefficients:
 797    std::cout << std::setprecision(35);
 798    print_levels(generate_constants<cpp_bin_float_100>(10, 8), "L");
 799
 800 #else
 801
 802 #ifdef TEST1
 803
 804     test_right_limit_infinite<float>();
 805     test_left_limit_infinite<float>();
 806     test_linear<float>();
 807     test_quadratic<float>();
 808     test_singular<float>();
 809     test_ca<float>();
 810     test_three_quadrature_schemes_examples<float>();
 811     test_horrible<float>();
 812     test_integration_over_real_line<float>();
 813     test_nr_examples<float>();
 814 #endif
 815 #ifdef TEST1A
 816     test_early_termination<float>();
 817     test_crc<float>();
 818     test_2_arg<float>();
 819
 820 #endif
 821 #ifdef TEST2
 822     test_right_limit_infinite<double>();
 823     test_left_limit_infinite<double>();
 824     test_linear<double>();
 825     test_quadratic<double>();
 826     test_singular<double>();
 827     test_ca<double>();
 828     test_three_quadrature_schemes_examples<double>();
 829     test_horrible<double>();
 830     test_integration_over_real_line<double>();
 831     test_nr_examples<double>();
 832     test_early_termination<double>();
 833     test_sf<double>();
 834     test_2_arg<double>();
 835 #endif
 836 #ifdef TEST2A
 837     test_crc<double>();
 838 #endif
 839
 840 #ifdef TEST3
 841
 842     test_right_limit_infinite<long double>();
 843     test_left_limit_infinite<long double>();
 844     test_linear<long double>();
 845     test_quadratic<long double>();
 846     test_singular<long double>();
 847     test_ca<long double>();
 848     test_three_quadrature_schemes_examples<long double>();
 849     test_horrible<long double>();
 850     test_integration_over_real_line<long double>();
 851     test_nr_examples<long double>();
 852     test_early_termination<long double>();
 853     test_sf<long double>();
 854     test_2_arg<long double>();
 855 #endif
 856 #ifdef TEST3A
 857     test_crc<long double>();
 858
 859 #endif
 860
 861 #ifdef TEST4
 862
 863     test_right_limit_infinite<cpp_bin_float_quad>();
 864     test_left_limit_infinite<cpp_bin_float_quad>();
 865     test_linear<cpp_bin_float_quad>();
 866     test_quadratic<cpp_bin_float_quad>();
 867     test_singular<cpp_bin_float_quad>();
 868     test_ca<cpp_bin_float_quad>();
 869     test_three_quadrature_schemes_examples<cpp_bin_float_quad>();
 870     test_horrible<cpp_bin_float_quad>();
 871     test_nr_examples<cpp_bin_float_quad>();
 872     test_early_termination<cpp_bin_float_quad>();
 873     test_crc<cpp_bin_float_quad>();
 874     test_sf<cpp_bin_float_quad>();
 875     test_2_arg<cpp_bin_float_quad>();
 876
 877 #endif
 878 #ifdef TEST5
 879
 880     test_sf<cpp_bin_float_50>();
 881     test_sf<cpp_bin_float_100>();
 882     test_sf<boost::multiprecision::number<boost::multiprecision::cpp_bin_float<150> > >();
 883
 884 #endif
 885 #ifdef TEST6
 886
 887     test_right_limit_infinite<boost::math::concepts::real_concept>();
 888     test_left_limit_infinite<boost::math::concepts::real_concept>();
 889     test_linear<boost::math::concepts::real_concept>();
 890     test_quadratic<boost::math::concepts::real_concept>();
 891     test_singular<boost::math::concepts::real_concept>();
 892     test_ca<boost::math::concepts::real_concept>();
 893     test_three_quadrature_schemes_examples<boost::math::concepts::real_concept>();
 894     test_horrible<boost::math::concepts::real_concept>();
 895     test_integration_over_real_line<boost::math::concepts::real_concept>();
 896     test_nr_examples<boost::math::concepts::real_concept>();
 897     test_early_termination<boost::math::concepts::real_concept>();
 898     test_sf<boost::math::concepts::real_concept>();
 899     test_2_arg<boost::math::concepts::real_concept>();
 900 #endif
 901 #ifdef TEST6A
 902     test_crc<boost::math::concepts::real_concept>();
 903
 904 #endif
 905 #ifdef TEST7
 906     test_sf<cpp_dec_float_50>();
 907 #endif
 908 #if defined(TEST8) && defined(BOOST_HAS_FLOAT128)
 909
 910     test_right_limit_infinite<boost::multiprecision::float128>();
 911     test_left_limit_infinite<boost::multiprecision::float128>();
 912     test_linear<boost::multiprecision::float128>();
 913     test_quadratic<boost::multiprecision::float128>();
 914     test_singular<boost::multiprecision::float128>();
 915     test_ca<boost::multiprecision::float128>();
 916     test_three_quadrature_schemes_examples<boost::multiprecision::float128>();
 917     test_horrible<boost::multiprecision::float128>();
 918     test_integration_over_real_line<boost::multiprecision::float128>();
 919     test_nr_examples<boost::multiprecision::float128>();
 920     test_early_termination<boost::multiprecision::float128>();
 921     test_crc<boost::multiprecision::float128>();
 922     test_sf<boost::multiprecision::float128>();
 923     test_2_arg<boost::multiprecision::float128>();
 924
 925 #endif
 926 #endif
 927 }
 928
 929 #else
 930
 931 int main() { return 0; }
 932
 933 #endif