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

   1 // Copyright Paul Bristow 2007, 2011.
   2 // Copyright John Maddock 2006, 2011.
   3
   4 // Use, modification and distribution are subject to the
   5 // Boost Software License, Version 1.0.
   6 // (See accompanying file LICENSE_1_0.txt
   7 // or copy at http://www.boost.org/LICENSE_1_0.txt)
   8
   9 // test_constants.cpp
  10
  11 // Check values of constants are drawn from an independent source, or calculated.
  12 // Both must be at long double precision for the most precise compilers floating-point implementation.
  13 // So all values use static_cast<RealType>() of values at least 40 decimal digits
  14 // and that have suffix L to ensure floating-point type is long double.
  15
  16 // Steve Moshier's command interpreter V1.3 100 digits calculator used for some values.
  17
  18 #ifdef _MSC_VER
  19 #  pragma warning(disable : 4127) // conditional expression is constant.
  20 #endif
  21
  22 #include <boost/math/concepts/real_concept.hpp> // for real_concept
  23 #define BOOST_TEST_MAIN
  24 #include <boost/test/unit_test.hpp> // Boost.Test
  25 #include <boost/test/floating_point_comparison.hpp>
  26
  27 #include <boost/math/constants/constants.hpp>
  28 #include <boost/math/tools/test.hpp>
  29 #include <boost/static_assert.hpp>
  30 #include <boost/utility/enable_if.hpp>
  31
  32 // Check at compile time that the construction method for constants of type float, is "construct from a float", or "construct from a double", ...
  33 BOOST_STATIC_ASSERT((boost::is_same<boost::math::constants::construction_traits<float, boost::math::policies::policy<> >::type, boost::mpl::int_<boost::math::constants::construct_from_float> >::value));
  34 BOOST_STATIC_ASSERT((boost::is_same<boost::math::constants::construction_traits<double, boost::math::policies::policy<> >::type, boost::mpl::int_<boost::math::constants::construct_from_double> >::value));
  35 BOOST_STATIC_ASSERT((boost::is_same<boost::math::constants::construction_traits<long double, boost::math::policies::policy<> >::type, boost::mpl::int_<(sizeof(double) == sizeof(long double) ? boost::math::constants::construct_from_double : boost::math::constants::construct_from_long_double)> >::value));
  36 BOOST_STATIC_ASSERT((boost::is_same<boost::math::constants::construction_traits<boost::math::concepts::real_concept, boost::math::policies::policy<> >::type, boost::mpl::int_<0> >::value));
  37
  38 // Policy to set precision at maximum possible using long double.
  39 typedef boost::math::policies::policy<boost::math::policies::digits2<std::numeric_limits<long double>::digits> > real_concept_policy_1;
  40 // Policy with precision +2 (could be any reasonable value),
  41 // forces the precision of the policy to be greater than
  42 // that of a long double, and therefore triggers different code (construct from string).
  43 #ifdef BOOST_MATH_USE_FLOAT128
  44 typedef boost::math::policies::policy<boost::math::policies::digits2<115> > real_concept_policy_2;
  45 #else
  46 typedef boost::math::policies::policy<boost::math::policies::digits2<std::numeric_limits<long double>::digits + 2> > real_concept_policy_2;
  47 #endif
  48 // Policy with precision greater than the string representations, forces computation of values (i.e. different code path):
  49 typedef boost::math::policies::policy<boost::math::policies::digits2<400> > real_concept_policy_3;
  50
  51 BOOST_STATIC_ASSERT((boost::is_same<boost::math::constants::construction_traits<boost::math::concepts::real_concept, real_concept_policy_1 >::type, boost::mpl::int_<(sizeof(double) == sizeof(long double) ? boost::math::constants::construct_from_double : boost::math::constants::construct_from_long_double) > >::value));
  52 BOOST_STATIC_ASSERT((boost::is_same<boost::math::constants::construction_traits<boost::math::concepts::real_concept, real_concept_policy_2 >::type, boost::mpl::int_<boost::math::constants::construct_from_string> >::value));
  53 BOOST_STATIC_ASSERT((boost::math::constants::construction_traits<boost::math::concepts::real_concept, real_concept_policy_3>::type::value >= 5));
  54
  55 #ifndef BOOST_NO_CXX11_CONSTEXPR
  56
  57 constexpr float fval = boost::math::constants::pi<float>();
  58 constexpr double dval = boost::math::constants::pi<double>();
  59 constexpr long double ldval = boost::math::constants::pi<long double>();
  60
  61 constexpr float fval2 = boost::math::float_constants::pi;
  62 constexpr double dval2 = boost::math::double_constants::pi;
  63 constexpr long double ldval2 = boost::math::long_double_constants::pi;
  64
  65 #endif
  66
  67 // We need to declare a conceptual type whose precision is unknown at
  68 // compile time, and is so enormous when checked at runtime,
  69 // that we're forced to calculate the values of the constants ourselves.
  70
  71 namespace boost{ namespace math{ namespace concepts{
  72
  73 class big_real_concept : public real_concept
  74 {
  75 public:
  76    big_real_concept() {}
  77    template <class T>
  78    big_real_concept(const T& t, typename enable_if<is_convertible<T, real_concept> >::type* = 0) : real_concept(t) {}
  79 };
  80
  81 inline int itrunc(const big_real_concept& val)
  82 {
  83    BOOST_MATH_STD_USING
  84    return itrunc(val.value());
  85 }
  86
  87 }
  88 namespace tools{
  89
  90 template <>
  91 inline BOOST_MATH_CONSTEXPR int digits<concepts::big_real_concept>(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(T)) BOOST_NOEXCEPT
  92 {
  93    return 2 * boost::math::constants::max_string_digits;
  94 }
  95
  96 }}}
  97
  98 template <class RealType>
  99 void test_spots(RealType)
 100 {
 101    // Basic sanity checks for constants,
 102    // where template parameter RealType can be float, double, long double,
 103    // or real_concept, a prototype for user-defined floating-point types.
 104
 105    // Parameter RealType is only used to communicate the RealType,
 106   //  and is an arbitrary zero for all tests.
 107    //
 108    // Actual tolerance is never really smaller than epsilon for long double,
 109    // because it's just a wrapper around a long double,
 110    // so although it's pretending to be something else (in order to exercise our code),
 111    // it can never really have precision greater than a long double.
 112
 113   typedef typename boost::math::constants::construction_traits<RealType, boost::math::policies::policy<> >::type construction_type;
 114    RealType tolerance = (std::max)(static_cast<RealType>(boost::math::tools::epsilon<long double>()), boost::math::tools::epsilon<RealType>()) * 2;  // double
 115    if((construction_type::value == 0) && (boost::math::tools::digits<RealType>() > boost::math::constants::max_string_digits))
 116       tolerance *= 30;  // Allow a little extra tolerance
 117    // for calculated (perhaps using a series representation) constants.
 118    std::cout << "Tolerance for type " << typeid(RealType).name()  << " is " << tolerance << "." << std::endl;
 119
 120    //typedef typename boost::math::policies::precision<RealType, boost::math::policies::policy<> >::type t1;
 121    // A precision of zero means we don't know what the precision of this type is until runtime.
 122    //std::cout << "Precision for type " << typeid(RealType).name()  << " is " << t1::value << "." << std::endl;
 123
 124    using namespace boost::math::constants;
 125    BOOST_MATH_STD_USING
 126
 127    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L, pi<RealType>(), tolerance);
 128    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L), root_pi<RealType>(), tolerance);
 129    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L/2), root_half_pi<RealType>(), tolerance);
 130    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L * 2), root_two_pi<RealType>(), tolerance);
 131    BOOST_CHECK_CLOSE_FRACTION(sqrt(log(4.0L)), root_ln_four<RealType>(), tolerance);
 132    BOOST_CHECK_CLOSE_FRACTION(2.71828182845904523536028747135266249775724709369995L, e<RealType>(), tolerance);
 133    BOOST_CHECK_CLOSE_FRACTION(0.5L, half<RealType>(), tolerance);
 134    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104259335L, euler<RealType>(), tolerance);
 135    BOOST_CHECK_CLOSE_FRACTION(sqrt(2.0L), root_two<RealType>(), tolerance);
 136    BOOST_CHECK_CLOSE_FRACTION(log(2.0L), ln_two<RealType>(), tolerance);
 137    BOOST_CHECK_CLOSE_FRACTION(log(10.0L), ln_ten<RealType>(), tolerance);
 138    BOOST_CHECK_CLOSE_FRACTION(log(log(2.0L)), ln_ln_two<RealType>(), tolerance);
 139
 140    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1)/3, third<RealType>(), tolerance);
 141    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(2)/3, twothirds<RealType>(), tolerance);
 142    BOOST_CHECK_CLOSE_FRACTION(0.14159265358979323846264338327950288419716939937510L, pi_minus_three<RealType>(), tolerance);
 143    BOOST_CHECK_CLOSE_FRACTION(4.L - 3.14159265358979323846264338327950288419716939937510L, four_minus_pi<RealType>(), tolerance);
 144 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 145    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510L), 2.71828182845904523536028747135266249775724709369995L), pi_pow_e<RealType>(), tolerance);
 146    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510L), 0.33333333333333333333333333333333333333333333333333L), cbrt_pi<RealType>(), tolerance);
 147    BOOST_CHECK_CLOSE_FRACTION(exp(-0.5L), exp_minus_half<RealType>(), tolerance);
 148    BOOST_CHECK_CLOSE_FRACTION(pow(2.71828182845904523536028747135266249775724709369995L, 3.14159265358979323846264338327950288419716939937510L), e_pow_pi<RealType>(), tolerance);
 149
 150
 151 #else // Only double, so no suffix L.
 152    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995), pi_pow_e<RealType>(), tolerance);
 153    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510), 0.33333333333333333333333333333333333333333333333333), cbrt_pi<RealType>(), tolerance);
 154    BOOST_CHECK_CLOSE_FRACTION(exp(-0.5), exp_minus_half<RealType>(), tolerance);
 155 #endif
 156    // Rational fractions.
 157    BOOST_CHECK_CLOSE_FRACTION(0.333333333333333333333333333333333333333L, third<RealType>(), tolerance);
 158    BOOST_CHECK_CLOSE_FRACTION(0.666666666666666666666666666666666666667L, two_thirds<RealType>(), tolerance);
 159    BOOST_CHECK_CLOSE_FRACTION(0.75L, three_quarters<RealType>(), tolerance);
 160    // Two and related.
 161    BOOST_CHECK_CLOSE_FRACTION(sqrt(2.L), root_two<RealType>(), tolerance);
 162    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.L), root_three<RealType>(), tolerance);
 163    BOOST_CHECK_CLOSE_FRACTION(sqrt(2.L)/2, half_root_two<RealType>(), tolerance);
 164    BOOST_CHECK_CLOSE_FRACTION(log(2.L), ln_two<RealType>(), tolerance);
 165    BOOST_CHECK_CLOSE_FRACTION(log(log(2.0L)), ln_ln_two<RealType>(), tolerance);
 166    BOOST_CHECK_CLOSE_FRACTION(sqrt(log(4.0L)), root_ln_four<RealType>(), tolerance);
 167    BOOST_CHECK_CLOSE_FRACTION(1/sqrt(2.0L), one_div_root_two<RealType>(), tolerance);
 168
 169    // pi.
 170    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L, pi<RealType>(), tolerance);
 171    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L/2, half_pi<RealType>(), tolerance);
 172    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L/3, third_pi<RealType>(), tolerance);
 173    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L/6, sixth_pi<RealType>(), tolerance);
 174    BOOST_CHECK_CLOSE_FRACTION(2 * 3.14159265358979323846264338327950288419716939937510L, two_pi<RealType>(), tolerance);
 175    BOOST_CHECK_CLOSE_FRACTION(3 * 3.14159265358979323846264338327950288419716939937510L / 4, three_quarters_pi<RealType>(), tolerance);
 176    BOOST_CHECK_CLOSE_FRACTION(4 * 3.14159265358979323846264338327950288419716939937510L / 3, four_thirds_pi<RealType>(), tolerance);
 177    BOOST_CHECK_CLOSE_FRACTION(1 / (2 * 3.14159265358979323846264338327950288419716939937510L), one_div_two_pi<RealType>(), tolerance);
 178    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L), root_pi<RealType>(), tolerance);
 179    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L / 2), root_half_pi<RealType>(), tolerance);
 180    BOOST_CHECK_CLOSE_FRACTION(sqrt(2 * 3.14159265358979323846264338327950288419716939937510L), root_two_pi<RealType>(), tolerance);
 181    BOOST_CHECK_CLOSE_FRACTION(1 / sqrt(3.14159265358979323846264338327950288419716939937510L), one_div_root_pi<RealType>(), tolerance);
 182    BOOST_CHECK_CLOSE_FRACTION(1 / sqrt(2 * 3.14159265358979323846264338327950288419716939937510L), one_div_root_two_pi<RealType>(), tolerance);
 183    BOOST_CHECK_CLOSE_FRACTION(sqrt(1. / 3.14159265358979323846264338327950288419716939937510L), root_one_div_pi<RealType>(), tolerance);
 184    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L - 3.L, pi_minus_three<RealType>(), tolerance * 4 ); // tolerance * 2 because of cancellation loss.
 185    BOOST_CHECK_CLOSE_FRACTION(4.L - 3.14159265358979323846264338327950288419716939937510L, four_minus_pi<RealType>(), tolerance );
 186    //
 187    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510L), 2.71828182845904523536028747135266249775724709369995L), pi_pow_e<RealType>(), tolerance);  // See above.
 188    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L, pi_sqr<RealType>(), tolerance);  // See above.
 189    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L/6, pi_sqr_div_six<RealType>(), tolerance);  // See above.
 190    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L, pi_cubed<RealType>(), tolerance);  // See above.
 191
 192    // BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L, cbrt_pi<RealType>(), tolerance);  // See above.
 193    BOOST_CHECK_CLOSE_FRACTION(cbrt_pi<RealType>() * cbrt_pi<RealType>() * cbrt_pi<RealType>(), pi<RealType>(), tolerance);
 194    BOOST_CHECK_CLOSE_FRACTION((1)/cbrt_pi<RealType>(), one_div_cbrt_pi<RealType>(), tolerance);
 195
 196    // Euler
 197    BOOST_CHECK_CLOSE_FRACTION(2.71828182845904523536028747135266249775724709369995L, e<RealType>(), tolerance);
 198
 199    //BOOST_CHECK_CLOSE_FRACTION(exp(-0.5L), exp_minus_half<RealType>(), tolerance);  // See above.
 200    BOOST_CHECK_CLOSE_FRACTION(pow(e<RealType>(), pi<RealType>()), e_pow_pi<RealType>(), tolerance); // See also above.
 201    BOOST_CHECK_CLOSE_FRACTION(sqrt(e<RealType>()), root_e<RealType>(), tolerance);
 202    BOOST_CHECK_CLOSE_FRACTION(log10(e<RealType>()), log10_e<RealType>(), tolerance);
 203    BOOST_CHECK_CLOSE_FRACTION(1/log10(e<RealType>()), one_div_log10_e<RealType>(), tolerance);
 204
 205    // Trigonmetric
 206    BOOST_CHECK_CLOSE_FRACTION(pi<RealType>()/180, degree<RealType>(), tolerance);
 207    BOOST_CHECK_CLOSE_FRACTION(180 / pi<RealType>(), radian<RealType>(), tolerance);
 208    BOOST_CHECK_CLOSE_FRACTION(sin(1.L), sin_one<RealType>(), tolerance);
 209    BOOST_CHECK_CLOSE_FRACTION(cos(1.L), cos_one<RealType>(), tolerance);
 210    BOOST_CHECK_CLOSE_FRACTION(sinh(1.L), sinh_one<RealType>(), tolerance);
 211    BOOST_CHECK_CLOSE_FRACTION(cosh(1.L), cosh_one<RealType>(), tolerance);
 212
 213    // Phi
 214    BOOST_CHECK_CLOSE_FRACTION((1.L + sqrt(5.L)) /2, phi<RealType>(), tolerance);
 215    BOOST_CHECK_CLOSE_FRACTION(log((1.L + sqrt(5.L)) /2), ln_phi<RealType>(), tolerance);
 216    BOOST_CHECK_CLOSE_FRACTION(1.L / log((1.L + sqrt(5.L)) /2), one_div_ln_phi<RealType>(), tolerance);
 217
 218    //Euler's Gamma
 219    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992L, euler<RealType>(), tolerance); // (sequence A001620 in OEIS).
 220    BOOST_CHECK_CLOSE_FRACTION(1.L/ 0.57721566490153286060651209008240243104215933593992L, one_div_euler<RealType>(), tolerance); // (from sequence A001620 in OEIS).
 221    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992L * 0.57721566490153286060651209008240243104215933593992L, euler_sqr<RealType>(), tolerance); // (from sequence A001620 in OEIS).
 222
 223    // Misc
 224    BOOST_CHECK_CLOSE_FRACTION(1.644934066848226436472415166646025189218949901206L, zeta_two<RealType>(), tolerance); // A013661 as a constant (usually base 10) in OEIS.
 225    BOOST_CHECK_CLOSE_FRACTION(1.20205690315959428539973816151144999076498629234049888179227L, zeta_three<RealType>(), tolerance); // (sequence A002117 in OEIS)
 226    BOOST_CHECK_CLOSE_FRACTION(.91596559417721901505460351493238411077414937428167213L, catalan<RealType>(), tolerance); // A006752 as a constant in OEIS.
 227    BOOST_CHECK_CLOSE_FRACTION(1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150L, extreme_value_skewness<RealType>(), tolerance); //  Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
 228    BOOST_CHECK_CLOSE_FRACTION(0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067L, rayleigh_skewness<RealType>(), tolerance); // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
 229    BOOST_CHECK_CLOSE_FRACTION(2.450893006876380628486604106197544154e-01L, rayleigh_kurtosis_excess<RealType>(), tolerance * 2);
 230    BOOST_CHECK_CLOSE_FRACTION(2.68545200106530644530971483548179569382038229399446295305115234555721885953715200280114117493184769799515L, khinchin<RealType>(), tolerance ); // A002210 as a constant https://oeis.org/A002210/constant
 231    BOOST_CHECK_CLOSE_FRACTION(1.2824271291006226368753425688697917277676889273250011L, glaisher<RealType>(), tolerance ); // https://oeis.org/A074962/constant
 232
 233    //
 234    // Last of all come the test cases that behave differently if we're calculating the constants on the fly:
 235    //
 236    if(boost::math::tools::digits<RealType>() > boost::math::constants::max_string_digits)
 237    {
 238       // This suffers from cancellation error, so increased tolerance:
 239       BOOST_CHECK_CLOSE_FRACTION(static_cast<RealType>(4. - 3.14159265358979323846264338327950288419716939937510L), four_minus_pi<RealType>(), tolerance * 3);
 240       BOOST_CHECK_CLOSE_FRACTION(static_cast<RealType>(0.14159265358979323846264338327950288419716939937510L), pi_minus_three<RealType>(), tolerance * 3);
 241    }
 242    else
 243    {
 244       BOOST_CHECK_CLOSE_FRACTION(static_cast<RealType>(4. - 3.14159265358979323846264338327950288419716939937510L), four_minus_pi<RealType>(), tolerance);
 245       BOOST_CHECK_CLOSE_FRACTION(static_cast<RealType>(0.14159265358979323846264338327950288419716939937510L), pi_minus_three<RealType>(), tolerance);
 246    }
 247 } // template <class RealType>void test_spots(RealType)
 248
 249 void test_float_spots()
 250 {
 251    // Basic sanity checks for constants in boost::math::float_constants::
 252    // for example: boost::math::float_constants::pi
 253    // (rather than boost::math::constants::pi<float>() ).
 254
 255    float tolerance = boost::math::tools::epsilon<float>() * 2;
 256
 257    using namespace boost::math::float_constants;
 258    BOOST_MATH_STD_USING
 259
 260    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F), pi, tolerance);
 261    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(3.14159265358979323846264338327950288419716939937510F)), root_pi, tolerance);
 262    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(3.14159265358979323846264338327950288419716939937510F/2)), root_half_pi, tolerance);
 263    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(3.14159265358979323846264338327950288419716939937510F * 2)), root_two_pi, tolerance);
 264    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(log(4.0F))), root_ln_four, tolerance);
 265    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(2.71828182845904523536028747135266249775724709369995F), e, tolerance);
 266    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(0.5), half, tolerance);
 267    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(0.57721566490153286060651209008240243104259335F), euler, tolerance);
 268    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(2.0F)), root_two, tolerance);
 269    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(log(2.0F)), ln_two, tolerance);
 270    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(log(log(2.0F))), ln_ln_two, tolerance);
 271    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(1)/3, third, tolerance);
 272    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(2)/3, twothirds, tolerance);
 273    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(0.14159265358979323846264338327950288419716939937510F), pi_minus_three, tolerance);
 274    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(4.F - 3.14159265358979323846264338327950288419716939937510F), four_minus_pi, tolerance);
 275 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 276    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(pow((3.14159265358979323846264338327950288419716939937510F), 2.71828182845904523536028747135266249775724709369995F)), pi_pow_e, tolerance);
 277    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(pow((3.14159265358979323846264338327950288419716939937510F), 0.33333333333333333333333333333333333333333333333333F)), cbrt_pi, tolerance);
 278    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(exp(-0.5F)), exp_minus_half, tolerance);
 279    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(pow(2.71828182845904523536028747135266249775724709369995F, 3.14159265358979323846264338327950288419716939937510F)), e_pow_pi, tolerance);
 280
 281
 282 #else // Only double, so no suffix F.
 283    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995)), pi_pow_e, tolerance);
 284    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(pow((3.14159265358979323846264338327950288419716939937510), 0.33333333333333333333333333333333333333333333333333)), cbrt_pi, tolerance);
 285    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(exp(-0.5)), exp_minus_half, tolerance);
 286 #endif
 287    // Rational fractions.
 288    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(0.333333333333333333333333333333333333333F), third, tolerance);
 289    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(0.666666666666666666666666666666666666667F), two_thirds, tolerance);
 290    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(0.75F), three_quarters, tolerance);
 291    // Two and related.
 292    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(2.F)), root_two, tolerance);
 293    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(3.F)), root_three, tolerance);
 294    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(2.F)/2), half_root_two, tolerance);
 295    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(log(2.F)), ln_two, tolerance);
 296    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(log(log(2.0F))), ln_ln_two, tolerance);
 297    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(log(4.0F))), root_ln_four, tolerance);
 298    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(1/sqrt(2.0F)), one_div_root_two, tolerance);
 299
 300    // pi.
 301    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F), pi, tolerance);
 302    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F/2), half_pi, tolerance);
 303    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F/3), third_pi, tolerance);
 304    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F/6), sixth_pi, tolerance);
 305    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(2 * 3.14159265358979323846264338327950288419716939937510F), two_pi, tolerance);
 306    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3 * 3.14159265358979323846264338327950288419716939937510F / 4), three_quarters_pi, tolerance);
 307    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(4 * 3.14159265358979323846264338327950288419716939937510F / 3), four_thirds_pi, tolerance);
 308    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(1 / (2 * 3.14159265358979323846264338327950288419716939937510F)), one_div_two_pi, tolerance);
 309    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(3.14159265358979323846264338327950288419716939937510F)), root_pi, tolerance);
 310    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(3.14159265358979323846264338327950288419716939937510F / 2)), root_half_pi, tolerance);
 311    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(2 * 3.14159265358979323846264338327950288419716939937510F)), root_two_pi, tolerance);
 312    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(1 / sqrt(3.14159265358979323846264338327950288419716939937510F)), one_div_root_pi, tolerance);
 313    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(1 / sqrt(2 * 3.14159265358979323846264338327950288419716939937510F)), one_div_root_two_pi, tolerance);
 314    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(sqrt(1. / 3.14159265358979323846264338327950288419716939937510F)), root_one_div_pi, tolerance);
 315    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510L - 3.L), pi_minus_three, tolerance * 2 ); // tolerance * 2 because of cancellation loss.
 316    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(4.L - 3.14159265358979323846264338327950288419716939937510L), four_minus_pi, tolerance );
 317    //
 318    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(pow((3.14159265358979323846264338327950288419716939937510F), 2.71828182845904523536028747135266249775724709369995F)), pi_pow_e, tolerance);  // See above.
 319    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F * 3.14159265358979323846264338327950288419716939937510F), pi_sqr, tolerance);  // See above.
 320    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F * 3.14159265358979323846264338327950288419716939937510F/6), pi_sqr_div_six, tolerance);  // See above.
 321    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F * 3.14159265358979323846264338327950288419716939937510F * 3.14159265358979323846264338327950288419716939937510F), pi_cubed, tolerance);  // See above.
 322
 323    // BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(3.14159265358979323846264338327950288419716939937510F * 3.14159265358979323846264338327950288419716939937510F), cbrt_pi, tolerance);  // See above.
 324    BOOST_CHECK_CLOSE_FRACTION(cbrt_pi * cbrt_pi * cbrt_pi, pi, tolerance);
 325    BOOST_CHECK_CLOSE_FRACTION((static_cast<float>(1)/cbrt_pi), one_div_cbrt_pi, tolerance);
 326
 327    // Euler
 328    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(2.71828182845904523536028747135266249775724709369995F), e, tolerance);
 329
 330    //BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(exp(-0.5F)), exp_minus_half, tolerance);  // See above.
 331    BOOST_CHECK_CLOSE_FRACTION(pow(e, pi), e_pow_pi, tolerance); // See also above.
 332    BOOST_CHECK_CLOSE_FRACTION(sqrt(e), root_e, tolerance);
 333    BOOST_CHECK_CLOSE_FRACTION(log10(e), log10_e, tolerance);
 334    BOOST_CHECK_CLOSE_FRACTION(static_cast<float>(1)/log10(e), one_div_log10_e, tolerance);
 335
 336    // Trigonmetric
 337    BOOST_CHECK_CLOSE_FRACTION(pi/180, degree, tolerance);
 338    BOOST_CHECK_CLOSE_FRACTION(180 / pi, radian, tolerance);
 339    BOOST_CHECK_CLOSE_FRACTION(sin(1.F), sin_one, tolerance);
 340    BOOST_CHECK_CLOSE_FRACTION(cos(1.F), cos_one, tolerance);
 341    BOOST_CHECK_CLOSE_FRACTION(sinh(1.F), sinh_one, tolerance);
 342    BOOST_CHECK_CLOSE_FRACTION(cosh(1.F), cosh_one, tolerance);
 343
 344    // Phi
 345    BOOST_CHECK_CLOSE_FRACTION((1.F + sqrt(5.F)) /2, phi, tolerance);
 346    BOOST_CHECK_CLOSE_FRACTION(log((1.F + sqrt(5.F)) /2), ln_phi, tolerance);
 347    BOOST_CHECK_CLOSE_FRACTION(1.F / log((1.F + sqrt(5.F)) /2), one_div_ln_phi, tolerance);
 348
 349    // Euler's Gamma
 350    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992F, euler, tolerance); // (sequence A001620 in OEIS).
 351    BOOST_CHECK_CLOSE_FRACTION(1.F/ 0.57721566490153286060651209008240243104215933593992F, one_div_euler, tolerance); // (from sequence A001620 in OEIS).
 352    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992F * 0.57721566490153286060651209008240243104215933593992F, euler_sqr, tolerance); // (from sequence A001620 in OEIS).
 353
 354    // Misc
 355    BOOST_CHECK_CLOSE_FRACTION(1.644934066848226436472415166646025189218949901206F, zeta_two, tolerance); // A013661 as a constant (usually base 10) in OEIS.
 356    BOOST_CHECK_CLOSE_FRACTION(1.20205690315959428539973816151144999076498629234049888179227F, zeta_three, tolerance); // (sequence A002117 in OEIS)
 357    BOOST_CHECK_CLOSE_FRACTION(.91596559417721901505460351493238411077414937428167213F, catalan, tolerance); // A006752 as a constant in OEIS.
 358    BOOST_CHECK_CLOSE_FRACTION(1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150F, extreme_value_skewness, tolerance); //  Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
 359    BOOST_CHECK_CLOSE_FRACTION(0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067F, rayleigh_skewness, tolerance); // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
 360    BOOST_CHECK_CLOSE_FRACTION(2.450893006876380628486604106197544154e-01F, rayleigh_kurtosis_excess, tolerance);
 361    BOOST_CHECK_CLOSE_FRACTION(2.68545200106530644530971483548179569382038229399446295305115234555721885953715200280114117493184769799515F, khinchin, tolerance ); // A002210 as a constant https://oeis.org/A002210/constant
 362    BOOST_CHECK_CLOSE_FRACTION(1.2824271291006226368753425688697917277676889273250011F, glaisher, tolerance ); // https://oeis.org/A074962/constant
 363
 364 } // template <class RealType>void test_spots(RealType)
 365
 366 void test_double_spots()
 367 {
 368    // Basic sanity checks for constants in boost::math::double_constants::
 369    // for example: boost::math::double_constants::pi
 370    // (rather than boost::math::constants::pi<double>() ).
 371
 372    double tolerance = boost::math::tools::epsilon<double>() * 2;
 373
 374    using namespace boost::math::double_constants;
 375    BOOST_MATH_STD_USING
 376
 377    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510), pi, tolerance);
 378    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(3.14159265358979323846264338327950288419716939937510)), root_pi, tolerance);
 379    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(3.14159265358979323846264338327950288419716939937510/2)), root_half_pi, tolerance);
 380    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(3.14159265358979323846264338327950288419716939937510 * 2)), root_two_pi, tolerance);
 381    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(log(4.0))), root_ln_four, tolerance);
 382    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(2.71828182845904523536028747135266249775724709369995), e, tolerance);
 383    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(0.5), half, tolerance);
 384    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(0.57721566490153286060651209008240243104259335), euler, tolerance);
 385    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(2.0)), root_two, tolerance);
 386    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(log(2.0)), ln_two, tolerance);
 387    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(log(log(2.0))), ln_ln_two, tolerance);
 388    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(1)/3, third, tolerance);
 389    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(2)/3, twothirds, tolerance);
 390    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(0.14159265358979323846264338327950288419716939937510), pi_minus_three, tolerance);
 391    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(4. - 3.14159265358979323846264338327950288419716939937510), four_minus_pi, tolerance);
 392 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 393    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995)), pi_pow_e, tolerance);
 394    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(pow((3.14159265358979323846264338327950288419716939937510), 0.33333333333333333333333333333333333333333333333333)), cbrt_pi, tolerance);
 395    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(exp(-0.5)), exp_minus_half, tolerance);
 396    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(pow(2.71828182845904523536028747135266249775724709369995, 3.14159265358979323846264338327950288419716939937510)), e_pow_pi, tolerance);
 397
 398
 399 #else // Only double, so no suffix .
 400    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995)), pi_pow_e, tolerance);
 401    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(pow((3.14159265358979323846264338327950288419716939937510), 0.33333333333333333333333333333333333333333333333333)), cbrt_pi, tolerance);
 402    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(exp(-0.5)), exp_minus_half, tolerance);
 403 #endif
 404    // Rational fractions.
 405    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(0.333333333333333333333333333333333333333), third, tolerance);
 406    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(0.666666666666666666666666666666666666667), two_thirds, tolerance);
 407    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(0.75), three_quarters, tolerance);
 408    // Two and related.
 409    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(2.)), root_two, tolerance);
 410    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(3.)), root_three, tolerance);
 411    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(2.)/2), half_root_two, tolerance);
 412    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(log(2.)), ln_two, tolerance);
 413    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(log(log(2.0))), ln_ln_two, tolerance);
 414    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(log(4.0))), root_ln_four, tolerance);
 415    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(1/sqrt(2.0)), one_div_root_two, tolerance);
 416
 417    // pi.
 418    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510), pi, tolerance);
 419    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510/2), half_pi, tolerance);
 420    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510/3), third_pi, tolerance);
 421    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510/6), sixth_pi, tolerance);
 422    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(2 * 3.14159265358979323846264338327950288419716939937510), two_pi, tolerance);
 423    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3 * 3.14159265358979323846264338327950288419716939937510 / 4), three_quarters_pi, tolerance);
 424    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(4 * 3.14159265358979323846264338327950288419716939937510 / 3), four_thirds_pi, tolerance);
 425    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(1 / (2 * 3.14159265358979323846264338327950288419716939937510)), one_div_two_pi, tolerance);
 426    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(3.14159265358979323846264338327950288419716939937510)), root_pi, tolerance);
 427    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(3.14159265358979323846264338327950288419716939937510 / 2)), root_half_pi, tolerance);
 428    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(2 * 3.14159265358979323846264338327950288419716939937510)), root_two_pi, tolerance);
 429    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(1 / sqrt(3.14159265358979323846264338327950288419716939937510)), one_div_root_pi, tolerance);
 430    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(1 / sqrt(2 * 3.14159265358979323846264338327950288419716939937510)), one_div_root_two_pi, tolerance);
 431    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(sqrt(1. / 3.14159265358979323846264338327950288419716939937510)), root_one_div_pi, tolerance);
 432    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510 - 3.), pi_minus_three, tolerance * 2 ); // tolerance * 2 because of cancellation loss.
 433    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(4. - 3.14159265358979323846264338327950288419716939937510), four_minus_pi, tolerance );
 434    //
 435    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995)), pi_pow_e, tolerance);  // See above.
 436    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510 * 3.14159265358979323846264338327950288419716939937510), pi_sqr, tolerance);  // See above.
 437    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510 * 3.14159265358979323846264338327950288419716939937510/6), pi_sqr_div_six, tolerance);  // See above.
 438    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510 * 3.14159265358979323846264338327950288419716939937510 * 3.14159265358979323846264338327950288419716939937510), pi_cubed, tolerance);  // See above.
 439
 440    // BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(3.14159265358979323846264338327950288419716939937510 * 3.14159265358979323846264338327950288419716939937510), cbrt_pi, tolerance);  // See above.
 441    BOOST_CHECK_CLOSE_FRACTION(cbrt_pi * cbrt_pi * cbrt_pi, pi, tolerance);
 442    BOOST_CHECK_CLOSE_FRACTION((static_cast<double>(1)/cbrt_pi), one_div_cbrt_pi, tolerance);
 443
 444    // Euler
 445    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(2.71828182845904523536028747135266249775724709369995), e, tolerance);
 446
 447    //BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(exp(-0.5)), exp_minus_half, tolerance);  // See above.
 448    BOOST_CHECK_CLOSE_FRACTION(pow(e, pi), e_pow_pi, tolerance); // See also above.
 449    BOOST_CHECK_CLOSE_FRACTION(sqrt(e), root_e, tolerance);
 450    BOOST_CHECK_CLOSE_FRACTION(log10(e), log10_e, tolerance);
 451    BOOST_CHECK_CLOSE_FRACTION(static_cast<double>(1)/log10(e), one_div_log10_e, tolerance);
 452
 453    // Trigonmetric
 454    BOOST_CHECK_CLOSE_FRACTION(pi/180, degree, tolerance);
 455    BOOST_CHECK_CLOSE_FRACTION(180 / pi, radian, tolerance);
 456    BOOST_CHECK_CLOSE_FRACTION(sin(1.), sin_one, tolerance);
 457    BOOST_CHECK_CLOSE_FRACTION(cos(1.), cos_one, tolerance);
 458    BOOST_CHECK_CLOSE_FRACTION(sinh(1.), sinh_one, tolerance);
 459    BOOST_CHECK_CLOSE_FRACTION(cosh(1.), cosh_one, tolerance);
 460
 461    // Phi
 462    BOOST_CHECK_CLOSE_FRACTION((1. + sqrt(5.)) /2, phi, tolerance);
 463    BOOST_CHECK_CLOSE_FRACTION(log((1. + sqrt(5.)) /2), ln_phi, tolerance);
 464    BOOST_CHECK_CLOSE_FRACTION(1. / log((1. + sqrt(5.)) /2), one_div_ln_phi, tolerance);
 465
 466    //Euler's Gamma
 467    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992, euler, tolerance); // (sequence A001620 in OEIS).
 468    BOOST_CHECK_CLOSE_FRACTION(1./ 0.57721566490153286060651209008240243104215933593992, one_div_euler, tolerance); // (from sequence A001620 in OEIS).
 469    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992 * 0.57721566490153286060651209008240243104215933593992, euler_sqr, tolerance); // (from sequence A001620 in OEIS).
 470
 471    // Misc
 472    BOOST_CHECK_CLOSE_FRACTION(1.644934066848226436472415166646025189218949901206, zeta_two, tolerance); // A013661 as a constant (usually base 10) in OEIS.
 473    BOOST_CHECK_CLOSE_FRACTION(1.20205690315959428539973816151144999076498629234049888179227, zeta_three, tolerance); // (sequence A002117 in OEIS)
 474    BOOST_CHECK_CLOSE_FRACTION(.91596559417721901505460351493238411077414937428167213, catalan, tolerance); // A006752 as a constant in OEIS.
 475    BOOST_CHECK_CLOSE_FRACTION(1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150, extreme_value_skewness, tolerance); //  Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
 476    BOOST_CHECK_CLOSE_FRACTION(0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067, rayleigh_skewness, tolerance); // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
 477    BOOST_CHECK_CLOSE_FRACTION(2.450893006876380628486604106197544154e-01, rayleigh_kurtosis_excess, tolerance);
 478    BOOST_CHECK_CLOSE_FRACTION(2.68545200106530644530971483548179569382038229399446295305115234555721885953715200280114117493184769799515, khinchin, tolerance ); // A002210 as a constant https://oeis.org/A002210/constant
 479    BOOST_CHECK_CLOSE_FRACTION(1.2824271291006226368753425688697917277676889273250011, glaisher, tolerance ); // https://oeis.org/A074962/constant
 480
 481 } // template <class RealType>void test_spots(RealType)
 482
 483 void test_long_double_spots()
 484 {
 485    // Basic sanity checks for constants in boost::math::long double_constants::
 486    // for example: boost::math::long_double_constants::pi
 487    // (rather than boost::math::constants::pi<long double>() ).
 488
 489   // All constants are tested here using at least long double precision
 490   // with independent calculated or listed values,
 491   // or calculations using long double (sometime a little less accurate).
 492
 493    long double tolerance = boost::math::tools::epsilon<long double>() * 2;
 494
 495    using namespace boost::math::long_double_constants;
 496    BOOST_MATH_STD_USING
 497
 498    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L), pi, tolerance);
 499    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(3.14159265358979323846264338327950288419716939937510L)), root_pi, tolerance);
 500    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(3.14159265358979323846264338327950288419716939937510L/2)), root_half_pi, tolerance);
 501    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(3.14159265358979323846264338327950288419716939937510L * 2)), root_two_pi, tolerance);
 502    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(log(4.0L))), root_ln_four, tolerance);
 503    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(2.71828182845904523536028747135266249775724709369995L), e, tolerance);
 504    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(0.5), half, tolerance);
 505    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(0.57721566490153286060651209008240243104259335L), euler, tolerance);
 506    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(2.0L)), root_two, tolerance);
 507    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(log(2.0L)), ln_two, tolerance);
 508    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(log(log(2.0L))), ln_ln_two, tolerance);
 509    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1)/3, third, tolerance);
 510    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(2)/3, twothirds, tolerance);
 511    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(0.14159265358979323846264338327950288419716939937510L), pi_minus_three, tolerance);
 512    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(4.L - 3.14159265358979323846264338327950288419716939937510L), four_minus_pi, tolerance);
 513 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 514    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(pow((3.14159265358979323846264338327950288419716939937510L), 2.71828182845904523536028747135266249775724709369995L)), pi_pow_e, tolerance);
 515    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(pow((3.14159265358979323846264338327950288419716939937510L), 0.33333333333333333333333333333333333333333333333333L)), cbrt_pi, tolerance);
 516    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(exp(-0.5L)), exp_minus_half, tolerance);
 517    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(pow(2.71828182845904523536028747135266249775724709369995L, 3.14159265358979323846264338327950288419716939937510L)), e_pow_pi, tolerance);
 518
 519
 520 #else // Only double, so no suffix L.
 521    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995)), pi_pow_e, tolerance);
 522    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(pow((3.14159265358979323846264338327950288419716939937510), 0.33333333333333333333333333333333333333333333333333)), cbrt_pi, tolerance);
 523    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(exp(-0.5)), exp_minus_half, tolerance);
 524 #endif
 525    // Rational fractions.
 526    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(0.333333333333333333333333333333333333333L), third, tolerance);
 527    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(0.666666666666666666666666666666666666667L), two_thirds, tolerance);
 528    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(0.75L), three_quarters, tolerance);
 529    // Two and related.
 530    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(2.L)), root_two, tolerance);
 531    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(3.L)), root_three, tolerance);
 532    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(2.L)/2), half_root_two, tolerance);
 533    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(log(2.L)), ln_two, tolerance);
 534    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(log(log(2.0L))), ln_ln_two, tolerance);
 535    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(log(4.0L))), root_ln_four, tolerance);
 536    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1/sqrt(2.0L)), one_div_root_two, tolerance);
 537
 538    // pi.
 539    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L), pi, tolerance);
 540    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L/2), half_pi, tolerance);
 541    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L/3), third_pi, tolerance);
 542    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L/6), sixth_pi, tolerance);
 543    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(2 * 3.14159265358979323846264338327950288419716939937510L), two_pi, tolerance);
 544    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3 * 3.14159265358979323846264338327950288419716939937510L / 4), three_quarters_pi, tolerance);
 545    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(4 * 3.14159265358979323846264338327950288419716939937510L / 3), four_thirds_pi, tolerance);
 546    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1 / (2 * 3.14159265358979323846264338327950288419716939937510L)), one_div_two_pi, tolerance);
 547    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(3.14159265358979323846264338327950288419716939937510L)), root_pi, tolerance);
 548    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(3.14159265358979323846264338327950288419716939937510L / 2)), root_half_pi, tolerance);
 549    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(2 * 3.14159265358979323846264338327950288419716939937510L)), root_two_pi, tolerance);
 550    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1 / sqrt(3.14159265358979323846264338327950288419716939937510L)), one_div_root_pi, tolerance);
 551    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1 / sqrt(2 * 3.14159265358979323846264338327950288419716939937510L)), one_div_root_two_pi, tolerance);
 552    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(sqrt(1. / 3.14159265358979323846264338327950288419716939937510L)), root_one_div_pi, tolerance);
 553    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L - 3.L), pi_minus_three, tolerance * 4 ); // tolerance * 2 because of cancellation loss.
 554    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(4.L - 3.14159265358979323846264338327950288419716939937510L), four_minus_pi, tolerance );
 555    //
 556    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(pow((3.14159265358979323846264338327950288419716939937510L), 2.71828182845904523536028747135266249775724709369995L)), pi_pow_e, tolerance);  // See above.
 557    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L), pi_sqr, tolerance);  // See above.
 558    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L/6), pi_sqr_div_six, tolerance);  // See above.
 559    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L), pi_cubed, tolerance);  // See above.
 560
 561    // BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L), cbrt_pi, tolerance);  // See above.
 562    BOOST_CHECK_CLOSE_FRACTION(cbrt_pi * cbrt_pi * cbrt_pi, pi, tolerance);
 563    BOOST_CHECK_CLOSE_FRACTION((static_cast<long double>(1)/cbrt_pi), one_div_cbrt_pi, tolerance);
 564
 565    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(6.366197723675813430755350534900574481378385829618257E-1L), two_div_pi, tolerance * 3);  // 2/pi
 566    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(7.97884560802865355879892119868763736951717262329869E-1L), root_two_div_pi, tolerance * 3);  //  sqrt(2/pi)
 567
 568    // Euler
 569    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(2.71828182845904523536028747135266249775724709369995L), e, tolerance);
 570
 571    //BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(exp(-0.5L)), exp_minus_half, tolerance);  // See above.
 572    BOOST_CHECK_CLOSE_FRACTION(pow(e, pi), e_pow_pi, tolerance); // See also above.
 573    BOOST_CHECK_CLOSE_FRACTION(sqrt(e), root_e, tolerance);
 574    BOOST_CHECK_CLOSE_FRACTION(log10(e), log10_e, tolerance);
 575    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1)/log10(e), one_div_log10_e, tolerance);
 576
 577    // Trigonmetric
 578    BOOST_CHECK_CLOSE_FRACTION(pi/180, degree, tolerance);
 579    BOOST_CHECK_CLOSE_FRACTION(180 / pi, radian, tolerance);
 580    BOOST_CHECK_CLOSE_FRACTION(sin(1.L), sin_one, tolerance);
 581    BOOST_CHECK_CLOSE_FRACTION(cos(1.L), cos_one, tolerance);
 582    BOOST_CHECK_CLOSE_FRACTION(sinh(1.L), sinh_one, tolerance);
 583    BOOST_CHECK_CLOSE_FRACTION(cosh(1.L), cosh_one, tolerance);
 584
 585    // Phi
 586    BOOST_CHECK_CLOSE_FRACTION((1.L + sqrt(5.L)) /2, phi, tolerance);
 587    BOOST_CHECK_CLOSE_FRACTION(log((1.L + sqrt(5.L)) /2), ln_phi, tolerance);
 588    BOOST_CHECK_CLOSE_FRACTION(1.L / log((1.L + sqrt(5.L)) /2), one_div_ln_phi, tolerance);
 589
 590    //Euler's Gamma
 591    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992L, euler, tolerance); // (sequence A001620 in OEIS).
 592    BOOST_CHECK_CLOSE_FRACTION(1.L/ 0.57721566490153286060651209008240243104215933593992L, one_div_euler, tolerance); // (from sequence A001620 in OEIS).
 593    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992L * 0.57721566490153286060651209008240243104215933593992L, euler_sqr, tolerance); // (from sequence A001620 in OEIS).
 594
 595    // Misc
 596    BOOST_CHECK_CLOSE_FRACTION(1.644934066848226436472415166646025189218949901206L, zeta_two, tolerance); // A013661 as a constant (usually base 10) in OEIS.
 597    BOOST_CHECK_CLOSE_FRACTION(1.20205690315959428539973816151144999076498629234049888179227L, zeta_three, tolerance); // (sequence A002117 in OEIS)
 598    BOOST_CHECK_CLOSE_FRACTION(.91596559417721901505460351493238411077414937428167213L, catalan, tolerance); // A006752 as a constant in OEIS.
 599    BOOST_CHECK_CLOSE_FRACTION(1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150L, extreme_value_skewness, tolerance); //  Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
 600    BOOST_CHECK_CLOSE_FRACTION(0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067L, rayleigh_skewness, tolerance); // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
 601    BOOST_CHECK_CLOSE_FRACTION(2.450893006876380628486604106197544154e-01L, rayleigh_kurtosis_excess, tolerance);
 602    BOOST_CHECK_CLOSE_FRACTION(2.68545200106530644530971483548179569382038229399446295305115234555721885953715200280114117493184769799515L, khinchin, tolerance ); // A002210 as a constant https://oeis.org/A002210/constant
 603    BOOST_CHECK_CLOSE_FRACTION(1.2824271291006226368753425688697917277676889273250011L, glaisher, tolerance ); // https://oeis.org/A074962/constant
 604
 605 } // template <class RealType>void test_spots(RealType)
 606
 607 template <class Policy>
 608 void test_real_concept_policy(const Policy&)
 609 {
 610    // Basic sanity checks for constants using real_concept.
 611    // Parameter Policy is used to control precision.
 612
 613    using boost::math::concepts::real_concept;
 614
 615    boost::math::concepts::real_concept tolerance = boost::math::tools::epsilon<real_concept>() * 2;  // double
 616    if(Policy::precision_type::value > 200)
 617       tolerance *= 50;
 618    std::cout << "Tolerance for type " << typeid(real_concept).name()  << " is " << tolerance << "." << std::endl;
 619
 620    //typedef typename boost::math::policies::precision<boost::math::concepts::real_concept, boost::math::policies::policy<> >::type t1;
 621    // A precision of zero means we don't know what the precision of this type is until runtime.
 622    //std::cout << "Precision for type " << typeid(boost::math::concepts::real_concept).name()  << " is " << t1::value << "." << std::endl;
 623
 624    using namespace boost::math::constants;
 625    BOOST_MATH_STD_USING
 626
 627    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L, (pi<real_concept, Policy>)(), tolerance);
 628    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L), (root_pi<real_concept, Policy>)(), tolerance);
 629    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L/2), (root_half_pi<real_concept, Policy>)(), tolerance);
 630    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L * 2), (root_two_pi<real_concept, Policy>)(), tolerance);
 631    BOOST_CHECK_CLOSE_FRACTION(sqrt(log(4.0L)), (root_ln_four<real_concept, Policy>)(), tolerance);
 632    BOOST_CHECK_CLOSE_FRACTION(2.71828182845904523536028747135266249775724709369995L, (e<real_concept, Policy>)(), tolerance);
 633    BOOST_CHECK_CLOSE_FRACTION(0.5, (half<real_concept, Policy>)(), tolerance);
 634    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104259335L, (euler<real_concept, Policy>)(), tolerance);
 635    BOOST_CHECK_CLOSE_FRACTION(sqrt(2.0L), (root_two<real_concept, Policy>)(), tolerance);
 636    BOOST_CHECK_CLOSE_FRACTION(log(2.0L), (ln_two<real_concept, Policy>)(), tolerance);
 637    BOOST_CHECK_CLOSE_FRACTION(log(log(2.0L)), (ln_ln_two<real_concept, Policy>)(), tolerance);
 638    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(1)/3, (third<real_concept, Policy>)(), tolerance);
 639    BOOST_CHECK_CLOSE_FRACTION(static_cast<long double>(2)/3, (twothirds<real_concept, Policy>)(), tolerance);
 640    BOOST_CHECK_CLOSE_FRACTION(0.14159265358979323846264338327950288419716939937510L, (pi_minus_three<real_concept, Policy>)(), tolerance);
 641    BOOST_CHECK_CLOSE_FRACTION(4.L - 3.14159265358979323846264338327950288419716939937510L, (four_minus_pi<real_concept, Policy>)(), tolerance);
 642 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 643    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510L), 2.71828182845904523536028747135266249775724709369995L), (pi_pow_e<real_concept, Policy>)(), tolerance);
 644    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510L), 0.33333333333333333333333333333333333333333333333333L), (cbrt_pi<real_concept, Policy>)(), tolerance);
 645    BOOST_CHECK_CLOSE_FRACTION(exp(-0.5L), (exp_minus_half<real_concept, Policy>)(), tolerance);
 646    BOOST_CHECK_CLOSE_FRACTION(pow(2.71828182845904523536028747135266249775724709369995L, 3.14159265358979323846264338327950288419716939937510L), (e_pow_pi<real_concept, Policy>)(), tolerance);
 647
 648
 649 #else // Only double, so no suffix L.
 650    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510), 2.71828182845904523536028747135266249775724709369995), (pi_pow_e<real_concept, Policy>)(), tolerance);
 651    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510), 0.33333333333333333333333333333333333333333333333333), (cbrt_pi<real_concept, Policy>)(), tolerance);
 652    BOOST_CHECK_CLOSE_FRACTION(exp(-0.5), (exp_minus_half<real_concept, Policy>)(), tolerance);
 653 #endif
 654    // Rational fractions.
 655    BOOST_CHECK_CLOSE_FRACTION(0.333333333333333333333333333333333333333L, (third<real_concept, Policy>)(), tolerance);
 656    BOOST_CHECK_CLOSE_FRACTION(0.666666666666666666666666666666666666667L, (two_thirds<real_concept, Policy>)(), tolerance);
 657    BOOST_CHECK_CLOSE_FRACTION(0.75L, (three_quarters<real_concept, Policy>)(), tolerance);
 658    // Two and related.
 659    BOOST_CHECK_CLOSE_FRACTION(sqrt(2.L), (root_two<real_concept, Policy>)(), tolerance);
 660    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.L), (root_three<real_concept, Policy>)(), tolerance);
 661    BOOST_CHECK_CLOSE_FRACTION(sqrt(2.L)/2, (half_root_two<real_concept, Policy>)(), tolerance);
 662    BOOST_CHECK_CLOSE_FRACTION(log(2.L), (ln_two<real_concept, Policy>)(), tolerance);
 663    BOOST_CHECK_CLOSE_FRACTION(log(log(2.0L)), (ln_ln_two<real_concept, Policy>)(), tolerance);
 664    BOOST_CHECK_CLOSE_FRACTION(sqrt(log(4.0L)), (root_ln_four<real_concept, Policy>)(), tolerance);
 665    BOOST_CHECK_CLOSE_FRACTION(1/sqrt(2.0L), (one_div_root_two<real_concept, Policy>)(), tolerance);
 666
 667    // pi.
 668    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L, (pi<real_concept, Policy>)(), tolerance);
 669    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L/2, (half_pi<real_concept, Policy>)(), tolerance);
 670    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L/3, (third_pi<real_concept, Policy>)(), tolerance);
 671    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L/6, (sixth_pi<real_concept, Policy>)(), tolerance);
 672    BOOST_CHECK_CLOSE_FRACTION(2 * 3.14159265358979323846264338327950288419716939937510L, (two_pi<real_concept, Policy>)(), tolerance);
 673    BOOST_CHECK_CLOSE_FRACTION(3 * 3.14159265358979323846264338327950288419716939937510L / 4, (three_quarters_pi<real_concept, Policy>)(), tolerance);
 674    BOOST_CHECK_CLOSE_FRACTION(4 * 3.14159265358979323846264338327950288419716939937510L / 3, (four_thirds_pi<real_concept, Policy>)(), tolerance);
 675    BOOST_CHECK_CLOSE_FRACTION(1 / (2 * 3.14159265358979323846264338327950288419716939937510L), (one_div_two_pi<real_concept, Policy>)(), tolerance);
 676    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L), (root_pi<real_concept, Policy>)(), tolerance);
 677    BOOST_CHECK_CLOSE_FRACTION(sqrt(3.14159265358979323846264338327950288419716939937510L / 2), (root_half_pi<real_concept, Policy>)(), tolerance);
 678    BOOST_CHECK_CLOSE_FRACTION(sqrt(2 * 3.14159265358979323846264338327950288419716939937510L), (root_two_pi<real_concept, Policy>)(), tolerance);
 679    BOOST_CHECK_CLOSE_FRACTION(1 / sqrt(3.14159265358979323846264338327950288419716939937510L), (one_div_root_pi<real_concept, Policy>)(), tolerance);
 680    BOOST_CHECK_CLOSE_FRACTION(1 / sqrt(2 * 3.14159265358979323846264338327950288419716939937510L), (one_div_root_two_pi<real_concept, Policy>)(), tolerance);
 681    BOOST_CHECK_CLOSE_FRACTION(sqrt(1. / 3.14159265358979323846264338327950288419716939937510L), (root_one_div_pi<real_concept, Policy>)(), tolerance);
 682    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L - 3.L, (pi_minus_three<real_concept, Policy>)(), tolerance * 4 ); // tolerance * 2 because of cancellation loss.
 683    BOOST_CHECK_CLOSE_FRACTION(4.L - 3.14159265358979323846264338327950288419716939937510L, (four_minus_pi<real_concept, Policy>)(), tolerance );
 684    //
 685    BOOST_CHECK_CLOSE_FRACTION(pow((3.14159265358979323846264338327950288419716939937510L), 2.71828182845904523536028747135266249775724709369995L), (pi_pow_e<real_concept, Policy>)(), tolerance);  // See above.
 686    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L, (pi_sqr<real_concept, Policy>)(), tolerance);  // See above.
 687    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L/6, (pi_sqr_div_six<real_concept, Policy>)(), tolerance);  // See above.
 688    BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L, (pi_cubed<real_concept, Policy>)(), tolerance);  // See above.
 689
 690    // BOOST_CHECK_CLOSE_FRACTION(3.14159265358979323846264338327950288419716939937510L * 3.14159265358979323846264338327950288419716939937510L, (cbrt_pi<real_concept, Policy>)(), tolerance);  // See above.
 691    BOOST_CHECK_CLOSE_FRACTION((cbrt_pi<real_concept, Policy>)() * (cbrt_pi<real_concept, Policy>)() * (cbrt_pi<real_concept, Policy>)(), (pi<real_concept, Policy>)(), tolerance);
 692    BOOST_CHECK_CLOSE_FRACTION((1)/(cbrt_pi<real_concept, Policy>)(), (one_div_cbrt_pi<real_concept, Policy>)(), tolerance);
 693
 694    // Euler
 695    BOOST_CHECK_CLOSE_FRACTION(2.71828182845904523536028747135266249775724709369995L, (e<real_concept, Policy>)(), tolerance);
 696
 697    //BOOST_CHECK_CLOSE_FRACTION(exp(-0.5L), (exp_minus_half<real_concept, Policy>)(), tolerance);  // See above.
 698    BOOST_CHECK_CLOSE_FRACTION(pow(e<real_concept, Policy>(), (pi<real_concept, Policy>)()), (e_pow_pi<real_concept, Policy>)(), tolerance); // See also above.
 699    BOOST_CHECK_CLOSE_FRACTION(sqrt(e<real_concept, Policy>()), (root_e<real_concept, Policy>)(), tolerance);
 700    BOOST_CHECK_CLOSE_FRACTION(log10(e<real_concept, Policy>()), (log10_e<real_concept, Policy>)(), tolerance);
 701    BOOST_CHECK_CLOSE_FRACTION(1/log10(e<real_concept, Policy>()), (one_div_log10_e<real_concept, Policy>)(), tolerance);
 702
 703    // Trigonmetric
 704    BOOST_CHECK_CLOSE_FRACTION((pi<real_concept, Policy>)()/180, (degree<real_concept, Policy>)(), tolerance);
 705    BOOST_CHECK_CLOSE_FRACTION(180 / (pi<real_concept, Policy>)(), (radian<real_concept, Policy>)(), tolerance);
 706    BOOST_CHECK_CLOSE_FRACTION(sin(1.L), (sin_one<real_concept, Policy>)(), tolerance);
 707    BOOST_CHECK_CLOSE_FRACTION(cos(1.L), (cos_one<real_concept, Policy>)(), tolerance);
 708    BOOST_CHECK_CLOSE_FRACTION(sinh(1.L), (sinh_one<real_concept, Policy>)(), tolerance);
 709    BOOST_CHECK_CLOSE_FRACTION(cosh(1.L), (cosh_one<real_concept, Policy>)(), tolerance);
 710
 711    // Phi
 712    BOOST_CHECK_CLOSE_FRACTION((1.L + sqrt(5.L)) /2, (phi<real_concept, Policy>)(), tolerance);
 713    BOOST_CHECK_CLOSE_FRACTION(log((1.L + sqrt(5.L)) /2), (ln_phi<real_concept, Policy>)(), tolerance);
 714    BOOST_CHECK_CLOSE_FRACTION(1.L / log((1.L + sqrt(5.L)) /2), (one_div_ln_phi<real_concept, Policy>)(), tolerance);
 715
 716    //Euler's Gamma
 717    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992L, (euler<real_concept, Policy>)(), tolerance); // (sequence A001620 in OEIS).
 718    BOOST_CHECK_CLOSE_FRACTION(1.L/ 0.57721566490153286060651209008240243104215933593992L, (one_div_euler<real_concept, Policy>)(), tolerance); // (from sequence A001620 in OEIS).
 719    BOOST_CHECK_CLOSE_FRACTION(0.57721566490153286060651209008240243104215933593992L * 0.57721566490153286060651209008240243104215933593992L, (euler_sqr<real_concept, Policy>)(), tolerance); // (from sequence A001620 in OEIS).
 720
 721    // Misc
 722    BOOST_CHECK_CLOSE_FRACTION(1.644934066848226436472415166646025189218949901206L, (zeta_two<real_concept, Policy>)(), tolerance); // A013661 as a constant (usually base 10) in OEIS.
 723    BOOST_CHECK_CLOSE_FRACTION(1.20205690315959428539973816151144999076498629234049888179227L, (zeta_three<real_concept, Policy>)(), tolerance); // (sequence A002117 in OEIS)
 724    BOOST_CHECK_CLOSE_FRACTION(.91596559417721901505460351493238411077414937428167213L, (catalan<real_concept, Policy>)(), tolerance); // A006752 as a constant in OEIS.
 725    BOOST_CHECK_CLOSE_FRACTION(1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150L, (extreme_value_skewness<real_concept, Policy>)(), tolerance); //  Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
 726    BOOST_CHECK_CLOSE_FRACTION(0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067L, (rayleigh_skewness<real_concept, Policy>)(), tolerance); // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
 727    BOOST_CHECK_CLOSE_FRACTION(2.450893006876380628486604106197544154e-01L, (rayleigh_kurtosis_excess<real_concept, Policy>)(), tolerance);
 728    BOOST_CHECK_CLOSE_FRACTION(2.68545200106530644530971483548179569382038229399446295305115234555721885953715200280114117493184769799515L, (khinchin<real_concept, Policy>)(), tolerance ); // A002210 as a constant https://oeis.org/A002210/constant
 729    BOOST_CHECK_CLOSE_FRACTION(1.2824271291006226368753425688697917277676889273250011L, (glaisher<real_concept, Policy>)(), tolerance ); // https://oeis.org/A074962/constant
 730
 731    //
 732    // Last of all come the test cases that behave differently if we're calculating the constants on the fly:
 733    //
 734    if(boost::math::tools::digits<real_concept>() > boost::math::constants::max_string_digits)
 735    {
 736       // This suffers from cancellation error, so increased tolerance:
 737       BOOST_CHECK_CLOSE_FRACTION((static_cast<real_concept>(4. - 3.14159265358979323846264338327950288419716939937510L)), (four_minus_pi<real_concept, Policy>)(), tolerance * 3);
 738       BOOST_CHECK_CLOSE_FRACTION((static_cast<real_concept>(0.14159265358979323846264338327950288419716939937510L)), (pi_minus_three<real_concept, Policy>)(), tolerance * 3);
 739    }
 740    else
 741    {
 742       BOOST_CHECK_CLOSE_FRACTION((static_cast<real_concept>(4. - 3.14159265358979323846264338327950288419716939937510L)), (four_minus_pi<real_concept, Policy>)(), tolerance);
 743       BOOST_CHECK_CLOSE_FRACTION((static_cast<real_concept>(0.14159265358979323846264338327950288419716939937510L)), (pi_minus_three<real_concept, Policy>)(), tolerance);
 744    }
 745
 746 } // template <class boost::math::concepts::real_concept>void test_spots(boost::math::concepts::real_concept)
 747
 748 #ifdef BOOST_MATH_USE_FLOAT128
 749 void test_float128()
 750 {
 751    static const __float128 eps = 1.92592994438723585305597794258492732e-34Q;
 752
 753    __float128 p = boost::math::constants::pi<__float128>();
 754    __float128 r = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651Q;
 755    __float128 err = (p - r) / r;
 756    if(err < 0)
 757       err = -err;
 758    BOOST_CHECK(err < 2 * eps);
 759 }
 760 #endif
 761
 762 void test_constexpr()
 763 {
 764 #ifndef BOOST_NO_CXX11_CONSTEXPR
 765    constexpr float f1 = boost::math::constants::pi<float>();
 766    constexpr double f2 = boost::math::constants::pi<double>();
 767    constexpr long double f3 = boost::math::constants::pi<long double>();
 768    (void)f1;
 769    (void)f2;
 770    (void)f3;
 771 #ifdef BOOST_MATH_USE_FLOAT128
 772    constexpr __float128 f4 = boost::math::constants::pi<__float128>();
 773    (void)f4;
 774 #endif
 775 #endif
 776 }
 777
 778 BOOST_AUTO_TEST_CASE( test_main )
 779 {
 780    // Basic sanity-check spot values.
 781
 782    test_float_spots(); // Test float_constants, like boost::math::float_constants::pi;
 783    test_double_spots(); // Test double_constants.
 784 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 785    test_long_double_spots(); // Test long_double_constants.
 786 #ifdef BOOST_MATH_USE_FLOAT128
 787    test_float128();
 788 #endif
 789    test_constexpr();
 790
 791    test_real_concept_policy(real_concept_policy_1());
 792    test_real_concept_policy(real_concept_policy_2()); // Increased precision forcing construction from string.
 793    test_real_concept_policy(real_concept_policy_3()); // Increased precision forcing caching of computed values.
 794    test_real_concept_policy(boost::math::policies::policy<>()); // Default.
 795 #endif
 796    // (Parameter value, arbitrarily zero, only communicates the floating-point type).
 797    test_spots(0.0F); // Test float.
 798    test_spots(0.0); // Test double.
 799 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
 800    test_spots(0.0L); // Test long double.
 801 #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582))
 802    test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
 803    test_spots(boost::math::concepts::big_real_concept(0.)); // Test real concept.
 804 #endif
 805 #else
 806   std::cout << "<note>The long double tests have been disabled on this platform "
 807     "either because the long double overloads of the usual math functions are "
 808     "not available at all, or because they are too inaccurate for these tests "
 809     "to pass.</note>" << std::endl;
 810 #endif
 811
 812 } // BOOST_AUTO_TEST_CASE( test_main )
 813
 814 /*
 815
 816 Output:
 817
 818   1 Feb 2012
 819
 820 test_constants.cpp
 821   test_constants.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Debug\test_constants.exe
 822   Running 1 test case...
 823   Tolerance for type class boost::math::concepts::real_concept is 4.44089e-016.
 824   Tolerance for type class boost::math::concepts::real_concept is 4.44089e-016.
 825   Tolerance for type class boost::math::concepts::real_concept is 4.44089e-016.
 826   Tolerance for type float is 2.38419e-007.
 827   Tolerance for type double is 4.44089e-016.
 828   Tolerance for type long double is 4.44089e-016.
 829   Tolerance for type class boost::math::concepts::real_concept is 4.44089e-016.
 830   Tolerance for type class boost::math::concepts::big_real_concept is 1.33227e-014.
 831
 832   *** No errors detected
 833
 834 */
 835