| // Copyright Paul Bristow 2007. |
| // Copyright John Maddock 2006. |
| |
| // Use, modification and distribution are subject to the |
| // Boost Software License, Version 1.0. |
| // (See accompanying file LICENSE_1_0.txt |
| // or copy at http://www.boost.org/LICENSE_1_0.txt) |
| |
| // test_uniform.cpp |
| |
| #include <pch.hpp> |
| |
| #ifdef _MSC_VER |
| # pragma warning(disable: 4127) // conditional expression is constant. |
| # pragma warning(disable: 4100) // unreferenced formal parameter. |
| #endif |
| |
| #include <boost/math/concepts/real_concept.hpp> // for real_concept |
| #include <boost/test/test_exec_monitor.hpp> // Boost.Test |
| #include <boost/test/floating_point_comparison.hpp> |
| |
| #include <boost/math/distributions/uniform.hpp> |
| using boost::math::uniform_distribution; |
| #include <boost/math/tools/test.hpp> |
| |
| #include <iostream> |
| using std::cout; |
| using std::endl; |
| using std::setprecision; |
| #include <limits> |
| using std::numeric_limits; |
| |
| template <class RealType> |
| void check_uniform(RealType lower, RealType upper, RealType x, RealType p, RealType q, RealType tol) |
| { |
| BOOST_CHECK_CLOSE_FRACTION( |
| ::boost::math::cdf( |
| uniform_distribution<RealType>(lower, upper), // distribution. |
| x), // random variable. |
| p, // probability. |
| tol); // tolerance. |
| BOOST_CHECK_CLOSE_FRACTION( |
| ::boost::math::cdf( |
| complement( |
| uniform_distribution<RealType>(lower, upper), // distribution. |
| x)), // random variable. |
| q, // probability complement. |
| tol); // tolerance. |
| BOOST_CHECK_CLOSE_FRACTION( |
| ::boost::math::quantile( |
| uniform_distribution<RealType>(lower, upper), // distribution. |
| p), // probability. |
| x, // random variable. |
| tol); // tolerance. |
| BOOST_CHECK_CLOSE_FRACTION( |
| ::boost::math::quantile( |
| complement( |
| uniform_distribution<RealType>(lower, upper), // distribution. |
| q)), // probability complement. |
| x, // random variable. |
| tol); // tolerance. |
| } // void check_uniform |
| |
| template <class RealType> |
| void test_spots(RealType) |
| { |
| // Basic sanity checks |
| // |
| // These test values were generated for the normal distribution |
| // using the online calculator at |
| // http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm |
| // |
| // Tolerance is just over 5 decimal digits expressed as a fraction: |
| // that's the limit of the test data. |
| RealType tolerance = 2e-5f; |
| cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << "." << endl; |
| |
| using std::exp; |
| |
| // Tests for PDF |
| // |
| BOOST_CHECK_CLOSE_FRACTION( // x == upper |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0)), |
| static_cast<RealType>(1), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( // x == lower |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(1)), |
| static_cast<RealType>(1), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( // x > upper |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(-1)), |
| static_cast<RealType>(0), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( // x < lower |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(2)), |
| static_cast<RealType>(0), |
| tolerance); |
| |
| if(std::numeric_limits<RealType>::has_infinity) |
| { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() |
| // Note that infinity is not implemented for real_concept, so these tests |
| // are only done for types, like built-in float, double.. that have infinity. |
| // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. |
| // #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here. |
| // #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path |
| // of error handling is tested below with BOOST_CHECK_THROW tests. |
| |
| BOOST_CHECK_THROW( // x == infinity should NOT be OK. |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(std::numeric_limits<RealType>::infinity())), |
| std::domain_error); |
| |
| BOOST_CHECK_THROW( // x == minus infinity should be OK too. |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(-std::numeric_limits<RealType>::infinity())), |
| std::domain_error); |
| } |
| if(std::numeric_limits<RealType>::has_quiet_NaN) |
| { // BOOST_CHECK tests for NaN using std::numeric_limits<>::has_quiet_NaN() - should throw. |
| BOOST_CHECK_THROW( |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(std::numeric_limits<RealType>::quiet_NaN())), |
| std::domain_error); |
| BOOST_CHECK_THROW( |
| pdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(-std::numeric_limits<RealType>::quiet_NaN())), |
| std::domain_error); |
| } // test for x = NaN using std::numeric_limits<>::quiet_NaN() |
| |
| // cdf |
| BOOST_CHECK_EQUAL( // x < lower |
| cdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(-1)), |
| static_cast<RealType>(0) ); |
| BOOST_CHECK_CLOSE_FRACTION( |
| cdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0)), |
| static_cast<RealType>(0), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| cdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.5)), |
| static_cast<RealType>(0.5), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| cdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.1)), |
| static_cast<RealType>(0.1), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| cdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.9)), |
| static_cast<RealType>(0.9), |
| tolerance); |
| BOOST_CHECK_EQUAL( // x > upper |
| cdf(uniform_distribution<RealType>(0, 1), static_cast<RealType>(2)), |
| static_cast<RealType>(1)); |
| |
| // cdf complement |
| BOOST_CHECK_EQUAL( // x < lower |
| cdf(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0))), |
| static_cast<RealType>(1)); |
| BOOST_CHECK_EQUAL( // x == 0 |
| cdf(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0))), |
| static_cast<RealType>(1)); |
| BOOST_CHECK_CLOSE_FRACTION( // x = 0.1 |
| cdf(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.1))), |
| static_cast<RealType>(0.9), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( // x = 0.5 |
| cdf(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.5))), |
| static_cast<RealType>(0.5), |
| tolerance); |
| BOOST_CHECK_EQUAL( // x == 1 |
| cdf(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(1))), |
| static_cast<RealType>(0)); |
| BOOST_CHECK_EQUAL( // x > upper |
| cdf(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(2))), |
| static_cast<RealType>(1)); |
| |
| // quantile |
| |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.9)), |
| static_cast<RealType>(0.9), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.1)), |
| static_cast<RealType>(0.1), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.5)), |
| static_cast<RealType>(0.5), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0)), |
| static_cast<RealType>(0), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(uniform_distribution<RealType>(0, 1), static_cast<RealType>(1)), |
| static_cast<RealType>(1), |
| tolerance); |
| |
| // quantile complement |
| |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.1))), |
| static_cast<RealType>(0.9), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.9))), |
| static_cast<RealType>(0.1), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0.5))), |
| static_cast<RealType>(0.5), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(0))), |
| static_cast<RealType>(1), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| quantile(complement(uniform_distribution<RealType>(0, 1), static_cast<RealType>(1))), |
| static_cast<RealType>(1), |
| tolerance); |
| |
| // Some tests using a different location & scale, neight zero or unity. |
| BOOST_CHECK_CLOSE_FRACTION( // x == mid |
| pdf(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(1)), |
| static_cast<RealType>(0.3333333333333333333333333333333333333333333333333333), |
| tolerance); |
| |
| BOOST_CHECK_CLOSE_FRACTION( // x == upper |
| pdf(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(+2)), |
| static_cast<RealType>(0.3333333333333333333333333333333333333333333333333333), // 1 / (2 - -1) = 1/3 |
| tolerance); |
| |
| BOOST_CHECK_CLOSE_FRACTION( // x == lower |
| cdf(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(-1)), |
| static_cast<RealType>(0), |
| tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( // x == upper |
| cdf(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(0)), |
| static_cast<RealType>(0.3333333333333333333333333333333333333333333333333333), |
| tolerance); |
| |
| BOOST_CHECK_CLOSE_FRACTION( // x == upper |
| cdf(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(1)), |
| static_cast<RealType>(0.6666666666666666666666666666666666666666666666666667), |
| tolerance); |
| |
| BOOST_CHECK_CLOSE_FRACTION( // x == lower |
| cdf(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(2)), |
| static_cast<RealType>(1), |
| tolerance); |
| |
| BOOST_CHECK_CLOSE_FRACTION( // x == upper |
| quantile(uniform_distribution<RealType>(-1, 2), static_cast<RealType>(0.6666666666666666666666666666666666666666666666666667)), |
| static_cast<RealType>(1), |
| tolerance); |
| |
| check_uniform( |
| static_cast<RealType>(0), // lower |
| static_cast<RealType>(1), // upper |
| static_cast<RealType>(0.5), // x |
| static_cast<RealType>(0.5), // p |
| static_cast<RealType>(1 - 0.5), // q |
| tolerance); |
| |
| // Some Not-standard uniform tests. |
| check_uniform( |
| static_cast<RealType>(-1), // lower |
| static_cast<RealType>(1), // upper |
| static_cast<RealType>(0), // x |
| static_cast<RealType>(0.5), // p |
| static_cast<RealType>(1 - 0.5), // q = 1 - p |
| tolerance); |
| |
| check_uniform( |
| static_cast<RealType>(1), // lower |
| static_cast<RealType>(3), // upper |
| static_cast<RealType>(2), // x |
| static_cast<RealType>(0.5), // p |
| static_cast<RealType>(1 - 0.5), // q = 1 - p |
| tolerance); |
| |
| check_uniform( |
| static_cast<RealType>(-1), // lower |
| static_cast<RealType>(2), // upper |
| static_cast<RealType>(1), // x |
| static_cast<RealType>(0.66666666666666666666666666666666666666666667), // p |
| static_cast<RealType>(0.33333333333333333333333333333333333333333333), // q = 1 - p |
| tolerance); |
| tolerance = (std::max)( |
| boost::math::tools::epsilon<RealType>(), |
| static_cast<RealType>(boost::math::tools::epsilon<double>())) * 5; // 5 eps as a fraction. |
| cout << "Tolerance (as fraction) for type " << typeid(RealType).name() << " is " << tolerance << "." << endl; |
| uniform_distribution<RealType> distu01(0, 1); |
| RealType x = static_cast<RealType>(0.5); |
| using namespace std; // ADL of std names. |
| // mean: |
| BOOST_CHECK_CLOSE_FRACTION( |
| mean(distu01), static_cast<RealType>(0.5), tolerance); |
| // variance: |
| BOOST_CHECK_CLOSE_FRACTION( |
| variance(distu01), static_cast<RealType>(0.0833333333333333333333333333333333333333333), tolerance); |
| // std deviation: |
| BOOST_CHECK_CLOSE_FRACTION( |
| standard_deviation(distu01), sqrt(variance(distu01)), tolerance); |
| // hazard: |
| BOOST_CHECK_CLOSE_FRACTION( |
| hazard(distu01, x), pdf(distu01, x) / cdf(complement(distu01, x)), tolerance); |
| // cumulative hazard: |
| BOOST_CHECK_CLOSE_FRACTION( |
| chf(distu01, x), -log(cdf(complement(distu01, x))), tolerance); |
| // coefficient_of_variation: |
| BOOST_CHECK_CLOSE_FRACTION( |
| coefficient_of_variation(distu01), standard_deviation(distu01) / mean(distu01), tolerance); |
| // mode: |
| BOOST_CHECK_CLOSE_FRACTION( |
| mode(distu01), static_cast<RealType>(0), tolerance); |
| BOOST_CHECK_CLOSE_FRACTION( |
| median(distu01), static_cast<RealType>(0.5), tolerance); |
| // skewness: |
| BOOST_CHECK_EQUAL( |
| skewness(distu01), static_cast<RealType>(0)); |
| // kertosis: |
| BOOST_CHECK_CLOSE_FRACTION( |
| kurtosis(distu01), kurtosis_excess(distu01) + static_cast<RealType>(3), tolerance); |
| // kertosis excess: |
| BOOST_CHECK_CLOSE_FRACTION( |
| kurtosis_excess(distu01), static_cast<RealType>(-1.2), tolerance); |
| |
| if(std::numeric_limits<RealType>::has_infinity) |
| { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() |
| // Note that infinity is not implemented for real_concept, so these tests |
| // are only done for types, like built-in float, double, long double, that have infinity. |
| // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. |
| // #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here. |
| // #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path |
| // of error handling is tested below with BOOST_CHECK_THROW tests. |
| |
| BOOST_CHECK_THROW(pdf(distu01, std::numeric_limits<RealType>::infinity()), std::domain_error); |
| BOOST_CHECK_THROW(pdf(distu01, -std::numeric_limits<RealType>::infinity()), std::domain_error); |
| } // test for infinity using std::numeric_limits<>::infinity() |
| else |
| { // real_concept case, does has_infinfity == false, so can't check it throws. |
| // cout << std::numeric_limits<RealType>::infinity() << ' ' |
| // << boost::math::fpclassify(std::numeric_limits<RealType>::infinity()) << endl; |
| // value of std::numeric_limits<RealType>::infinity() is zero, so FPclassify is zero, |
| // so (boost::math::isfinite)(std::numeric_limits<RealType>::infinity()) does not detect infinity. |
| // so these tests would never throw. |
| //BOOST_CHECK_THROW(pdf(distu01, std::numeric_limits<RealType>::infinity()), std::domain_error); |
| //BOOST_CHECK_THROW(pdf(distu01, std::numeric_limits<RealType>::quiet_NaN()), std::domain_error); |
| // BOOST_CHECK_THROW(pdf(distu01, boost::math::tools::max_value<RealType>() * 2), std::domain_error); // Doesn't throw. |
| BOOST_CHECK_EQUAL(pdf(distu01, boost::math::tools::max_value<RealType>()), 0); |
| } |
| // Special cases: |
| BOOST_CHECK(pdf(distu01, 0) == 1); |
| BOOST_CHECK(cdf(distu01, 0) == 0); |
| BOOST_CHECK(pdf(distu01, 1) == 1); |
| BOOST_CHECK(cdf(distu01, 1) == 1); |
| BOOST_CHECK(cdf(complement(distu01, 0)) == 1); |
| BOOST_CHECK(cdf(complement(distu01, 1)) == 0); |
| BOOST_CHECK(quantile(distu01, 0) == 0); |
| BOOST_CHECK(quantile(complement(distu01, 0)) == 1); |
| BOOST_CHECK(quantile(distu01, 1) == 1); |
| BOOST_CHECK(quantile(complement(distu01, 1)) == 1); |
| |
| // Error checks: |
| if(std::numeric_limits<RealType>::has_quiet_NaN) |
| { // BOOST_CHECK tests for constructing with quiet_NaN (not for real_concept, for example - see notes above). |
| BOOST_CHECK_THROW(uniform_distribution<RealType>(0, std::numeric_limits<RealType>::quiet_NaN()), std::domain_error); |
| BOOST_CHECK_THROW(uniform_distribution<RealType>(0, -std::numeric_limits<RealType>::quiet_NaN()), std::domain_error); |
| } |
| BOOST_CHECK_THROW(uniform_distribution<RealType>(1, 0), std::domain_error); // lower > upper! |
| BOOST_CHECK_THROW(uniform_distribution<RealType>(1, 1), std::domain_error); // lower == upper! |
| |
| } // template <class RealType>void test_spots(RealType) |
| |
| int test_main(int, char* []) |
| { |
| // Check that can construct uniform distribution using the two convenience methods: |
| using namespace boost::math; |
| uniform unistd; // Using typedef |
| // == uniform_distribution<double> unistd; |
| BOOST_CHECK_EQUAL(unistd.lower(), 0); // Check defaults. |
| BOOST_CHECK_EQUAL(unistd.upper(), 1); |
| uniform_distribution<> myu01(0, 1); // Using default RealType double. |
| BOOST_CHECK_EQUAL(myu01.lower(), 0); // Check defaults again. |
| BOOST_CHECK_EQUAL(myu01.upper(), 1); |
| |
| // Test on extreme values of random variate x, using just double because it has numeric_limit infinity etc.. |
| // No longer allow x to be + or - infinity, then these tests should throw. |
| BOOST_CHECK_THROW(pdf(unistd, +std::numeric_limits<double>::infinity()), std::domain_error); // x = + infinity |
| BOOST_CHECK_THROW(pdf(unistd, -std::numeric_limits<double>::infinity()), std::domain_error); // x = - infinity |
| BOOST_CHECK_THROW(cdf(unistd, +std::numeric_limits<double>::infinity()), std::domain_error); // x = + infinity |
| BOOST_CHECK_THROW(cdf(unistd, -std::numeric_limits<double>::infinity()), std::domain_error); // x = - infinity |
| |
| BOOST_CHECK_EQUAL(pdf(unistd, +(std::numeric_limits<double>::max)()), 0); // x = + max |
| BOOST_CHECK_EQUAL(pdf(unistd, -(std::numeric_limits<double>::min)()), 0); // x = - min |
| BOOST_CHECK_EQUAL(cdf(unistd, +(std::numeric_limits<double>::max)()), 1); // x = + max |
| BOOST_CHECK_EQUAL(cdf(unistd, -(std::numeric_limits<double>::min)()), 0); // x = - min |
| |
| BOOST_CHECK_THROW(uniform_distribution<> zinf(0, +std::numeric_limits<double>::infinity()), std::domain_error); // zero to infinity using default RealType double. |
| |
| uniform_distribution<> zmax(0, +(std::numeric_limits<double>::max)()); // zero to max using default RealType double. |
| BOOST_CHECK_EQUAL(zmax.lower(), 0); // Check defaults again. |
| BOOST_CHECK_EQUAL(zmax.upper(), +(std::numeric_limits<double>::max)()); |
| |
| BOOST_CHECK_EQUAL(pdf(zmax, -1), 0); // pdf is 1/(0 - max) = almost zero for all x |
| BOOST_CHECK_EQUAL(pdf(zmax, 0), (std::numeric_limits<double>::min)()/4); // x = |
| BOOST_CHECK_EQUAL(pdf(zmax, 1), (std::numeric_limits<double>::min)()/4); // x = |
| BOOST_CHECK_THROW(pdf(zmax, +std::numeric_limits<double>::infinity()), std::domain_error); // pdf is 1/(0 - infinity) = zero for all x |
| BOOST_CHECK_THROW(pdf(zmax, -std::numeric_limits<double>::infinity()), std::domain_error); |
| BOOST_CHECK_EQUAL(pdf(zmax, +(std::numeric_limits<double>::max)()), (std::numeric_limits<double>::min)()/4); // x = |
| BOOST_CHECK_EQUAL(pdf(zmax, -(std::numeric_limits<double>::max)()), 0); // x = |
| |
| // Ensure NaN throws an exception. |
| BOOST_CHECK_THROW(uniform_distribution<> zNaN(0, std::numeric_limits<double>::quiet_NaN()), std::domain_error); |
| BOOST_CHECK_THROW(pdf(unistd, std::numeric_limits<double>::quiet_NaN()), std::domain_error); |
| |
| // Basic sanity-check spot values. |
| // (Parameter value, arbitrarily zero, only communicates the floating point type). |
| test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % |
| test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % |
| #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS |
| test_spots(0.0L); // Test long double. |
| #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) |
| test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. |
| #endif |
| #else |
| std::cout << "<note>The long double tests have been disabled on this platform " |
| "either because the long double overloads of the usual math functions are " |
| "not available at all, or because they are too inaccurate for these tests " |
| "to pass.</note>" << std::cout; |
| #endif |
| |
| return 0; |
| } // int test_main(int, char* []) |
| |
| /* |
| |
| Output: |
| |
| Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_uniform.exe" |
| Running 1 test case... |
| Tolerance for type float is 2e-005. |
| Tolerance (as fraction) for type float is 5.96046e-007. |
| Tolerance for type double is 2e-005. |
| Tolerance (as fraction) for type double is 1.11022e-015. |
| Tolerance for type long double is 2e-005. |
| Tolerance (as fraction) for type long double is 1.11022e-015. |
| Tolerance for type class boost::math::concepts::real_concept is 2e-005. |
| Tolerance (as fraction) for type class boost::math::concepts::real_concept is 1.11022e-015. |
| *** No errors detected |
| |
| */ |
| |
| |
| |
