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Subject: [Boost-commit] svn:boost r64967 - trunk/libs/math/test
From: pbristow_at_[hidden]
Date: 2010-08-23 07:54:35

Author: pbristow
Date: 2010-08-23 07:54:27 EDT (Mon, 23 Aug 2010)
New Revision: 64967
URL: http://svn.boost.org/trac/boost/changeset/64967

Log:
1st version of test
Added:
   trunk/libs/math/test/test_inverse_gamma_distribution.cpp (contents, props changed)

Added: trunk/libs/math/test/test_inverse_gamma_distribution.cpp
==============================================================================
--- (empty file)
+++ trunk/libs/math/test/test_inverse_gamma_distribution.cpp 2010-08-23 07:54:27 EDT (Mon, 23 Aug 2010)
@@ -0,0 +1,442 @@
+// test_inverse_gamma.cpp
+
+// Copyright Paul A. Bristow 2010.
+// Copyright John Maddock 2010.
+
+// 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)
+
+
+#ifdef _MSC_VER
+# pragma warning (disable : 4224) // nonstandard extension used : formal parameter 'type' was previously defined as a type
+// in Boost.test and lexical_cast
+# pragma warning (disable : 4310) // cast truncates constant value
+#endif
+
+#include <boost/math/concepts/real_concept.hpp> // for real_concept
+using ::boost::math::concepts::real_concept;
+
+//#include <boost/math/tools/test.hpp>
+#include <boost/test/test_exec_monitor.hpp> // for test_main
+#include <boost/test/floating_point_comparison.hpp> // for BOOST_CHECK_CLOSE_FRACTION
+
+#include <boost/math/distributions/inverse_gamma.hpp> // for inverse_gamma_distribution
+
+#include <boost/math/special_functions/gamma.hpp> // for tgamma for naive pdf.
+
+using boost::math::inverse_gamma_distribution;
+using ::boost::math::inverse_gamma;
+// using ::boost::math::cdf;
+// using ::boost::math::pdf;
+
+#include <iostream>
+using std::cout;
+using std::endl;
+#include <limits>
+using std::numeric_limits;
+
+template <class RealType>
+RealType naive_pdf(RealType shape, RealType scale, RealType x)
+{ // Formula from Wikipedia
+ using namespace std; // For ADL of std functions.
+ using boost::math::tgamma;
+ RealType result = (pow(scale, shape) * pow(x, (-shape -1)) * exp(-scale/x) ) / tgamma(shape);
+ return result;
+}
+
+// Test using a spot value from some other reference source,
+// in this case test values from output from R provided by Thomas Mang.
+
+template <class RealType>
+void test_spot(
+ RealType shape, // shape,
+ RealType scale, // scale,
+ RealType x, // random variate x,
+ RealType pd, // expected pdf,
+ RealType P, // expected CDF,
+ RealType Q, // expected complement of CDF,
+ RealType tol) // test tolerance.
+{
+ boost::math::inverse_gamma_distribution<RealType> dist(shape, scale);
+
+ BOOST_CHECK_CLOSE_FRACTION
+ ( // Compare to expected PDF.
+ pdf(dist, x), // calculated.
+ pd, // expected
+ tol);
+
+ BOOST_CHECK_CLOSE_FRACTION( // Compare to naive formula (might be less accurate).
+ pdf(dist, x), naive_pdf(dist.shape(), dist.scale(), x), tol);
+
+ BOOST_CHECK_CLOSE_FRACTION( // Compare to expected CDF.
+ cdf(dist, x), P, tol);
+
+ if((P < 0.999) && (Q < 0.999))
+ { // We can only check this if P is not too close to 1,
+ // so that we can guarantee Q is accurate:
+ BOOST_CHECK_CLOSE_FRACTION(
+ cdf(complement(dist, x)), Q, tol);
+ BOOST_CHECK_CLOSE_FRACTION(
+ quantile(dist, P), x, tol); // quantile(pdf) = x
+ BOOST_CHECK_CLOSE_FRACTION(
+ quantile(complement(dist, Q)), x, tol);
+ }
+} // test_spot
+
+// Test using a spot value from some other reference source.
+
+template <class RealType> // Any floating-point type RealType.
+void test_spots(RealType)
+{
+ // Basic sanity checks, test data is to six decimal places only
+ // so set tolerance to 0.000001 expressed as a percentage = 0.0001%.
+
+ RealType tolerance = 0.000001f * 100;
+ cout << "Tolerance = " << tolerance << "%." << endl;
+
+// This test values from output from R provided by Thomas Mang.
+ test_spot(static_cast<RealType>(2), static_cast<RealType>(1), // shape, scale
+ static_cast<RealType>(2.L), // x
+ static_cast<RealType>(0.075816332464079136L), // pdf
+ static_cast<RealType>(0.90979598956895047L), // cdf
+ static_cast<RealType>(1 - 0.90979598956895047L), // cdf complement
+ tolerance // tol
+ );
+
+ test_spot(static_cast<RealType>(1.593), static_cast<RealType>( 0.5), // shape, scale
+ static_cast<RealType>( 0.5), // x
+ static_cast<RealType>(0.82415241749687074L), // pdf
+ static_cast<RealType>(0.60648042700409865L), // cdf
+ static_cast<RealType>(1 - 0.60648042700409865), // cdf complement
+ tolerance // tol
+ );
+
+ test_spot(static_cast<RealType>(13.319), static_cast<RealType>(0.5), // shape, scale
+ static_cast<RealType>(0.5), // x
+ static_cast<RealType>(0.00000000068343206235379223), // pdf
+ static_cast<RealType>(0.99999999997242739L), // cdf
+ static_cast<RealType>(1 - 0.99999999997242739), // cdf complement
+ tolerance // tol
+ );
+
+
+ test_spot(static_cast<RealType>(1.593), static_cast<RealType>(1), // shape, scale
+ static_cast<RealType>(1.977), // x
+ static_cast<RealType>(0.11535946773398653L), // pdf
+ static_cast<RealType>(0.82449794420341549L), // cdf
+ static_cast<RealType>(1 - 0.82449794420341549), // cdf complement
+ tolerance // tol
+ );
+
+
+ test_spot(static_cast<RealType>(6.666), static_cast<RealType>(1.411), // shape, scale
+ static_cast<RealType>(5), // x
+ static_cast<RealType>(0.000000084415758206386872), // pdf
+ static_cast<RealType>(0.99999993427280998L), // cdf
+ static_cast<RealType>(1 - 0.99999993427280998L), // cdf complement
+ tolerance // tol
+ );
+
+
+ // Check some bad parameters to the distribution,
+ BOOST_CHECK_THROW(boost::math::inverse_gamma_distribution<RealType> igbad1(-1, 0), std::domain_error); // negative shape.
+ BOOST_CHECK_THROW(boost::math::inverse_gamma_distribution<RealType> igbad2(0, -1), std::domain_error); // negative scale.
+ BOOST_CHECK_THROW(boost::math::inverse_gamma_distribution<RealType> igbad2(-1, -1), std::domain_error); // negative scale and shape.
+
+ inverse_gamma_distribution<RealType> ig21(2, 1);
+
+ if(std::numeric_limits<RealType>::has_infinity)
+ {
+ BOOST_CHECK_THROW(pdf(ig21, +std::numeric_limits<RealType>::infinity()), std::domain_error); // x = + infinity, pdf = 0
+ BOOST_CHECK_THROW(pdf(ig21, -std::numeric_limits<RealType>::infinity()), std::domain_error); // x = - infinity, pdf = 0
+ BOOST_CHECK_THROW(cdf(ig21, +std::numeric_limits<RealType>::infinity()),std::domain_error ); // x = + infinity, cdf = 1
+ BOOST_CHECK_THROW(cdf(ig21, -std::numeric_limits<RealType>::infinity()), std::domain_error); // x = - infinity, cdf = 0
+ BOOST_CHECK_THROW(cdf(complement(ig21, +std::numeric_limits<RealType>::infinity())), std::domain_error); // x = + infinity, c cdf = 0
+ BOOST_CHECK_THROW(cdf(complement(ig21, -std::numeric_limits<RealType>::infinity())), std::domain_error); // x = - infinity, c cdf = 1
+ BOOST_CHECK_THROW(boost::math::inverse_gamma_distribution<RealType> nbad1(std::numeric_limits<RealType>::infinity(), static_cast<RealType>(1)), std::domain_error); // +infinite mean
+ BOOST_CHECK_THROW(boost::math::inverse_gamma_distribution<RealType> nbad1(-std::numeric_limits<RealType>::infinity(), static_cast<RealType>(1)), std::domain_error); // -infinite mean
+ BOOST_CHECK_THROW(boost::math::inverse_gamma_distribution<RealType> nbad1(static_cast<RealType>(0), std::numeric_limits<RealType>::infinity()), std::domain_error); // infinite sd
+ }
+
+ if (std::numeric_limits<RealType>::has_quiet_NaN)
+ {
+ // No longer allow x to be NaN, then these tests should throw.
+ BOOST_CHECK_THROW(pdf(ig21, +std::numeric_limits<RealType>::quiet_NaN()), std::domain_error); // x = NaN
+ BOOST_CHECK_THROW(cdf(ig21, +std::numeric_limits<RealType>::quiet_NaN()), std::domain_error); // x = NaN
+ BOOST_CHECK_THROW(cdf(complement(ig21, +std::numeric_limits<RealType>::quiet_NaN())), std::domain_error); // x = + infinity
+ BOOST_CHECK_THROW(quantile(ig21, +std::numeric_limits<RealType>::quiet_NaN()), std::domain_error); // p = + infinity
+ BOOST_CHECK_THROW(quantile(complement(ig21, +std::numeric_limits<RealType>::quiet_NaN())), std::domain_error); // p = + infinity
+ }
+ // Spot check for pdf using 'naive pdf' function
+ for(RealType x = 0.5; x < 5; x += 0.5)
+ {
+ BOOST_CHECK_CLOSE_FRACTION(
+ pdf(inverse_gamma_distribution<RealType>(5, 6), x),
+ naive_pdf(RealType(5), RealType(6), x),
+ tolerance);
+ } // Spot checks for parameters:
+
+ RealType tol_5eps = boost::math::tools::epsilon<RealType>() * 5; // 5 eps as a fraction.
+ inverse_gamma_distribution<RealType> dist51(5, 1);
+ inverse_gamma_distribution<RealType> dist52(5, 2);
+ inverse_gamma_distribution<RealType> dist31(3, 1);
+ inverse_gamma_distribution<RealType> dist111(11, 1);
+ // 11 mean 0.10000000000000001, variance 0.0011111111111111111, sd 0.033333333333333333
+
+ RealType x = static_cast<RealType>(0.125);
+ using namespace std; // ADL of std names.
+ using namespace boost::math;
+
+ // mean, variance etc
+ BOOST_CHECK_CLOSE_FRACTION(mean(dist52), static_cast<RealType>(0.5), tol_5eps);
+ BOOST_CHECK_CLOSE_FRACTION(mean(dist111), static_cast<RealType>(0.1L), tol_5eps);
+ inverse_gamma_distribution<RealType> igamma41(static_cast<RealType>(4.), static_cast<RealType>(1.) );
+ BOOST_CHECK_CLOSE_FRACTION(mean(igamma41), static_cast<RealType>(0.3333333333333333333333333333333333333333333333333333333L), tol_5eps);
+
+
+ // variance:
+ BOOST_CHECK_CLOSE_FRACTION(variance(dist51), static_cast<RealType>(0.0208333333333333333333333333333333333333333333333333L), tol_5eps);
+ BOOST_CHECK_CLOSE_FRACTION(variance(dist31), static_cast<RealType>(0.25), tol_5eps);
+ BOOST_CHECK_CLOSE_FRACTION(variance(dist111), static_cast<RealType>(0.001111111111111111111111111111111111111111111111111L), tol_5eps);
+ // std deviation:
+ BOOST_CHECK_CLOSE_FRACTION(standard_deviation(dist31), static_cast<RealType>(0.5), tol_5eps);
+ BOOST_CHECK_CLOSE_FRACTION(standard_deviation(dist111), static_cast<RealType>(0.0333333333333333333333333333333333333333333333333L), tol_5eps);
+ // hazard:
+ BOOST_CHECK_CLOSE_FRACTION(hazard(dist51, x), pdf(dist51, x) / cdf(complement(dist51, x)), tol_5eps);
+ // cumulative hazard:
+ BOOST_CHECK_CLOSE_FRACTION(chf(dist51, x), -log(cdf(complement(dist51, x))), tol_5eps);
+ // coefficient_of_variation:
+ BOOST_CHECK_CLOSE_FRACTION(coefficient_of_variation(dist51), standard_deviation(dist51) / mean(dist51), tol_5eps);
+ // mode:
+ BOOST_CHECK_CLOSE_FRACTION(mode(dist51), static_cast<RealType>(0.166666666666666666666666666666666666666666666666666L), tol_5eps);
+
+ //BOOST_CHECK_CLOSE_FRACTION(median(dist52), static_cast<RealType>(0), tol_5eps);
+ // need an exact median? Failing that a loop back test.
+
+ BOOST_CHECK_CLOSE_FRACTION(cdf(dist111, median(dist111)), 0.5, tol_5eps);
+
+
+ // skewness:
+ BOOST_CHECK_CLOSE_FRACTION(skewness(dist111), static_cast<RealType>(1.5), tol_5eps);
+ //kurtosis:
+ BOOST_CHECK_CLOSE_FRACTION(kurtosis(dist51), static_cast<RealType>(42 + 3), tol_5eps);
+ // kurtosis excess:
+ BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(dist51), static_cast<RealType>(42), tol_5eps);
+
+ tol_5eps = boost::math::tools::epsilon<RealType>() * 5; // 5 eps as a percentage.
+
+ // Special and limit cases:
+
+ RealType mx = (std::numeric_limits<RealType>::max)();
+ RealType mi = (std::numeric_limits<RealType>::min)();
+
+ BOOST_CHECK_EQUAL(
+ pdf(inverse_gamma_distribution<RealType>(1),
+ static_cast<RealType>(mx)), // max()
+ static_cast<RealType>(0)
+ );
+
+ BOOST_CHECK_EQUAL(
+ pdf(inverse_gamma_distribution<RealType>(1),
+ static_cast<RealType>(mi)), // min()
+ static_cast<RealType>(0)
+ );
+
+ BOOST_CHECK_EQUAL(
+ pdf(inverse_gamma_distribution<RealType>(1), static_cast<RealType>(0)), static_cast<RealType>(0));
+ BOOST_CHECK_EQUAL(
+ pdf(inverse_gamma_distribution<RealType>(3), static_cast<RealType>(0))
+ , static_cast<RealType>(0.0f));
+ BOOST_CHECK_EQUAL(
+ cdf(inverse_gamma_distribution<RealType>(1), static_cast<RealType>(0))
+ , static_cast<RealType>(0.0f));
+ BOOST_CHECK_EQUAL(
+ cdf(inverse_gamma_distribution<RealType>(2), static_cast<RealType>(0))
+ , static_cast<RealType>(0.0f));
+ BOOST_CHECK_EQUAL(
+ cdf(inverse_gamma_distribution<RealType>(3), static_cast<RealType>(0))
+ , static_cast<RealType>(0.0f));
+ BOOST_CHECK_EQUAL(
+ cdf(complement(inverse_gamma_distribution<RealType>(1), static_cast<RealType>(0)))
+ , static_cast<RealType>(1));
+ BOOST_CHECK_EQUAL(
+ cdf(complement(inverse_gamma_distribution<RealType>(2), static_cast<RealType>(0)))
+ , static_cast<RealType>(1));
+ BOOST_CHECK_EQUAL(
+ cdf(complement(inverse_gamma_distribution<RealType>(3), static_cast<RealType>(0)))
+ , static_cast<RealType>(1));
+
+ BOOST_CHECK_THROW(
+ pdf(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(-1)), // shape negative.
+ static_cast<RealType>(1)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ pdf(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(-1)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ cdf(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(-1)),
+ static_cast<RealType>(1)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ cdf(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(-1)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ cdf(complement(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(-1)),
+ static_cast<RealType>(1))), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ cdf(complement(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(-1))), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ quantile(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(-1)),
+ static_cast<RealType>(0.5)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ quantile(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(-1)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ quantile(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(1.1)), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ quantile(complement(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(-1)),
+ static_cast<RealType>(0.5))), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ quantile(complement(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(-1))), std::domain_error
+ );
+ BOOST_CHECK_THROW(
+ quantile(complement(
+ inverse_gamma_distribution<RealType>(static_cast<RealType>(8)),
+ static_cast<RealType>(1.1))), std::domain_error
+ );
+
+} // template <class RealType>void test_spots(RealType)
+
+int test_main(int, char* [])
+{
+ BOOST_MATH_CONTROL_FP;
+
+ // Check that can generate inverse_gamma distribution using the two convenience methods:
+ // inverse_gamma_distribution; // with default parameters, shape = 1, scale - 1
+ using boost::math::inverse_gamma;
+ inverse_gamma ig2(2.); // Using typedef and shape parameter (and default scale = 1).
+ BOOST_CHECK_EQUAL(ig2.shape(), 2.); // scale == 2.
+ BOOST_CHECK_EQUAL(ig2.scale(), 1.); // scale == 1 (default).
+ inverse_gamma ig; // Using typedef, type double and default values, shape = 1 and scale = 1
+ // check default is (1, 1)
+ BOOST_CHECK_EQUAL(ig.shape(), 1.); // shape == 1
+ BOOST_CHECK_EQUAL(ig.scale(), 1.); // scale == 1
+ BOOST_CHECK_EQUAL(mode(ig), 0.5); // mode = 1/2
+
+ // Used to find some 'exact' values for testing mean, variance ...
+ //for (int shape = 4; shape < 30; shape++)
+ // {
+ // inverse_gamma ig(shape, 1);
+ // cout.precision(17);
+ // cout << shape << ' ' << mean(ig) << ' ' << variance(ig) << ' ' << standard_deviation(ig)
+ // << ' ' << median(ig) << endl;
+ // }
+
+ // and "using boost::math::inverse_gamma_distribution;".
+ inverse_gamma_distribution<> ig23(2., 3.); // Using default RealType double.
+ BOOST_CHECK_EQUAL(ig23.shape(), 2.); //
+ BOOST_CHECK_EQUAL(ig23.scale(), 3.); //
+
+ inverse_gamma_distribution<float> igf23(1.f, 2.f); // Using explicit RealType float.
+ BOOST_CHECK_EQUAL(igf23.shape(), 1.f); //
+ BOOST_CHECK_EQUAL(igf23.scale(), 2.f); //
+ // Some tests using default double.
+ double tol5eps = boost::math::tools::epsilon<double>() * 5; // 5 eps as a fraction.
+ inverse_gamma_distribution<double> ig102(10., 2.); //
+ BOOST_CHECK_EQUAL(ig102.shape(), 10.); //
+ BOOST_CHECK_EQUAL(ig102.scale(), 2.); //
+ BOOST_CHECK_CLOSE_FRACTION(pdf(ig102, 0.5), 0.1058495335284024, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(cdf(ig102, 0.5), 0.99186775720306608, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(quantile(ig102, 0.05), 0.12734622346137681, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(quantile(ig102, 0.5), 0.20685272858879727, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(quantile(ig102, 0.95), 0.36863602680851204, tol5eps);
+ // Check mean, etc spot values.
+ inverse_gamma_distribution<double> ig51(5., 1.); // shape = 5, scale = 1
+ BOOST_CHECK_CLOSE_FRACTION(mean(ig51), 0.25, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(variance(ig51), 0.0208333333333333333333333333333333333333333, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(skewness(ig51), 2 * std::sqrt(3.), tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(ig51), 42, tol5eps);
+ // mode and median
+ inverse_gamma_distribution<double> ig21(1., 2.);
+ BOOST_CHECK_CLOSE_FRACTION(mode(ig21), 1, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(median(ig21), 2.8853900817779268, tol5eps);
+
+ BOOST_CHECK_CLOSE_FRACTION(quantile(ig21, 0.5), 2.8853900817779268, tol5eps);
+ BOOST_CHECK_CLOSE_FRACTION(cdf(ig21, median(ig21)), 0.5, tol5eps);
+
+ // Check throws from bad parameters.
+ inverse_gamma ig051(0.5, 1.); // shape < 1, so wrong for mean.
+ BOOST_CHECK_THROW(mean(ig051), std::domain_error);
+ inverse_gamma ig191(1.9999, 1.); // shape < 2, so wrong for variance.
+ BOOST_CHECK_THROW(variance(ig191), std::domain_error);
+ inverse_gamma ig291(2.9999, 1.); // shape < 3, so wrong for skewness.
+ BOOST_CHECK_THROW(skewness(ig291), std::domain_error);
+ inverse_gamma ig391(3.9999, 1.); // shape < 1, so wrong for kurtosis and kurtosis_excess.
+ BOOST_CHECK_THROW(kurtosis(ig391), std::domain_error);
+ BOOST_CHECK_THROW(kurtosis_excess(ig391), 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:
+
+------ Build started: Project: test_inverse_gamma_distribution, Configuration: Release Win32 ------
+ test_inverse_gamma_distribution.cpp
+ Generating code
+ Finished generating code
+ test_inverse_gamma_distribution.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\test_inverse_gamma_distribution.exe
+ Running 1 test case...
+ Tolerance = 0.0001%.
+ Tolerance = 0.0001%.
+ Tolerance = 0.0001%.
+ Tolerance = 0.0001%.
+
+ *** No errors detected
+========== Build: 1 succeeded, 0 failed, 0 up-to-date, 0 skipped ==========
+
+
+*/
+
+
+

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