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Boost-Commit : |
Subject: [Boost-commit] svn:boost r66732 - trunk/libs/math/example
From: pbristow_at_[hidden]
Date: 2010-11-24 11:54:19
Author: pbristow
Date: 2010-11-24 11:54:17 EST (Wed, 24 Nov 2010)
New Revision: 66732
URL: http://svn.boost.org/trac/boost/changeset/66732
Log:
New example
Added:
trunk/libs/math/example/inverse_gaussian_example.cpp (contents, props changed)
Added: trunk/libs/math/example/inverse_gaussian_example.cpp
==============================================================================
--- (empty file)
+++ trunk/libs/math/example/inverse_gaussian_example.cpp 2010-11-24 11:54:17 EST (Wed, 24 Nov 2010)
@@ -0,0 +1,510 @@
+// wald_example.cpp or inverse_gaussian_example.cpp
+
+// Copyright Paul A. Bristow 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)
+
+// Example of using the Inverse Gaussian (or Inverse Normal) distribution.
+// The Wald Distribution is
+
+
+// Note that this file contains Quickbook mark-up as well as code
+// and comments, don't change any of the special comment mark-ups!
+
+//[inverse_gaussian_basic1
+/*`
+First we need some includes to access the normal distribution
+(and some std output of course).
+*/
+
+#ifdef _MSC_VER
+# pragma warning (disable : 4224)
+# pragma warning (disable : 4189)
+# pragma warning (disable : 4100)
+# pragma warning (disable : 4224)
+# pragma warning (disable : 4512)
+# pragma warning (disable : 4702)
+# pragma warning (disable : 4127)
+#endif
+
+//#define BOOST_MATH_INSTRUMENT
+
+#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error
+#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
+
+#include <boost/math/distributions/inverse_gaussian.hpp> // for inverse_gaussian_distribution
+ using boost::math::inverse_gaussian; // typedef provides default type is double.
+ using boost::math::inverse_gaussian_distribution; // for inverse gaussian distribution.
+
+#include <boost/math/distributions/normal.hpp> // for normal_distribution
+using boost::math::normal; // typedef provides default type is double.
+
+#include <boost/array.hpp>
+using boost::array;
+
+#include <iostream>
+ using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;
+#include <iomanip>
+ using std::setw; using std::setprecision;
+#include <limits>
+ using std::numeric_limits;
+#include <sstream>
+ using std::string;
+#include <string>
+ using std::stringstream;
+
+// const double tol = 3 * numeric_limits<double>::epsilon();
+
+int main()
+{
+ cout << "Example: Inverse Gaussian Distribution."<< endl;
+
+ try
+ {
+
+ double tolfeweps = numeric_limits<double>::epsilon();
+ //cout << "Tolerance = " << tol << endl;
+
+ int precision = 17; // traditional tables are only computed to much lower precision.
+ cout.precision(17); // std::numeric_limits<double>::max_digits10; for 64-bit doubles.
+
+ // Traditional tables and values.
+ double step = 0.2; // in z
+ double range = 4; // min and max z = -range to +range.
+ // Construct a (standard) inverse gaussian distribution s
+ inverse_gaussian w11(1, 1);
+ // (default mean = units, and standard deviation = unity)
+ cout << "(Standard) Inverse Gaussian distribution, mean = "<< w11.mean()
+ << ", scale = " << w11.scale() << endl;
+
+/*` First the probability distribution function (pdf).
+ */
+ cout << "Probability distribution function (pdf) values" << endl;
+ cout << " z " " pdf " << endl;
+ cout.precision(5);
+ for (double z = numeric_limits<double>::min(); z < range + step; z += step)
+ {
+ cout << left << setprecision(3) << setw(6) << z << " "
+ << setprecision(precision) << setw(12) << pdf(w11, z) << endl;
+ }
+ cout.precision(6); // default
+ /*`And the area under the normal curve from -[infin] up to z,
+ the cumulative distribution function (cdf).
+*/
+
+ // For a (default) inverse gaussian distribution.
+ cout << "Integral (area under the curve) from 0 up to z (cdf) " << endl;
+ cout << " z " " cdf " << endl;
+ for (double z = numeric_limits<double>::min(); z < range + step; z += step)
+ {
+ cout << left << setprecision(3) << setw(6) << z << " "
+ << setprecision(precision) << setw(12) << cdf(w11, z) << endl;
+ }
+ /*`giving the following table:
+[pre
+ z pdf
+ 2.23e-308 -1.#IND
+ 0.2 0.90052111680384117
+ 0.4 1.0055127039453111
+ 0.6 0.75123750098955733
+ 0.8 0.54377310461643302
+ 1 0.3989422804014327
+ 1.2 0.29846949816803292
+ 1.4 0.2274579835638664
+ 1.6 0.17614566625628389
+ 1.8 0.13829083543591469
+ 2 0.10984782236693062
+ 2.2 0.088133964251182237
+ 2.4 0.071327382959107177
+ 2.6 0.058162562161661699
+ 2.8 0.047742223328567722
+ 3 0.039418357969819712
+ 3.2 0.032715223861241892
+ 3.4 0.027278388940958308
+ 3.6 0.022840312999395804
+ 3.8 0.019196657941016954
+ 4 0.016189699458236451
+ Integral (area under the curve) from 0 up to z (cdf)
+ z cdf
+ 2.23e-308 0
+ 0.2 0.063753567519976254
+ 0.4 0.2706136704424541
+ 0.6 0.44638391340412931
+ 0.8 0.57472390962590925
+ 1 0.66810200122317065
+ 1.2 0.73724578422952536
+ 1.4 0.78944214237790356
+ 1.6 0.82953458108474554
+ 1.8 0.86079282968276671
+ 2 0.88547542598600626
+ 2.2 0.90517870624273966
+ 2.4 0.92105495653509362
+ 2.6 0.93395164268166786
+ 2.8 0.94450240360053817
+ 3 0.95318792074278835
+ 3.2 0.96037753019309191
+ 3.4 0.96635823989417369
+ 3.6 0.97135533107998406
+ 3.8 0.97554722413538364
+ 4 0.97907636417888622
+]
+
+/*`We can get the inverse, the quantile, percentile, percentage point, or critical value
+for a probability for a few probability from the above table, for z = 0.4, 1.0, 2.0:
+*/
+ cout << quantile(w11, 0.27061367044245421 ) << endl; // 0.4
+ cout << quantile(w11, 0.66810200122317065) << endl; // 1.0
+ cout << quantile(w11, 0.88547542598600615) << endl; // 2.0
+/*`turning the expect values apart from some 'computational noise' in the least significant bit or two.
+
+[pre
+ 0.40000000000000008
+ 0.99999999999999967
+ 1.9999999999999973
+]
+
+*/
+
+ // cout << "pnorm01(-0.406053) " << pnorm01(-0.406053) << ", cdfn01(-0.406053) = " << cdf(n01, -0.406053) << endl;
+ //cout << "pnorm01(0.5) = " << pnorm01(0.5) << endl; // R pnorm(0.5,0,1) = 0.6914625 == 0.69146246127401312
+ // R qnorm(0.6914625,0,1) = 0.5
+
+ // formatC(SuppDists::qinvGauss(0.3649755481729598, 1, 1), digits=17) [1] "0.50000000969034875"
+
+
+
+ // formatC(SuppDists::dinvGauss(0.01, 1, 1), digits=17) [1] "2.0811768202028392e-19"
+ // formatC(SuppDists::pinvGauss(0.01, 1, 1), digits=17) [1] "4.122313403318778e-23"
+
+
+
+ //cout << " qinvgauss(0.3649755481729598, 1, 1) = " << qinvgauss(0.3649755481729598, 1, 1) << endl; // 0.5
+ // cout << quantile(s, 0.66810200122317065) << endl; // expect 1, get 0.50517388467190727
+ //cout << " qinvgauss(0.62502320258649202, 1, 1) = " << qinvgauss(0.62502320258649202, 1, 1) << endl; // 0.9
+ //cout << " qinvgauss(0.063753567519976254, 1, 1) = " << qinvgauss(0.063753567519976254, 1, 1) << endl; // 0.2
+ //cout << " qinvgauss(0.0040761113207110162, 1, 1) = " << qinvgauss(0.0040761113207110162, 1, 1) << endl; // 0.1
+
+ //double x = 1.; // SuppDists::pinvGauss(0.4, 1,1) [1] 0.2706137
+ //double c = pinvgauss(x, 1, 1); // 0.3649755481729598 == cdf(x, 1,1) 0.36497554817295974
+ //cout << " pinvgauss(x, 1, 1) = " << c << endl; // pinvgauss(x, 1, 1) = 0.27061367044245421
+ //double p = pdf(w11, x);
+ //double c = cdf(w11, x); // cdf(1, 1, 1) = 0.66810200122317065
+ //cout << "cdf(" << x << ", " << w11.mean() << ", "<< w11.scale() << ") = " << c << endl; // cdf(x, 1, 1) 0.27061367044245421
+ //cout << "pdf(" << x << ", " << w11.mean() << ", "<< w11.scale() << ") = " << p << endl;
+ //double q = quantile(w11, c);
+ //cout << "quantile(w11, " << c << ") = " << q << endl;
+
+ //cout << "quantile(w11, 4.122313403318778e-23) = "<< quantile(w11, 4.122313403318778e-23) << endl; // quantile
+ //cout << "quantile(w11, 4.8791443010851493e-219) = " << quantile(w11, 4.8791443010851493e-219) << endl; // quantile
+
+ //double c1 = 1 - cdf(w11, x); // 1 - cdf(1, 1, 1) = 0.33189799877682935
+ //cout << "1 - cdf(" << x << ", " << w11.mean() << ", " << w11.scale() << ") = " << c1 << endl; // cdf(x, 1, 1) 0.27061367044245421
+ //double cc = cdf(complement(w11, x));
+ //cout << "cdf(complement(" << x << ", " << w11.mean() << ", "<< w11.scale() << ")) = " << cc << endl; // cdf(x, 1, 1) 0.27061367044245421
+ //// 1 - cdf(1000, 1, 1) = 0
+ //// cdf(complement(1000, 1, 1)) = 4.8694344366900402e-222
+
+ //cout << "quantile(w11, " << c << ") = "<< quantile(w11, c) << endl; // quantile = 0.99999999999999978 == x = 1
+ //cout << "quantile(w11, " << c << ") = "<< quantile(w11, 1 - c) << endl; // quantile complement. quantile(w11, 0.66810200122317065) = 0.46336593652340152
+// cout << "quantile(complement(w11, " << c << ")) = " << quantile(complement(w11, c)) << endl; // quantile complement = 0.46336593652340163
+
+ // cdf(1, 1, 1) = 0.66810200122317065
+ // 1 - cdf(1, 1, 1) = 0.33189799877682935
+ // cdf(complement(1, 1, 1)) = 0.33189799877682929
+
+ // quantile(w11, 0.66810200122317065) = 0.99999999999999978
+ // 1 - quantile(w11, 0.66810200122317065) = 2.2204460492503131e-016
+ // quantile(complement(w11, 0.33189799877682929)) = 0.99999999999999989
+
+
+ // qinvgauss(c, 1, 1) = 0.3999999999999998
+ // SuppDists::qinvGauss(0.270613670442454, 1, 1) [1] 0.4
+
+
+ /*
+ double qs = pinvgaussU(c, 1, 1); //
+ cout << "qinvgaussU(c, 1, 1) = " << qs << endl; // qinvgaussU(c, 1, 1) = 0.86567442459240929
+ // > z=q - exp(c) * p [1] 0.8656744 qs 0.86567442459240929 double
+ // Is this the complement?
+ cout << "qgamma(0.2, 0.5, 1) expect 0.0320923 = " << qgamma(0.2, 0.5, 1) << endl;
+ // qgamma(0.2, 0.5, 1) expect 0.0320923 = 0.032092377333650807
+
+
+ cout << "qinvgauss(pinvgauss(x, 1, 1) = " << q
+ << ", diff = " << x - q << ", fraction = " << (x - q) /x << endl; // 0.5
+
+ */ // > SuppDists::pinvGauss(0.02, 1,1) [1] 4.139176e-12
+ // > SuppDists::qinvGauss(4.139176e-12, 1,1) [1] 0.02000000
+
+
+ // pinvGauss(1,1,1) = 0.668102 C++ == 0.66810200122317065
+ // qinvGauss(0.668102,1,1) = 1
+
+ // SuppDists::pinvGauss(0.3,1,1) = 0.1657266
+ // cout << "qinvGauss(0.0040761113207110162, 1, 1) = " << qinvgauss(0.0040761113207110162, 1, 1) << endl;
+ //cout << "quantile(s, 0.1657266) = " << quantile(s, 0.1657266) << endl; // expect 1.
+
+ //wald s12(2, 1);
+ //cout << "qinvGauss(0.3, 2, 1) = " << qinvgauss(0.3, 2, 1) << endl; // SuppDists::qinvGauss(0.3,2,1) == 0.58288065635052944
+ //// but actually get qinvGauss(0.3, 2, 1) = 0.58288064777632187
+ //cout << "cdf(s12, 0.3) = " << cdf(s12, 0.3) << endl; // cdf(s12, 0.3) = 0.10895339868447573
+
+ // using boost::math::wald;
+ //cout.precision(6);
+
+ /*
+ double m = 1;
+ double l = 1;
+ double x = 0.1;
+ //c = cdf(w, x);
+ double p = pinvgauss(x, m, l);
+ cout << "x = " << x << ", pinvgauss(x, m, l) = " << p << endl; // R 0.4 0.2706137
+ double qg = qgamma(1.- p, 0.5, 1.0, true, false);
+ cout << " qgamma(1.- p, 0.5, 1.0, true, false) = " << qg << endl; // 0.606817
+ double g = guess_whitmore(p, m, l);
+ cout << "m = " << m << ", l = " << l << ", x = " << x << ", guess = " << g
+ << ", diff = " << (x - g) << endl;
+
+ g = guess_wheeler(p, m, l);
+ cout << "m = " << m << ", l = " << l << ", x = " << x << ", guess = " << g
+ << ", diff = " << (x - g) << endl;
+
+ g = guess_bagshaw(p, m, l);
+ cout << "m = " << m << ", l = " << l << ", x = " << x << ", guess = " << g
+ << ", diff = " << (x - g) << endl;
+
+ // m = 1, l = 10, x = 0.9, guess = 0.89792, diff = 0.00231075 so a better fit.
+ // x = 0.9, guess = 0.887907
+ // x = 0.5, guess = 0.474977
+ // x = 0.4, guess = 0.369597
+ // x = 0.2, guess = 0.155196
+
+ // m = 1, l = 2, x = 0.9, guess = 1.0312, diff = -0.145778
+ // m = 1, l = 2, x = 0.1, guess = 0.122201, diff = -0.222013
+ // m = 1, l = 2, x = 0.2, guess = 0.299326, diff = -0.49663
+ // m = 1, l = 2, x = 0.5, guess = 1.00437, diff = -1.00875
+ // m = 1, l = 2, x = 0.7, guess = 1.01517, diff = -0.450247
+
+ double ls[7] = {0.1, 0.2, 0.5, 1., 2., 10, 100}; // scale values.
+ double ms[10] = {0.001, 0.02, 0.1, 0.2, 0.5, 0.9, 1., 2., 10, 100}; // mean values.
+ */
+
+ cout.precision(6); // Restore to default.
+ } // try
+ catch(const std::exception& e)
+ { // Always useful to include try & catch blocks because default policies
+ // are to throw exceptions on arguments that cause errors like underflow, overflow.
+ // Lacking try & catch blocks, the program will abort without a message below,
+ // which may give some helpful clues as to the cause of the exception.
+ std::cout <<
+ "\n""Message from thrown exception was:\n " << e.what() << std::endl;
+ }
+ return 0;
+} // int main()
+
+
+/*
+
+Output is:
+
+inverse_gaussian_example.cpp
+ inverse_gaussian_example.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Debug\inverse_gaussian_example.exe
+ Example: Inverse Gaussian Distribution.
+ (Standard) Inverse Gaussian distribution, mean = 1, scale = 1
+ Probability distribution function (pdf) values
+ z pdf
+ 2.23e-308 -1.#IND
+ 0.2 0.90052111680384117
+ 0.4 1.0055127039453111
+ 0.6 0.75123750098955733
+ 0.8 0.54377310461643302
+ 1 0.3989422804014327
+ 1.2 0.29846949816803292
+ 1.4 0.2274579835638664
+ 1.6 0.17614566625628389
+ 1.8 0.13829083543591469
+ 2 0.10984782236693062
+ 2.2 0.088133964251182237
+ 2.4 0.071327382959107177
+ 2.6 0.058162562161661699
+ 2.8 0.047742223328567722
+ 3 0.039418357969819712
+ 3.2 0.032715223861241892
+ 3.4 0.027278388940958308
+ 3.6 0.022840312999395804
+ 3.8 0.019196657941016954
+ 4 0.016189699458236451
+ Integral (area under the curve) from 0 up to z (cdf)
+ z cdf
+ 2.23e-308 0
+ 0.2 0.063753567519976254
+ 0.4 0.2706136704424541
+ 0.6 0.44638391340412931
+ 0.8 0.57472390962590925
+ 1 0.66810200122317065
+ 1.2 0.73724578422952536
+ 1.4 0.78944214237790356
+ 1.6 0.82953458108474554
+ 1.8 0.86079282968276671
+ 2 0.88547542598600626
+ 2.2 0.90517870624273966
+ 2.4 0.92105495653509362
+ 2.6 0.93395164268166786
+ 2.8 0.94450240360053817
+ 3 0.95318792074278835
+ 3.2 0.96037753019309191
+ 3.4 0.96635823989417369
+ 3.6 0.97135533107998406
+ 3.8 0.97554722413538364
+ 4 0.97907636417888622
+ 0.40000000000000008
+ 0.99999999999999967
+ 1.9999999999999973
+
+
+
+> SuppDists::dinvGauss(2, 1, 1) [1] 0.1098478
+> SuppDists::dinvGauss(0.4, 1, 1) [1] 1.005513
+> SuppDists::dinvGauss(0.5, 1, 1) [1] 0.8787826
+> SuppDists::dinvGauss(0.39, 1, 1) [1] 1.016559
+> SuppDists::dinvGauss(0.38, 1, 1) [1] 1.027006
+> SuppDists::dinvGauss(0.37, 1, 1) [1] 1.036748
+> SuppDists::dinvGauss(0.36, 1, 1) [1] 1.045661
+> SuppDists::dinvGauss(0.35, 1, 1) [1] 1.053611
+> SuppDists::dinvGauss(0.3, 1, 1) [1] 1.072888
+> SuppDists::dinvGauss(0.1, 1, 1) [1] 0.2197948
+> SuppDists::dinvGauss(0.2, 1, 1) [1] 0.9005211
+>
+x = 0.3 [1, 1] 1.0728879234594337 // R SuppDists::dinvGauss(0.3, 1, 1) [1] 1.072888
+
+x = 1 [1, 1] 0.3989422804014327
+
+
+ 0 " NA"
+ 1 "0.3989422804014327"
+ 2 "0.10984782236693059"
+ 3 "0.039418357969819733"
+ 4 "0.016189699458236468"
+ 5 "0.007204168934430732"
+ 6 "0.003379893528659049"
+ 7 "0.0016462878258114036"
+ 8 "0.00082460931140859956"
+ 9 "0.00042207355643694234"
+10 "0.00021979480031862676"
+
+
+[1] " NA" " 0.690988298942671" "0.11539974210409144"
+ [4] "0.01799698883772935" "0.0029555399206496469" "0.00050863023587406587"
+ [7] "9.0761842931362914e-05" "1.6655279133132795e-05" "3.1243174913715429e-06"
+[10] "5.96530227727434e-07" "1.1555606328299836e-07"
+
+
+matC(dinvGauss(0:10, 1, 3), digits=17) df = 3
+[1] " NA" " 0.690988298942671" "0.11539974210409144"
+ [4] "0.01799698883772935" "0.0029555399206496469" "0.00050863023587406587"
+ [7] "9.0761842931362914e-05" "1.6655279133132795e-05" "3.1243174913715429e-06"
+[10] "5.96530227727434e-07" "1.1555606328299836e-07"
+$title
+[1] "Inverse Gaussian"
+
+$nu
+[1] 1
+
+$lambda
+[1] 3
+
+$Mean
+[1] 1
+
+$Median
+[1] 0.8596309
+
+$Mode
+[1] 0.618034
+
+$Variance
+[1] 0.3333333
+
+$SD
+[1] 0.5773503
+
+$ThirdCentralMoment
+[1] 0.3333333
+
+$FourthCentralMoment
+[1] 0.8888889
+
+$PearsonsSkewness...mean.minus.mode.div.SD
+[1] 0.6615845
+
+$Skewness...sqrtB1
+[1] 1.732051
+
+$Kurtosis...B2.minus.3
+[1] 5
+
+ Example: Wald distribution.
+ (Standard) Wald distribution, mean = 1, scale = 1
+ 1 dx = 0.24890250442652451, x = 0.70924622051646713
+ 2 dx = -0.038547954953794553, x = 0.46034371608994262
+ 3 dx = -0.0011074090830291131, x = 0.49889167104373716
+ 4 dx = -9.1987259926368029e-007, x = 0.49999908012676625
+ 5 dx = -6.346513344581067e-013, x = 0.49999999999936551
+ dx = 6.3168242705156857e-017 at i = 6
+ qinvgauss(0.3649755481729598, 1, 1) = 0.50000000000000011
+ 1 dx = 0.6719944578376621, x = 1.3735318786222666
+ 2 dx = -0.16997432635769361, x = 0.70153742078460446
+ 3 dx = -0.027865119206495724, x = 0.87151174714229807
+ 4 dx = -0.00062283290009492603, x = 0.89937686634879377
+ 5 dx = -3.0075104108208687e-007, x = 0.89999969924888867
+ 6 dx = -7.0485322513588089e-014, x = 0.89999999999992975
+ 7 dx = 9.557331866250277e-016, x = 0.90000000000000024
+ dx = 0 at i = 8
+ qinvgauss(0.62502320258649202, 1, 1) = 0.89999999999999925
+ 1 dx = -0.0052296256747447678, x = 0.19483508278446249
+ 2 dx = 6.4699046853900505e-005, x = 0.20006470845920726
+ 3 dx = 9.4123530465288137e-009, x = 0.20000000941235335
+ 4 dx = 2.7739513919147025e-016, x = 0.20000000000000032
+ dx = 1.5410841066192808e-016 at i = 5
+ qinvgauss(0.063753567519976254, 1, 1) = 0.20000000000000004
+ 1 dx = -1, x = -0.46073286697416105
+ 2 dx = 0.47665501251497061, x = 0.53926713302583895
+ 3 dx = -0.171105768635964, x = 0.062612120510868341
+ 4 dx = 0.091490360797512563, x = 0.23371788914683234
+ 5 dx = 0.029410311722649803, x = 0.14222752834931979
+ 6 dx = 0.010761845493592421, x = 0.11281721662666999
+ 7 dx = 0.0019864890597643035, x = 0.10205537113307757
+ 8 dx = 6.8800383732599561e-005, x = 0.10006888207331327
+ 9 dx = 8.1689466405590418e-008, x = 0.10000008168958067
+ 10 dx = 1.133634672475146e-013, x = 0.10000000000011428
+ 11 dx = 5.9588135045224526e-016, x = 0.10000000000000091
+ 12 dx = 3.433223674791152e-016, x = 0.10000000000000031
+ dx = 9.0763384505974048e-017 at i = 13
+ qinvgauss(0.0040761113207110162, 1, 1) = 0.099999999999999964
+
+
+ wald_example.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Debug\wald_example.exe
+ Example: Wald distribution.
+ Tolerance = 6.66134e-016
+ (Standard) Wald distribution, mean = 1, scale = 1
+ cdf(x, 1,1) 4.1390252102096375e-012
+ qinvgauss(pinvgauss(x, 1, 1) = 0.020116801973767886, diff = -0.00011680197376788548, fraction = -0.005840098688394274
+ ____________________________________________________________
+ wald 1, 1
+ x = 0.02, diff x - qinvgauss(cdf) = -0.00011680197376788548
+ x = 0.10000000000000001, diff x - qinvgauss(cdf) = 8.7430063189231078e-016
+ x = 0.20000000000000001, diff x - qinvgauss(cdf) = -1.1102230246251565e-016
+ x = 0.29999999999999999, diff x - qinvgauss(cdf) = 0
+ x = 0.40000000000000002, diff x - qinvgauss(cdf) = 2.2204460492503131e-016
+ x = 0.5, diff x - qinvgauss(cdf) = -1.1102230246251565e-016
+ x = 0.59999999999999998, diff x - qinvgauss(cdf) = 1.1102230246251565e-016
+ x = 0.80000000000000004, diff x - qinvgauss(cdf) = 1.1102230246251565e-016
+ x = 0.90000000000000002, diff x - qinvgauss(cdf) = 0
+ x = 0.98999999999999999, diff x - qinvgauss(cdf) = -1.1102230246251565e-016
+ x = 0.999, diff x - qinvgauss(cdf) = -1.1102230246251565e-016
+
+
+*/
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