libs/math/example/normal_tables.cpp
// normal_misc_examples.cpp // Copyright Paul A. Bristow 2007, 2010, 2014, 2016. // 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 normal distribution. // Note that this file contains Quickbook mark-up as well as code // and comments, don't change any of the special comment mark-ups! /*` First we need some includes to access the normal distribution (and some std output of course). */ #include <boost/cstdfloat.hpp> // MUST be first include!!! // See Implementation of Float128 type, Overloading template functions with float128_t. #include <boost/math/distributions/normal.hpp> // for normal_distribution. using boost::math::normal; // typedef provides default type of double. #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; /*! Function max_digits10 Returns maximum number of possibly significant decimal digits for a floating-point type FPT, even for older compilers/standard libraries that lack support for std::std::numeric_limits<FPT>::max_digits10, when the Kahan formula 2 + binary_digits * 0.3010 is used instead. Also provides the correct result for Visual Studio 2010 where the max_digits10 provided for float is wrong. */ namespace boost { namespace math { template <typename FPT> int max_digits10() { // Since max_digits10 is not defined (or wrong) on older systems, define a local max_digits10. // Usage: int m = max_digits10<boost::float64_t>(); const int m = #if (defined BOOST_NO_CXX11_NUMERIC_LIMITS) || (_MSC_VER == 1600) // is wrongly 8 not 9 for VS2010. 2 + std::numeric_limits<FPT>::digits * 3010/10000; #else std::numeric_limits<FPT>::max_digits10; #endif return m; } } // namespace math } // namespace boost template <typename FPT> void normal_table() { using namespace boost::math; FPT step = static_cast<FPT>(1.); // step in z. FPT range = static_cast<FPT>(10.); // min and max z = -range to +range. // Traditional tables are only computed to much lower precision. // but @c std::std::numeric_limits<double>::max_digits10; // on new Standard Libraries gives 17, // the maximum number of digits from 64-bit double that can possibly be significant. // @c std::std::numeric_limits<double>::digits10; == 15 // is number of @b guaranteed digits, the other two digits being 'noisy'. // Here we use a custom version of max_digits10 which deals with those platforms // where @c std::numeric_limits is not specialized, // or @c std::numeric_limits<>::max_digits10 not implemented, or wrong. int precision = boost::math::max_digits10<FPT>(); // std::cout << typeid(FPT).name() << std::endl; // demo_normal.cpp:85: undefined reference to `typeinfo for __float128' // [@http://gcc.gnu.org/bugzilla/show_bug.cgi?id=43622 GCC 43622] // typeinfo for __float128 was missing GCC 4.9 Mar 2014, but OK for GCC 6.1.1. // Construct a standard normal distribution s, with // (default mean = zero, and standard deviation = unity) normal s; std::cout << "\nStandard normal distribution, mean = "<< s.mean() << ", standard deviation = " << s.standard_deviation() << std::endl; std::cout << "maxdigits_10 is " << precision << ", digits10 is " << std::numeric_limits<FPT>::digits10 << std::endl; std::cout << "Probability distribution function values" << std::endl; std::cout << " z " " PDF " << std::endl; for (FPT z = -range; z < range + step; z += step) { std::cout << std::left << std::setprecision(3) << std::setw(6) << z << " " << std::setprecision(precision) << std::setw(12) << pdf(s, z) << std::endl; } std::cout.precision(6); // Restore to default precision. /*`And the area under the normal curve from -[infin] up to z, the cumulative distribution function (CDF). */ // For a standard normal distribution: std::cout << "Standard normal mean = "<< s.mean() << ", standard deviation = " << s.standard_deviation() << std::endl; std::cout << "Integral (area under the curve) from - infinity up to z." << std::endl; std::cout << " z " " CDF " << std::endl; for (FPT z = -range; z < range + step; z += step) { std::cout << std::left << std::setprecision(3) << std::setw(6) << z << " " << std::setprecision(precision) << std::setw(12) << cdf(s, z) << std::endl; } std::cout.precision(6); // Reset to default precision. } // template <typename FPT> void normal_table() int main() { std::cout << "\nExample: Normal distribution tables." << std::endl; using namespace boost::math; try {// Tip - always use try'n'catch blocks to ensure that messages from thrown exceptions are shown. //[normal_table_1 #ifdef BOOST_FLOAT32_C normal_table<boost::float32_t>(); // Usually type float #endif normal_table<boost::float64_t>(); // Uusually type double. Assume that float64_t is always available. #ifdef BOOST_FLOAT80_C normal_table<boost::float80_t>(); // Type long double on some X86 platforms. #endif #ifdef BOOST_FLOAT128_C normal_table<boost::float128_t>(); // Type _Quad on some Intel and __float128 on some GCC platforms. #endif normal_table<boost::floatmax_t>(); //] [/normal_table_1 ] } catch(std::exception ex) { std::cout << "exception thrown " << ex.what() << std::endl; } return 0; } // int main() /* GCC 4.8.1 with quadmath Example: Normal distribution tables. Standard normal distribution, mean = 0, standard deviation = 1 maxdigits_10 is 9, digits10 is 6 Probability distribution function values z PDF -10 7.69459863e-023 -9 1.02797736e-018 -8 5.05227108e-015 -7 9.13472041e-012 -6 6.07588285e-009 -5 1.48671951e-006 -4 0.000133830226 -3 0.00443184841 -2 0.0539909665 -1 0.241970725 0 0.39894228 1 0.241970725 2 0.0539909665 3 0.00443184841 4 0.000133830226 5 1.48671951e-006 6 6.07588285e-009 7 9.13472041e-012 8 5.05227108e-015 9 1.02797736e-018 10 7.69459863e-023 Standard normal mean = 0, standard deviation = 1 Integral (area under the curve) from - infinity up to z. z CDF -10 7.61985302e-024 -9 1.12858841e-019 -8 6.22096057e-016 -7 1.27981254e-012 -6 9.86587645e-010 -5 2.86651572e-007 -4 3.16712418e-005 -3 0.00134989803 -2 0.0227501319 -1 0.158655254 0 0.5 1 0.841344746 2 0.977249868 3 0.998650102 4 0.999968329 5 0.999999713 6 0.999999999 7 1 8 1 9 1 10 1 Standard normal distribution, mean = 0, standard deviation = 1 maxdigits_10 is 17, digits10 is 15 Probability distribution function values z PDF -10 7.6945986267064199e-023 -9 1.0279773571668917e-018 -8 5.0522710835368927e-015 -7 9.1347204083645953e-012 -6 6.0758828498232861e-009 -5 1.4867195147342979e-006 -4 0.00013383022576488537 -3 0.0044318484119380075 -2 0.053990966513188063 -1 0.24197072451914337 0 0.3989422804014327 1 0.24197072451914337 2 0.053990966513188063 3 0.0044318484119380075 4 0.00013383022576488537 5 1.4867195147342979e-006 6 6.0758828498232861e-009 7 9.1347204083645953e-012 8 5.0522710835368927e-015 9 1.0279773571668917e-018 10 7.6945986267064199e-023 Standard normal mean = 0, standard deviation = 1 Integral (area under the curve) from - infinity up to z. z CDF -10 7.6198530241605945e-024 -9 1.1285884059538422e-019 -8 6.2209605742718204e-016 -7 1.279812543885835e-012 -6 9.865876450377014e-010 -5 2.8665157187919455e-007 -4 3.1671241833119972e-005 -3 0.0013498980316300957 -2 0.022750131948179216 -1 0.15865525393145705 0 0.5 1 0.84134474606854293 2 0.97724986805182079 3 0.9986501019683699 4 0.99996832875816688 5 0.99999971334842808 6 0.9999999990134123 7 0.99999999999872013 8 0.99999999999999933 9 1 10 1 Standard normal distribution, mean = 0, standard deviation = 1 maxdigits_10 is 21, digits10 is 18 Probability distribution function values z PDF -10 7.69459862670641993759e-023 -9 1.0279773571668916523e-018 -8 5.05227108353689273243e-015 -7 9.13472040836459525705e-012 -6 6.07588284982328608733e-009 -5 1.48671951473429788965e-006 -4 0.00013383022576488536764 -3 0.00443184841193800752729 -2 0.0539909665131880628364 -1 0.241970724519143365328 0 0.398942280401432702863 1 0.241970724519143365328 2 0.0539909665131880628364 3 0.00443184841193800752729 4 0.00013383022576488536764 5 1.48671951473429788965e-006 6 6.07588284982328608733e-009 7 9.13472040836459525705e-012 8 5.05227108353689273243e-015 9 1.0279773571668916523e-018 10 7.69459862670641993759e-023 Standard normal mean = 0, standard deviation = 1 Integral (area under the curve) from - infinity up to z. z CDF -10 7.61985302416059451083e-024 -9 1.12858840595384222719e-019 -8 6.22096057427182035917e-016 -7 1.279812543885834962e-012 -6 9.86587645037701399241e-010 -5 2.86651571879194547129e-007 -4 3.16712418331199717608e-005 -3 0.00134989803163009566139 -2 0.0227501319481792155242 -1 0.158655253931457046468 0 0.5 1 0.841344746068542925777 2 0.977249868051820791415 3 0.998650101968369896532 4 0.999968328758166880021 5 0.999999713348428076465 6 0.999999999013412299576 7 0.999999999998720134897 8 0.999999999999999333866 9 1 10 1 Standard normal distribution, mean = 0, standard deviation = 1 maxdigits_10 is 36, digits10 is 34 Probability distribution function values z PDF -10 7.69459862670641993759264402330435296e-023 -9 1.02797735716689165230378750485667109e-018 -8 5.0522710835368927324337437844893081e-015 -7 9.13472040836459525705208369548147081e-012 -6 6.07588284982328608733411870229841611e-009 -5 1.48671951473429788965346931561839483e-006 -4 0.00013383022576488536764006964663309418 -3 0.00443184841193800752728870762098267733 -2 0.0539909665131880628363703067407186609 -1 0.241970724519143365327522587904240936 0 0.398942280401432702863218082711682655 1 0.241970724519143365327522587904240936 2 0.0539909665131880628363703067407186609 3 0.00443184841193800752728870762098267733 4 0.00013383022576488536764006964663309418 5 1.48671951473429788965346931561839483e-006 6 6.07588284982328608733411870229841611e-009 7 9.13472040836459525705208369548147081e-012 8 5.0522710835368927324337437844893081e-015 9 1.02797735716689165230378750485667109e-018 10 7.69459862670641993759264402330435296e-023 Standard normal mean = 0, standard deviation = 1 Integral (area under the curve) from - infinity up to z. z CDF -10 7.61985302416059451083278826816793623e-024 -9 1.1285884059538422271881384555435713e-019 -8 6.22096057427182035917417257601387863e-016 -7 1.27981254388583496200054074948511201e-012 -6 9.86587645037701399241244820583623953e-010 -5 2.86651571879194547128505464808623238e-007 -4 3.16712418331199717608064048146587766e-005 -3 0.001349898031630095661392854111682027 -2 0.0227501319481792155241528519127314212 -1 0.158655253931457046467912164189328905 0 0.5 1 0.841344746068542925776512220181757584 2 0.977249868051820791414741051994496956 3 0.998650101968369896532351503992686048 4 0.999968328758166880021462930017150939 5 0.999999713348428076464813329948810861 6 0.999999999013412299575520592043176293 7 0.999999999998720134897212119540199637 8 0.999999999999999333866185224906075746 9 1 10 1 Standard normal distribution, mean = 0, standard deviation = 1 maxdigits_10 is 36, digits10 is 34 Probability distribution function values z PDF -10 7.69459862670641993759264402330435296e-023 -9 1.02797735716689165230378750485667109e-018 -8 5.0522710835368927324337437844893081e-015 -7 9.13472040836459525705208369548147081e-012 -6 6.07588284982328608733411870229841611e-009 -5 1.48671951473429788965346931561839483e-006 -4 0.00013383022576488536764006964663309418 -3 0.00443184841193800752728870762098267733 -2 0.0539909665131880628363703067407186609 -1 0.241970724519143365327522587904240936 0 0.398942280401432702863218082711682655 1 0.241970724519143365327522587904240936 2 0.0539909665131880628363703067407186609 3 0.00443184841193800752728870762098267733 4 0.00013383022576488536764006964663309418 5 1.48671951473429788965346931561839483e-006 6 6.07588284982328608733411870229841611e-009 7 9.13472040836459525705208369548147081e-012 8 5.0522710835368927324337437844893081e-015 9 1.02797735716689165230378750485667109e-018 10 7.69459862670641993759264402330435296e-023 Standard normal mean = 0, standard deviation = 1 Integral (area under the curve) from - infinity up to z. z CDF -10 7.61985302416059451083278826816793623e-024 -9 1.1285884059538422271881384555435713e-019 -8 6.22096057427182035917417257601387863e-016 -7 1.27981254388583496200054074948511201e-012 -6 9.86587645037701399241244820583623953e-010 -5 2.86651571879194547128505464808623238e-007 -4 3.16712418331199717608064048146587766e-005 -3 0.001349898031630095661392854111682027 -2 0.0227501319481792155241528519127314212 -1 0.158655253931457046467912164189328905 0 0.5 1 0.841344746068542925776512220181757584 2 0.977249868051820791414741051994496956 3 0.998650101968369896532351503992686048 4 0.999968328758166880021462930017150939 5 0.999999713348428076464813329948810861 6 0.999999999013412299575520592043176293 7 0.999999999998720134897212119540199637 8 0.999999999999999333866185224906075746 9 1 10 1 MSVC 2013 64-bit 1> 1> Example: Normal distribution tables. 1> 1> Standard normal distribution, mean = 0, standard deviation = 1 1> maxdigits_10 is 9, digits10 is 6 1> Probability distribution function values 1> z PDF 1> -10 7.69459863e-023 1> -9 1.02797736e-018 1> -8 5.05227108e-015 1> -7 9.13472041e-012 1> -6 6.07588285e-009 1> -5 1.48671951e-006 1> -4 0.000133830226 1> -3 0.00443184841 1> -2 0.0539909665 1> -1 0.241970725 1> 0 0.39894228 1> 1 0.241970725 1> 2 0.0539909665 1> 3 0.00443184841 1> 4 0.000133830226 1> 5 1.48671951e-006 1> 6 6.07588285e-009 1> 7 9.13472041e-012 1> 8 5.05227108e-015 1> 9 1.02797736e-018 1> 10 7.69459863e-023 1> Standard normal mean = 0, standard deviation = 1 1> Integral (area under the curve) from - infinity up to z. 1> z CDF 1> -10 7.61985302e-024 1> -9 1.12858841e-019 1> -8 6.22096057e-016 1> -7 1.27981254e-012 1> -6 9.86587645e-010 1> -5 2.86651572e-007 1> -4 3.16712418e-005 1> -3 0.00134989803 1> -2 0.0227501319 1> -1 0.158655254 1> 0 0.5 1> 1 0.841344746 1> 2 0.977249868 1> 3 0.998650102 1> 4 0.999968329 1> 5 0.999999713 1> 6 0.999999999 1> 7 1 1> 8 1 1> 9 1 1> 10 1 1> 1> Standard normal distribution, mean = 0, standard deviation = 1 1> maxdigits_10 is 17, digits10 is 15 1> Probability distribution function values 1> z PDF 1> -10 7.6945986267064199e-023 1> -9 1.0279773571668917e-018 1> -8 5.0522710835368927e-015 1> -7 9.1347204083645953e-012 1> -6 6.0758828498232861e-009 1> -5 1.4867195147342979e-006 1> -4 0.00013383022576488537 1> -3 0.0044318484119380075 1> -2 0.053990966513188063 1> -1 0.24197072451914337 1> 0 0.3989422804014327 1> 1 0.24197072451914337 1> 2 0.053990966513188063 1> 3 0.0044318484119380075 1> 4 0.00013383022576488537 1> 5 1.4867195147342979e-006 1> 6 6.0758828498232861e-009 1> 7 9.1347204083645953e-012 1> 8 5.0522710835368927e-015 1> 9 1.0279773571668917e-018 1> 10 7.6945986267064199e-023 1> Standard normal mean = 0, standard deviation = 1 1> Integral (area under the curve) from - infinity up to z. 1> z CDF 1> -10 7.6198530241605813e-024 1> -9 1.1285884059538408e-019 1> -8 6.2209605742718292e-016 1> -7 1.2798125438858352e-012 1> -6 9.8658764503770161e-010 1> -5 2.8665157187919439e-007 1> -4 3.1671241833119979e-005 1> -3 0.0013498980316300957 1> -2 0.022750131948179219 1> -1 0.15865525393145707 1> 0 0.5 1> 1 0.84134474606854293 1> 2 0.97724986805182079 1> 3 0.9986501019683699 1> 4 0.99996832875816688 1> 5 0.99999971334842808 1> 6 0.9999999990134123 1> 7 0.99999999999872013 1> 8 0.99999999999999933 1> 9 1 1> 10 1 1> 1> Standard normal distribution, mean = 0, standard deviation = 1 1> maxdigits_10 is 17, digits10 is 15 1> Probability distribution function values 1> z PDF 1> -10 7.6945986267064199e-023 1> -9 1.0279773571668917e-018 1> -8 5.0522710835368927e-015 1> -7 9.1347204083645953e-012 1> -6 6.0758828498232861e-009 1> -5 1.4867195147342979e-006 1> -4 0.00013383022576488537 1> -3 0.0044318484119380075 1> -2 0.053990966513188063 1> -1 0.24197072451914337 1> 0 0.3989422804014327 1> 1 0.24197072451914337 1> 2 0.053990966513188063 1> 3 0.0044318484119380075 1> 4 0.00013383022576488537 1> 5 1.4867195147342979e-006 1> 6 6.0758828498232861e-009 1> 7 9.1347204083645953e-012 1> 8 5.0522710835368927e-015 1> 9 1.0279773571668917e-018 1> 10 7.6945986267064199e-023 1> Standard normal mean = 0, standard deviation = 1 1> Integral (area under the curve) from - infinity up to z. 1> z CDF 1> -10 7.6198530241605813e-024 1> -9 1.1285884059538408e-019 1> -8 6.2209605742718292e-016 1> -7 1.2798125438858352e-012 1> -6 9.8658764503770161e-010 1> -5 2.8665157187919439e-007 1> -4 3.1671241833119979e-005 1> -3 0.0013498980316300957 1> -2 0.022750131948179219 1> -1 0.15865525393145707 1> 0 0.5 1> 1 0.84134474606854293 1> 2 0.97724986805182079 1> 3 0.9986501019683699 1> 4 0.99996832875816688 1> 5 0.99999971334842808 1> 6 0.9999999990134123 1> 7 0.99999999999872013 1> 8 0.99999999999999933 1> 9 1 1> 10 1 */