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// 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>(); // Usually 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
*/
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