mirror of
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489 lines
20 KiB
C++
489 lines
20 KiB
C++
// Copyright John Maddock 2014
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// Copyright Christopher Kormanyos 2014.
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// Copyright Paul A. Bristow 2016.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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// Contains Quickbook snippets as C++ comments - do not remove.
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/*
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`This example shows use of a specified-width floating-point typedef
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to evaluate a moderately complex math function using
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[@http://en.wikipedia.org/wiki/Double-precision_floating-point_format IEEE754 64-bit double-precision].
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about 15 decimal digits `std::numeric_limits<boost::float64_t>::digits10`,
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(but never exceeding 17 decimal digits `std::numeric_limits<boost::float64_t>::max_digits10`).
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The Jahnke-Emden lambda function is described at
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Weisstein, Eric W. "Lambda Function." From MathWorld--A Wolfram Web Resource.
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http://mathworld.wolfram.com/LambdaFunction.html
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E. Jahnke and F. Emden, "Tables of Functions with Formulae and Curves,"
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Dover, New York, 4th ed., (1945), pages 180-188.
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*/
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//[cstdfloat_example_1
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#include <boost/cstdfloat.hpp> // For float_64_t, float128_t. Must be first include!
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#include <cmath> // for pow function.
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#include <boost/math/special_functions.hpp> // For gamma function.
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//] [/cstdfloat_example_1]
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#include <boost/type_traits/is_same.hpp>
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#include <iostream>
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/*!
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Function max_digits10
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Returns maximum number of possibly significant decimal digits for a floating-point type FPT,
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even for older compilers/standard libraries that
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lack support for std::numeric_limits<FPT>::max_digits10,
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when the Kahan formula 2 + binary_digits * 0.3010 is used instead.
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Also provides the correct result for Visual Studio 2010
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(where the value 8 provided for float is wrong).
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*/
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namespace boost
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{
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template <typename FPT>
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const int max_digits10()
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{
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// Since max_digits10 is not defined (or wrong) on older systems, define a local max_digits10.
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// Usage: int m = max_digits10<boost::float64_t>();
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const int m =
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#if (defined BOOST_NO_CXX11_NUMERIC_LIMITS) || (_MSC_VER == 1600) // is wrongly 8 not 9 for VS2010.
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2 + std::numeric_limits<FPT>::digits * 3010/10000;
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#else
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std::numeric_limits<FPT>::max_digits10;
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#endif
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return m;
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}
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} // namespace boost
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//`Define the Jahnke-Emden_lambda function.
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//[cstdfloat_example_2
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boost::float64_t jahnke_emden_lambda(boost::float64_t v, boost::float64_t x)
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{
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const boost::float64_t gamma_v_plus_one = boost::math::tgamma(v + 1);
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const boost::float64_t x_half_pow_v = std::pow(x /2, v);
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return gamma_v_plus_one * boost::math::cyl_bessel_j(x, v) / x_half_pow_v;
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}
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//] [/cstdfloat_example_2]
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int main()
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{
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std::cout.setf(std::ios::showpoint); // show all significant trailing zeros.
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long double p = 1.L;
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//std::cout.precision(std::numeric_limits<long double>::digits10);
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std::cout << "pi = " << p << std::endl;
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//[cstdfloat_example_3
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//`Ensure that all possibly significant digits (17) including trailing zeros are shown.
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std::cout.precision(std::numeric_limits<boost::float64_t>::max_digits10);
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std::cout.setf(std::ios::showpoint); // Show trailing zeros.
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try
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{ // Always use try'n'catch blocks to ensure any error messages are displayed.
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// Evaluate and display an evaluation of the Jahnke-Emden lambda function:
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boost::float64_t v = 1.;
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boost::float64_t x = 1.;
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std::cout << jahnke_emden_lambda(v, x) << std::endl; // 0.88010117148986700
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//] [/cstdfloat_example_3]
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// We can show some evaluations with various precisions:
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{ // float64_t
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for (int i = 0; i < 10; i++)
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{
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std::cout << std::setprecision(2) << boost::float64_t(i) << ' '
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<< std::setprecision(std::numeric_limits<boost::float64_t>::max_digits10)
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<< jahnke_emden_lambda(boost::float64_t(i), v) << std::endl; //
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}
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}
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{ // floatmax_t = the maximum available on this platform.
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for (int i = 0; i < 10; i++)
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{
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std::cout << std::setprecision(2) << boost::floatmax_t(i) << ' '
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<< std::setprecision(std::numeric_limits<boost::floatmax_t>::max_digits10)
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<< jahnke_emden_lambda(boost::floatmax_t(i), v) << std::endl; //
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}
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}
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// Show the precision of long double (this might be 64, 80 or 128 bits).
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std::cout << "Floating-point type long double is available with:" << std::endl;
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std::cout << " std::numeric_limits<long double>::digits10 == "
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<< std::numeric_limits<long double>::digits10 << std::endl; // 18
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std::cout << " std::numeric_limits<long double>::max_digits10 == "
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<< std::numeric_limits<long double>::max_digits10 << std::endl; // 21
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long double p = boost::math::constants::pi<double>();
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std::cout.precision(std::numeric_limits<long double>::digits10);
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std::cout << "pi = " << p << std::endl;
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//[cstdfloat_constant_2
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//`These allow floating-point [*constants of at least the specified width] to be declared:
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// Declare Archimedes' constant using float32_t with approximately 7 decimal digits of precision.
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static const boost::float32_t pi = BOOST_FLOAT32_C(3.1415926536);
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// Declare the Euler-gamma constant with approximately 15 decimal digits of precision.
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static const boost::float64_t euler =
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BOOST_FLOAT64_C(0.57721566490153286060651209008240243104216);
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// Declare the Golden Ratio constant with the maximum decimal digits of precision that the platform supports.
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static const boost::floatmax_t golden_ratio =
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BOOST_FLOATMAX_C(1.61803398874989484820458683436563811772);
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//] [/cstdfloat_constant_2]
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// http://www.boost.org/doc/libs/1_55_0/libs/multiprecision/doc/html/boost_multiprecision/tut/floats/float128.html
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//[cstdfloat_constant_1
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// Display the constant pi to the maximum available precision.
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boost::floatmax_t pi_max = boost::math::constants::pi<boost::floatmax_t>();
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std::cout.precision(std::numeric_limits<boost::floatmax_t>::digits10);
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std::cout << "Most precise pi = " << pi_max << std::endl;
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// If floatmax_t is float_128_t, then
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// Most precise pi = 3.141592653589793238462643383279503
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//] [/cstdfloat_constant_1]
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// Test all the floating-point precisions in turn, and if they are available
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// then display how many decimal digits of precision.
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#ifdef BOOST_FLOAT16_C
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std::cout << "Floating-point type boost::float16_t is available." << std::endl;
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#else
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std::cout << "Floating-point type boost::float16_t is NOT available." << std::endl;
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#endif
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#ifdef BOOST_FLOAT32_C
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std::cout << "Floating-point type boost::float32_t is available." << std::endl;
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std::cout << " std::numeric_limits<boost::float32_t>::digits10 == "
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<< std::numeric_limits<boost::float32_t>::digits10 << std::endl;
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std::cout << " std::numeric_limits<boost::float32_t>::max_digits10 == "
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<< std::numeric_limits<boost::float32_t>::max_digits10 << std::endl;
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#else
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std::cout << "Floating-point type boost::float32_t is NOT available." << std::endl;
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#endif
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#ifdef BOOST_FLOAT64_C
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std::cout << "Floating-point type boost::float64_t is available." << std::endl;
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std::cout << " std::numeric_limits<boost::float64_t>::digits10 == "
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<< std::numeric_limits<boost::float64_t>::digits10 << std::endl;
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std::cout << " std::numeric_limits<boost::float64_t>::max_digits10 == "
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<< std::numeric_limits<boost::float64_t>::max_digits10 << std::endl;
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#else
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std::cout << "Floating-point type boost::float64_t is NOT available." << std::endl;
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#endif
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#ifdef BOOST_FLOAT80_C
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std::cout << "Floating-point type boost::float80_t is available." << std::endl;
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std::cout << " std::numeric_limits<boost::float80_t>::digits10 == "
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<< std::numeric_limits<boost::float80_t>::digits10 << std::endl;
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std::cout << " std::numeric_limits<boost::float80_t>::max_digits10 == "
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<< std::numeric_limits<boost::float80_t>::max_digits10 << std::endl;
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#else
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std::cout << "Floating-point type boost::float80_t is NOT available." << std::endl;
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#endif
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#ifdef BOOST_FLOAT128_C
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std::cout << "Floating-point type boost::float128_t is available." << std::endl;
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std::cout << " std::numeric_limits<boost::float128_t>::digits10 == "
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<< std::numeric_limits<boost::float128_t>::digits10 << std::endl;
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std::cout << " std::numeric_limits<boost::float128_t>::max_digits10 == "
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<< std::numeric_limits<boost::float128_t>::max_digits10 << std::endl;
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#else
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std::cout << "Floating-point type boost::float128_t is NOT available." << std::endl;
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#endif
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// Show some constants at a precision depending on the available type(s).
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#ifdef BOOST_FLOAT16_C
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std::cout.precision(boost::max_digits10<boost::float16_t>()); // Show all significant decimal digits,
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std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
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std::cout << "BOOST_FLOAT16_C(123.456789012345678901234567890) = "
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<< BOOST_FLOAT16_C(123.456789012345678901234567890) << std::endl;
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// BOOST_FLOAT16_C(123.456789012345678901234567890) = 123.45678901234568
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#endif
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//[floatmax_widths_1
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#ifdef BOOST_FLOAT32_C
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std::cout.precision(boost::max_digits10<boost::float32_t>()); // Show all significant decimal digits,
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std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
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std::cout << "BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = "
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<< BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) << std::endl;
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// BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = 123.456787
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#endif
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//] [/floatmax_widths_1]
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#ifdef BOOST_FLOAT64_C
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std::cout.precision(boost::max_digits10<boost::float64_t>()); // Show all significant decimal digits,
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std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
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std::cout << "BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = "
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<< BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) << std::endl;
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// BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = 123.45678901234568
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#endif
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#ifdef BOOST_FLOAT80_C
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std::cout.precision(boost::max_digits10<boost::float80_t>()); // Show all significant decimal digits,
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std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
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std::cout << "BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = "
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<< BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) << std::endl;
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// BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = 123.456789012345678903
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#endif
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#ifdef BOOST_FLOAT128_C
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std::cout.precision(boost::max_digits10<boost::float128_t>()); // Show all significant decimal digits,
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std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
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std::cout << "BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = "
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<< BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) << std::endl;
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// BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = 123.456789012345678901234567890123453
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#endif
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/*
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//[floatmax_widths_2
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BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = 123.456787
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BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = 123.45678901234568
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BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = 123.456789012345678903
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BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = 123.456789012345678901234567890123453
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//] [/floatmax_widths_2]
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*/
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// Display the precisions available for floatmax_t
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#ifdef BOOST_FLOATMAX_C
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BOOST_ASSERT(std::numeric_limits<boost::floatmax_t>::is_specialized == true);
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BOOST_ASSERT(std::numeric_limits<boost::floatmax_t>::is_iec559 == true);
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BOOST_ASSERT(BOOST_FLOATMAX_C(0.) == 0);
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std::cout << "floatmax_t " << std::numeric_limits<boost::floatmax_t>::digits << " bits\n" // 113
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<< std::numeric_limits<boost::floatmax_t>::digits10 << " decimal digits\n" // 34
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<< std::numeric_limits<boost::floatmax_t>::max_digits10 << " max_digits\n" // 36
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<< std::numeric_limits<boost::floatmax_t>::radix << " radix\n"
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<< std::endl;
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int significand_bits = std::numeric_limits<boost::floatmax_t>::digits;
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int exponent_max = std::numeric_limits<boost::floatmax_t>::max_exponent;
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int exponent_min = std::numeric_limits<boost::floatmax_t>::min_exponent;
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int exponent_bits = 1 + static_cast<int>(std::log2(std::numeric_limits<boost::floatmax_t>::max_exponent));
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int sign_bits = std::numeric_limits<boost::floatmax_t>::is_signed;
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std::cout << "significand_bits (including one implicit bit)" << significand_bits
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<< ", exponent_bits " << exponent_bits
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<< ", sign_bits " << sign_bits << std::endl;
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// One can compute the total number of bits in the floatmax_t,
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// but probably not at compile time.
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std::cout << "bits = " << significand_bits + exponent_bits + sign_bits -1 << std::endl;
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// -1 to take account of the implicit bit that is not part of the physical layout.
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// One can compare typedefs (but, of course, only those that are defined for the platform in use.)
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std::cout.setf(std::ios::boolalpha);
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std::cout << "double, double: " << std::is_same<double, double>::value << std::endl;
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bool b = boost::is_same<boost::floatmax_t, boost::float64_t>::value;
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std::cout << "boost::is_same<boost::floatmax_t, boost::float64_t>::value; " << b << std::endl;
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std::cout << "floatmax_t, float64_t: "
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<< std::is_same<boost::floatmax_t, boost::float64_t>::value << std::endl;
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/*`So the simplest way of obtaining the total number of bits in the floatmax_t
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is to infer it from the std::numeric_limits<>::digits value.
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This is possible because the type, must be a IEEE754 layout. */
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//[floatmax_1
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const int fpbits =
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(std::numeric_limits<boost::floatmax_t>::digits == 113) ? 128 :
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(std::numeric_limits<boost::floatmax_t>::digits == 64) ? 80 :
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(std::numeric_limits<boost::floatmax_t>::digits == 53) ? 64 :
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(std::numeric_limits<boost::floatmax_t>::digits == 24) ? 32 :
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(std::numeric_limits<boost::floatmax_t>::digits == 11) ? 16 :
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0; // Unknown - not IEEE754 format.
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std::cout << fpbits << " bits." << std::endl;
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//] [/floatmax_1]
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#endif
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}
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catch (std::exception ex)
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{ // Display details about why any exceptions are thrown.
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std::cout << "Thrown exception " << ex.what() << std::endl;
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}
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} // int main()
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/*
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[cstdfloat_output
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GCC 4.8.1 with quadmath
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pi = 1.00000
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0.88010117148986700
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0.0 0.0000000000000000
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1.0 0.88010117148986700
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2.0 4.6137984620549872
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3.0 16.274830009244951
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4.0 -25.360637961042869
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5.0 -1257.9038883512264
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6.0 -12749.592182518225
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7.0 -3020.9830849309437
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8.0 2421897.6013183584
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9.0 45577595.449204877
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0.0 0.00000000000000000000000000000000000
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1.0 0.880101171489866995756301548681221902
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2.0 4.61379846205498722611082484945654869
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3.0 16.2748300092449511566883302293717861
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4.0 -25.3606379610428689375112298876047134
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5.0 -1257.90388835122644195507746189832687
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6.0 -12749.5921825182249449426308274269104
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7.0 -3020.98308493094373261556029319763184
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8.0 2421897.60131835844367742538452148438
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9.0 45577595.4492048770189285278320312500
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Floating-point type long double is available with:
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std::numeric_limits<long double>::digits10 == 18
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std::numeric_limits<long double>::max_digits10 == 21
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pi = 3.14159265358979312
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Most precise pi = 3.141592653589793238462643383279503
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Floating-point type boost::float16_t is NOT available.
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Floating-point type boost::float32_t is available.
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std::numeric_limits<boost::float32_t>::digits10 == 6
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std::numeric_limits<boost::float32_t>::max_digits10 == 9
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Floating-point type boost::float64_t is available.
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std::numeric_limits<boost::float64_t>::digits10 == 15
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std::numeric_limits<boost::float64_t>::max_digits10 == 17
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Floating-point type boost::float80_t is available.
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std::numeric_limits<boost::float80_t>::digits10 == 18
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std::numeric_limits<boost::float80_t>::max_digits10 == 21
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Floating-point type boost::float128_t is available.
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std::numeric_limits<boost::float128_t>::digits10 == 34
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std::numeric_limits<boost::float128_t>::max_digits10 == 36
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BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = 123.456787
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BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = 123.45678901234568
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BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = 123.456789012345678903
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BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = 123.456789012345678901234567890123453
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floatmax_t 113 bits
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34 decimal digits
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36 max_digits
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2 radix
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significand_bits (including one implicit bit)113, exponent_bits 15, sign_bits 1
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bits = 128
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double, double: true
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boost::is_same<boost::floatmax_t, boost::float64_t>::value; false
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floatmax_t, float64_t: false
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128 bits.
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RUN SUCCESSFUL (total time: 53ms)
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GCC 6.1.1
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pi = 1.00000
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0.88010117148986700
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0.0 0.0000000000000000
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1.0 0.88010117148986700
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2.0 4.6137984620549872
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3.0 16.274830009244951
|
|
4.0 -25.360637961042869
|
|
5.0 -1257.9038883512264
|
|
6.0 -12749.592182518225
|
|
7.0 -3020.9830849309437
|
|
8.0 2421897.6013183584
|
|
9.0 45577595.449204877
|
|
0.0 0.00000000000000000000000000000000000
|
|
1.0 0.880101171489866995756301548681221902
|
|
2.0 4.61379846205498722611082484945654869
|
|
3.0 16.2748300092449511566883302293717861
|
|
4.0 -25.3606379610428689375112298876047134
|
|
5.0 -1257.90388835122644195507746189832687
|
|
6.0 -12749.5921825182249449426308274269104
|
|
7.0 -3020.98308493094373261556029319763184
|
|
8.0 2421897.60131835844367742538452148438
|
|
9.0 45577595.4492048770189285278320312500
|
|
Floating-point type long double is available with:
|
|
std::numeric_limits<long double>::digits10 == 18
|
|
std::numeric_limits<long double>::max_digits10 == 21
|
|
pi = 3.14159265358979312
|
|
Most precise pi = 3.14159265358979323846264338327950
|
|
Floating-point type boost::float16_t is NOT available.
|
|
Floating-point type boost::float32_t is available.
|
|
std::numeric_limits<boost::float32_t>::digits10 == 6
|
|
std::numeric_limits<boost::float32_t>::max_digits10 == 9
|
|
Floating-point type boost::float64_t is available.
|
|
std::numeric_limits<boost::float64_t>::digits10 == 15
|
|
std::numeric_limits<boost::float64_t>::max_digits10 == 17
|
|
Floating-point type boost::float80_t is available.
|
|
std::numeric_limits<boost::float80_t>::digits10 == 18
|
|
std::numeric_limits<boost::float80_t>::max_digits10 == 21
|
|
Floating-point type boost::float128_t is available.
|
|
std::numeric_limits<boost::float128_t>::digits10 == 33
|
|
std::numeric_limits<boost::float128_t>::max_digits10 == 36
|
|
BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = 123.456787
|
|
BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = 123.45678901234568
|
|
BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = 123.456789012345678903
|
|
BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = 123.456789012345678901234567890123453
|
|
floatmax_t 113 bits
|
|
33 decimal digits
|
|
36 max_digits
|
|
2 radix
|
|
|
|
significand_bits (including one implicit bit)113, exponent_bits 15, sign_bits 1
|
|
bits = 128
|
|
double, double: true
|
|
boost::is_same<boost::floatmax_t, boost::float64_t>::value; false
|
|
floatmax_t, float64_t: false
|
|
128 bits.
|
|
|
|
|
|
MSVC 2013 64-bit
|
|
|
|
1> pi = 1.00000
|
|
1> 0.88010117148986700
|
|
1> 0.00 0.00000000000000000
|
|
1> 1.0 0.88010117148986700
|
|
1> 2.0 4.6137984620549854
|
|
1> 3.0 16.274830009244948
|
|
1> 4.0 -25.360637961042869
|
|
1> 5.0 -1257.9038883512258
|
|
1> 6.0 -12749.592182518225
|
|
1> 7.0 -3020.9830849309396
|
|
1> 8.0 2421897.6013183575
|
|
1> 9.0 45577595.449204892
|
|
1> 0.00 0.00000000000000000
|
|
1> 1.0 0.88010117148986700
|
|
1> 2.0 4.6137984620549854
|
|
1> 3.0 16.274830009244948
|
|
1> 4.0 -25.360637961042869
|
|
1> 5.0 -1257.9038883512258
|
|
1> 6.0 -12749.592182518225
|
|
1> 7.0 -3020.9830849309396
|
|
1> 8.0 2421897.6013183575
|
|
1> 9.0 45577595.449204892
|
|
1> Floating-point type long double is available with:
|
|
1> std::numeric_limits<long double>::digits10 == 15
|
|
1> std::numeric_limits<long double>::max_digits10 == 17
|
|
1> pi = 3.14159265358979
|
|
1> Most precise pi = 3.14159265358979
|
|
1> Floating-point type boost::float16_t is NOT available.
|
|
1> Floating-point type boost::float32_t is available.
|
|
1> std::numeric_limits<boost::float32_t>::digits10 == 6
|
|
1> std::numeric_limits<boost::float32_t>::max_digits10 == 9
|
|
1> Floating-point type boost::float64_t is available.
|
|
1> std::numeric_limits<boost::float64_t>::digits10 == 15
|
|
1> std::numeric_limits<boost::float64_t>::max_digits10 == 17
|
|
1> Floating-point type boost::float80_t is NOT available.
|
|
1> Floating-point type boost::float128_t is NOT available.
|
|
1> BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = 123.456787
|
|
1> BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = 123.45678901234568
|
|
1> floatmax_t 53 bits
|
|
1> 15 decimal digits
|
|
1> 17 max_digits
|
|
1> 2 radix
|
|
1>
|
|
1> significand_bits (including one implicit bit)53, exponent_bits 11, sign_bits 1
|
|
1> bits = 64
|
|
1> double, double: true
|
|
1> boost::is_same<boost::floatmax_t, boost::float64_t>::value; true
|
|
1> floatmax_t, float64_t: true
|
|
1> 64 bits.
|
|
] [/cstdfloat_output]
|
|
|
|
|
|
*/
|
|
|