123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488 |
- // Copyright John Maddock 2014
- // Copyright Christopher Kormanyos 2014.
- // Copyright Paul A. Bristow 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)
- // Contains Quickbook snippets as C++ comments - do not remove.
- /*
- `This example shows use of a specified-width floating-point typedef
- to evaluate a moderately complex math function using
- [@http://en.wikipedia.org/wiki/Double-precision_floating-point_format IEEE754 64-bit double-precision].
- about 15 decimal digits `std::numeric_limits<boost::float64_t>::digits10`,
- (but never exceeding 17 decimal digits `std::numeric_limits<boost::float64_t>::max_digits10`).
- The Jahnke-Emden lambda function is described at
- Weisstein, Eric W. "Lambda Function." From MathWorld--A Wolfram Web Resource.
- http://mathworld.wolfram.com/LambdaFunction.html
- E. Jahnke and F. Emden, "Tables of Functions with Formulae and Curves,"
- Dover, New York, 4th ed., (1945), pages 180-188.
- */
- //[cstdfloat_example_1
- #include <boost/cstdfloat.hpp> // For float_64_t, float128_t. Must be first include!
- #include <cmath> // for pow function.
- #include <boost/math/special_functions.hpp> // For gamma function.
- //] [/cstdfloat_example_1]
- #include <boost/type_traits/is_same.hpp>
- #include <iostream>
- /*!
- 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::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 value 8 provided for float is wrong).
- */
- namespace boost
- {
- template <typename FPT>
- const 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 boost
- //`Define the Jahnke-Emden_lambda function.
- //[cstdfloat_example_2
- boost::float64_t jahnke_emden_lambda(boost::float64_t v, boost::float64_t x)
- {
- const boost::float64_t gamma_v_plus_one = boost::math::tgamma(v + 1);
- const boost::float64_t x_half_pow_v = std::pow(x /2, v);
- return gamma_v_plus_one * boost::math::cyl_bessel_j(x, v) / x_half_pow_v;
- }
- //] [/cstdfloat_example_2]
- int main()
- {
- std::cout.setf(std::ios::showpoint); // show all significant trailing zeros.
- long double p = 1.L;
- //std::cout.precision(std::numeric_limits<long double>::digits10);
- std::cout << "pi = " << p << std::endl;
- //[cstdfloat_example_3
- //`Ensure that all possibly significant digits (17) including trailing zeros are shown.
- std::cout.precision(std::numeric_limits<boost::float64_t>::max_digits10);
- std::cout.setf(std::ios::showpoint); // Show trailing zeros.
- try
- { // Always use try'n'catch blocks to ensure any error messages are displayed.
- // Evaluate and display an evaluation of the Jahnke-Emden lambda function:
- boost::float64_t v = 1.;
- boost::float64_t x = 1.;
- std::cout << jahnke_emden_lambda(v, x) << std::endl; // 0.88010117148986700
- //] [/cstdfloat_example_3]
- // We can show some evaluations with various precisions:
- { // float64_t
- for (int i = 0; i < 10; i++)
- {
- std::cout << std::setprecision(2) << boost::float64_t(i) << ' '
- << std::setprecision(std::numeric_limits<boost::float64_t>::max_digits10)
- << jahnke_emden_lambda(boost::float64_t(i), v) << std::endl; //
- }
- }
- { // floatmax_t = the maximum available on this platform.
- for (int i = 0; i < 10; i++)
- {
- std::cout << std::setprecision(2) << boost::floatmax_t(i) << ' '
- << std::setprecision(std::numeric_limits<boost::floatmax_t>::max_digits10)
- << jahnke_emden_lambda(boost::floatmax_t(i), v) << std::endl; //
- }
- }
- // Show the precision of long double (this might be 64, 80 or 128 bits).
- std::cout << "Floating-point type long double is available with:" << std::endl;
- std::cout << " std::numeric_limits<long double>::digits10 == "
- << std::numeric_limits<long double>::digits10 << std::endl; // 18
- std::cout << " std::numeric_limits<long double>::max_digits10 == "
- << std::numeric_limits<long double>::max_digits10 << std::endl; // 21
- long double p = boost::math::constants::pi<double>();
- std::cout.precision(std::numeric_limits<long double>::digits10);
- std::cout << "pi = " << p << std::endl;
- //[cstdfloat_constant_2
- //`These allow floating-point [*constants of at least the specified width] to be declared:
- // Declare Archimedes' constant using float32_t with approximately 7 decimal digits of precision.
- static const boost::float32_t pi = BOOST_FLOAT32_C(3.1415926536);
- // Declare the Euler-gamma constant with approximately 15 decimal digits of precision.
- static const boost::float64_t euler =
- BOOST_FLOAT64_C(0.57721566490153286060651209008240243104216);
- // Declare the Golden Ratio constant with the maximum decimal digits of precision that the platform supports.
- static const boost::floatmax_t golden_ratio =
- BOOST_FLOATMAX_C(1.61803398874989484820458683436563811772);
- //] [/cstdfloat_constant_2]
- // http://www.boost.org/doc/libs/1_55_0/libs/multiprecision/doc/html/boost_multiprecision/tut/floats/float128.html
- //[cstdfloat_constant_1
- // Display the constant pi to the maximum available precision.
- boost::floatmax_t pi_max = boost::math::constants::pi<boost::floatmax_t>();
- std::cout.precision(std::numeric_limits<boost::floatmax_t>::digits10);
- std::cout << "Most precise pi = " << pi_max << std::endl;
- // If floatmax_t is float_128_t, then
- // Most precise pi = 3.141592653589793238462643383279503
- //] [/cstdfloat_constant_1]
- // Test all the floating-point precisions in turn, and if they are available
- // then display how many decimal digits of precision.
- #ifdef BOOST_FLOAT16_C
- std::cout << "Floating-point type boost::float16_t is available." << std::endl;
- #else
- std::cout << "Floating-point type boost::float16_t is NOT available." << std::endl;
- #endif
- #ifdef BOOST_FLOAT32_C
- std::cout << "Floating-point type boost::float32_t is available." << std::endl;
- std::cout << " std::numeric_limits<boost::float32_t>::digits10 == "
- << std::numeric_limits<boost::float32_t>::digits10 << std::endl;
- std::cout << " std::numeric_limits<boost::float32_t>::max_digits10 == "
- << std::numeric_limits<boost::float32_t>::max_digits10 << std::endl;
- #else
- std::cout << "Floating-point type boost::float32_t is NOT available." << std::endl;
- #endif
- #ifdef BOOST_FLOAT64_C
- std::cout << "Floating-point type boost::float64_t is available." << std::endl;
- std::cout << " std::numeric_limits<boost::float64_t>::digits10 == "
- << std::numeric_limits<boost::float64_t>::digits10 << std::endl;
- std::cout << " std::numeric_limits<boost::float64_t>::max_digits10 == "
- << std::numeric_limits<boost::float64_t>::max_digits10 << std::endl;
- #else
- std::cout << "Floating-point type boost::float64_t is NOT available." << std::endl;
- #endif
- #ifdef BOOST_FLOAT80_C
- std::cout << "Floating-point type boost::float80_t is available." << std::endl;
- std::cout << " std::numeric_limits<boost::float80_t>::digits10 == "
- << std::numeric_limits<boost::float80_t>::digits10 << std::endl;
- std::cout << " std::numeric_limits<boost::float80_t>::max_digits10 == "
- << std::numeric_limits<boost::float80_t>::max_digits10 << std::endl;
- #else
- std::cout << "Floating-point type boost::float80_t is NOT available." << std::endl;
- #endif
- #ifdef BOOST_FLOAT128_C
- std::cout << "Floating-point type boost::float128_t is available." << std::endl;
- std::cout << " std::numeric_limits<boost::float128_t>::digits10 == "
- << std::numeric_limits<boost::float128_t>::digits10 << std::endl;
- std::cout << " std::numeric_limits<boost::float128_t>::max_digits10 == "
- << std::numeric_limits<boost::float128_t>::max_digits10 << std::endl;
- #else
- std::cout << "Floating-point type boost::float128_t is NOT available." << std::endl;
- #endif
- // Show some constants at a precision depending on the available type(s).
- #ifdef BOOST_FLOAT16_C
- std::cout.precision(boost::max_digits10<boost::float16_t>()); // Show all significant decimal digits,
- std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
- std::cout << "BOOST_FLOAT16_C(123.456789012345678901234567890) = "
- << BOOST_FLOAT16_C(123.456789012345678901234567890) << std::endl;
- // BOOST_FLOAT16_C(123.456789012345678901234567890) = 123.45678901234568
- #endif
- //[floatmax_widths_1
- #ifdef BOOST_FLOAT32_C
- std::cout.precision(boost::max_digits10<boost::float32_t>()); // Show all significant decimal digits,
- std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
- std::cout << "BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = "
- << BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) << std::endl;
- // BOOST_FLOAT32_C(123.4567890123456789012345678901234567890) = 123.456787
- #endif
- //] [/floatmax_widths_1]
- #ifdef BOOST_FLOAT64_C
- std::cout.precision(boost::max_digits10<boost::float64_t>()); // Show all significant decimal digits,
- std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
- std::cout << "BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = "
- << BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) << std::endl;
- // BOOST_FLOAT64_C(123.4567890123456789012345678901234567890) = 123.45678901234568
- #endif
- #ifdef BOOST_FLOAT80_C
- std::cout.precision(boost::max_digits10<boost::float80_t>()); // Show all significant decimal digits,
- std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
- std::cout << "BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = "
- << BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) << std::endl;
- // BOOST_FLOAT80_C(123.4567890123456789012345678901234567890) = 123.456789012345678903
- #endif
- #ifdef BOOST_FLOAT128_C
- std::cout.precision(boost::max_digits10<boost::float128_t>()); // Show all significant decimal digits,
- std::cout.setf(std::ios::showpoint); // including all significant trailing zeros.
- std::cout << "BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = "
- << BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) << std::endl;
- // BOOST_FLOAT128_C(123.4567890123456789012345678901234567890) = 123.456789012345678901234567890123453
- #endif
- /*
- //[floatmax_widths_2
- 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_widths_2]
- */
- // Display the precisions available for floatmax_t
- #ifdef BOOST_FLOATMAX_C
- BOOST_ASSERT(std::numeric_limits<boost::floatmax_t>::is_specialized == true);
- BOOST_ASSERT(std::numeric_limits<boost::floatmax_t>::is_iec559 == true);
- BOOST_ASSERT(BOOST_FLOATMAX_C(0.) == 0);
- std::cout << "floatmax_t " << std::numeric_limits<boost::floatmax_t>::digits << " bits\n" // 113
- << std::numeric_limits<boost::floatmax_t>::digits10 << " decimal digits\n" // 34
- << std::numeric_limits<boost::floatmax_t>::max_digits10 << " max_digits\n" // 36
- << std::numeric_limits<boost::floatmax_t>::radix << " radix\n"
- << std::endl;
- int significand_bits = std::numeric_limits<boost::floatmax_t>::digits;
- int exponent_max = std::numeric_limits<boost::floatmax_t>::max_exponent;
- int exponent_min = std::numeric_limits<boost::floatmax_t>::min_exponent;
- int exponent_bits = 1 + static_cast<int>(std::log2(std::numeric_limits<boost::floatmax_t>::max_exponent));
- int sign_bits = std::numeric_limits<boost::floatmax_t>::is_signed;
- std::cout << "significand_bits (including one implicit bit)" << significand_bits
- << ", exponent_bits " << exponent_bits
- << ", sign_bits " << sign_bits << std::endl;
- // One can compute the total number of bits in the floatmax_t,
- // but probably not at compile time.
- std::cout << "bits = " << significand_bits + exponent_bits + sign_bits -1 << std::endl;
- // -1 to take account of the implicit bit that is not part of the physical layout.
- // One can compare typedefs (but, of course, only those that are defined for the platform in use.)
- std::cout.setf(std::ios::boolalpha);
- std::cout << "double, double: " << std::is_same<double, double>::value << std::endl;
- bool b = boost::is_same<boost::floatmax_t, boost::float64_t>::value;
- std::cout << "boost::is_same<boost::floatmax_t, boost::float64_t>::value; " << b << std::endl;
- std::cout << "floatmax_t, float64_t: "
- << std::is_same<boost::floatmax_t, boost::float64_t>::value << std::endl;
- /*`So the simplest way of obtaining the total number of bits in the floatmax_t
- is to infer it from the std::numeric_limits<>::digits value.
- This is possible because the type, must be a IEEE754 layout. */
- //[floatmax_1
- const int fpbits =
- (std::numeric_limits<boost::floatmax_t>::digits == 113) ? 128 :
- (std::numeric_limits<boost::floatmax_t>::digits == 64) ? 80 :
- (std::numeric_limits<boost::floatmax_t>::digits == 53) ? 64 :
- (std::numeric_limits<boost::floatmax_t>::digits == 24) ? 32 :
- (std::numeric_limits<boost::floatmax_t>::digits == 11) ? 16 :
- 0; // Unknown - not IEEE754 format.
- std::cout << fpbits << " bits." << std::endl;
- //] [/floatmax_1]
- #endif
- }
- catch (std::exception ex)
- { // Display details about why any exceptions are thrown.
- std::cout << "Thrown exception " << ex.what() << std::endl;
- }
- } // int main()
- /*
- [cstdfloat_output
- GCC 4.8.1 with quadmath
- pi = 1.00000
- 0.88010117148986700
- 0.0 0.0000000000000000
- 1.0 0.88010117148986700
- 2.0 4.6137984620549872
- 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 = 3.141592653589793238462643383279503
- 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 == 34
- 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
- 34 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.
- RUN SUCCESSFUL (total time: 53ms)
- GCC 6.1.1
- pi = 1.00000
- 0.88010117148986700
- 0.0 0.0000000000000000
- 1.0 0.88010117148986700
- 2.0 4.6137984620549872
- 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]
- */
|