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### Using Boost.Multiprecision

All new projects are recommended to use Boost.Multiprecision.

###### Using Boost.Multiprecision `cpp_float` for numerical calculations with high precision.

The Boost.Multiprecision library can be used for computations requiring precision exceeding that of standard built-in types such as float, double and long double. For extended-precision calculations, Boost.Multiprecision supplies a template data type called cpp_dec_float. The number of decimal digits of precision is fixed at compile-time via template parameter.

To use these floating-point types and constants, we need some includes:

    #include <boost/math/constants/constants.hpp>

    #include <boost/multiprecision/cpp_dec_float.hpp>
    // using boost::multiprecision::cpp_dec_float

    #include <iostream>
    #include <limits>

So now we can demonstrate with some trivial calculations:

    int main()
    {

Using `typedef cpp_dec_float_50` hides the complexity of multiprecision to allow us to define variables with 50 decimal digit precision just like built-in `double`.

    using boost::multiprecision::cpp_dec_float_50;

    cpp_dec_float_50 seventh = cpp_dec_float_50(1) / 7;

By default, output would only show the standard 6 decimal digits, so set precision to show all 50 significant digits.

    std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
    std::cout << seventh << std::endl;

which outputs:

    0.14285714285714285714285714285714285714285714285714

We can also use constants, guaranteed to be initialized with the very last bit of precision.

    cpp_dec_float_50 circumference = boost::math::constants::pi<cpp_dec_float_50>() * 2 * seventh;

    std::cout << circumference << std::endl;

which outputs

    0.89759790102565521098932668093700082405633411410717

###### Using Boost.Multiprecision to generate a high-precision array of sin coefficents for use with FFT.

The Boost.Multiprecision library can be used for computations requiring precision exceeding that of standard built-in types such as `float`, `double` and `long double`. For extended-precision calculations, Boost.Multiprecision supplies a template data type called `cpp_dec_float`. The number of decimal digits of precision is fixed at compile-time via template parameter.

To use these floating-point types and constants, we need some includes:

    #include <boost/math/constants/constants.hpp>
    // using boost::math::constants::pi;

    #include <boost/multiprecision/cpp_dec_float.hpp>
    // using boost::multiprecision::cpp_dec_float

    #include <iostream>
    #include <limits>
    #include <vector>
    #include <algorithm>
    #include <iomanip>
    #include <iterator>
    #include <fstream>

Define a text string which is a C++ comment with the program licence, copyright etc. You could of course, tailor this to your needs, including your copyright claim. There are versions of `array` provided by Boost.Array in `boost::array` or the C++11 std::array, but since not all platforms provide C++11 support, this program provides the Boost version as fallback.

    static const char* prolog =
    {
      "// Use, modification and distribution are subject to the\n"
      "// Boost Software License, Version 1.0.\n"
      "// (See accompanying file LICENSE_1_0.txt\n"
      "// or copy at ""http://www.boost.org/LICENSE_1_0.txt)\n\n"

      "// Copyright ???? 2013.\n\n"

      "// Use boost/array if std::array (C++11 feature) is not available.\n"
      "#ifdef  BOOST_NO_CXX11_HDR_ARRAY\n"
      "#include <boost/array/array.hpp>\n"
      "#else\n"
      "#include <array>\n"
      "#endif\n\n"
    };

    using boost::multiprecision::cpp_dec_float_50;
    using boost::math::constants::pi;
    // VS 2010 (wrongly) requires these at file scope, not local scope in `main`.
    // This program also requires `-std=c++11` option to compile using Clang and GCC.

    int main()
    {

One often needs to compute tables of numbers in mathematical software.

A fast Fourier transform (FFT), for example, may use a table of the values of sin((π/2n) in its implementation details. In order to maximize the precision in the FFT implementation, the precision of the tabulated trigonometric values should exceed that of the built-in floating-point type used in the FFT.

The sample below computes a table of the values of sin(π/2n) in the range 1 <= n <= 31.

This program makes use of, among other program elements, the data type `boost::multiprecision::cpp_dec_float_50` for a precision of 50 decimal digits from Boost.Multiprecision, the value of constant π retrieved from Boost.Math, guaranteed to be initialized with the very last bit of precision for the type, here `cpp_dec_float_50`, and a C++11 lambda function combined with `std::for_each()`.

define the number of values in the array.

    std::size_t size = 32U;
    cpp_dec_float_50 p = pi<cpp_dec_float_50>();
    cpp_dec_float_50 p2 = boost::math::constants::pi<cpp_dec_float_50>();

    std::vector <cpp_dec_float_50> sin_values (size);
    unsigned n = 1U;
    // Generate the sine values.
    std::for_each
    (
      sin_values.begin (),
      sin_values.end (),
      [&n](cpp_dec_float_50& y)
      {
        y = sin( pi<cpp_dec_float_50>() / pow(cpp_dec_float_50 (2), n));
        ++n;
      }
    );

Define the floating-point type for the generated file, either built-in `double,` float, or `long double`, or a user defined type like `cpp_dec_float_50`.

    std::string fp_type = "double";

    std::cout << "Generating an `std::array` or `boost::array` for floating-point type: "
      << fp_type << ". " << std::endl;

By default, output would only show the standard 6 decimal digits, so set precision to show enough significant digits for the chosen floating-point type. For `cpp_dec_float_50` is 50. (50 decimal digits should be ample for most applications).

    std::streamsize precision = std::numeric_limits<cpp_dec_float_50>::digits10;

    //  std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
    std::cout << precision << " decimal digits precision. " << std::endl;

Of course, one could also choose less, for example, 36 would be sufficient for the most precise current `long double` implementations using 128-bit. In general, it should be a couple of decimal digits more (guard digits) than `std::numeric_limits<RealType>::max_digits10` for the target system floating-point type. If the implementation does not provide `max_digits10`, the the Kahan formula `std::numeric_limits<RealType>::digits * 3010/10000 + 2` can be used instead.

The compiler will read these values as decimal digits strings and use the nearest representation for the floating-point type.

Now output all the sine table, to a file of your chosen name.

      const char sines_name[] = "sines.hpp";  // In same directory as .exe

      std::ofstream fout(sines_name, std::ios_base::out);  // Creates if no file exists,
      // & uses default overwrite/ ios::replace.
      if (fout.is_open() == false)
      {  // failed to open OK!
        std::cout << "Open file " << sines_name << " failed!" << std::endl;
        return EXIT_FAILURE;
      }
      else
      {
        std::cout << "Open file " << sines_name << " for output OK." << std::endl;
        fout << prolog << "// Table of " << sin_values.size() << " values with "
          << precision << " decimal digits precision,\n"
          "// generated by program fft_sines_table.cpp.\n" << std::endl;

        fout <<
    "#ifdef BOOST_NO_CXX11_HDR_ARRAY""\n"
     "  static const boost::array<double, " << size << "> sines =\n"
    "#else""\n"
    "  static const std::array<double, " << size << "> sines =\n"
    "#endif""\n"
        "{{\n"; // 2nd { needed for some GCC compiler versions.
        fout.precision(precision);

        for (unsigned int i = 0U; ;)
        {
          fout << "  " << sin_values[i];
          if (i == sin_values.size()-1)
          { // next is last value.
            fout << "\n}};\n"; // 2nd } needed for some earlier GCC compiler versions.
            break;
          }
          else
          {
            fout << ",\n";
            i++;
          }
        }

        fout.close();
        std::cout << "Close file " << sines_name << " for output OK." << std::endl;

      }

The output file generated can be seen at ../../example/sines.hpp

The table output is:

    The printed table is:

      1
      0.70710678118654752440084436210484903928483593768847
      0.38268343236508977172845998403039886676134456248563
      0.19509032201612826784828486847702224092769161775195
      0.098017140329560601994195563888641845861136673167501
      0.049067674327418014254954976942682658314745363025753
      0.024541228522912288031734529459282925065466119239451
      0.012271538285719926079408261951003212140372319591769
      0.0061358846491544753596402345903725809170578863173913
      0.003067956762965976270145365490919842518944610213452
      0.0015339801862847656123036971502640790799548645752374
      0.00076699031874270452693856835794857664314091945206328
      0.00038349518757139558907246168118138126339502603496474
      0.00019174759731070330743990956198900093346887403385916
      9.5873799095977345870517210976476351187065612851145e-05
      4.7936899603066884549003990494658872746866687685767e-05
      2.3968449808418218729186577165021820094761474895673e-05
      1.1984224905069706421521561596988984804731977538387e-05
      5.9921124526424278428797118088908617299871778780951e-06
      2.9960562263346607504548128083570598118251878683408e-06
      1.4980281131690112288542788461553611206917585861527e-06
      7.4901405658471572113049856673065563715595930217207e-07
      3.7450702829238412390316917908463317739740476297248e-07
      1.8725351414619534486882457659356361712045272098287e-07
      9.3626757073098082799067286680885620193236507169473e-08
      4.681337853654909269511551813854009695950362701667e-08
      2.3406689268274552759505493419034844037886207223779e-08
      1.1703344634137277181246213503238103798093456639976e-08
      5.8516723170686386908097901008341396943900085051757e-09
      2.9258361585343193579282304690689559020175857150074e-09
      1.4629180792671596805295321618659637103742615227834e-09
    */

The output can be copied as text and readily integrated into a given source code. Alternatively, the output can be written to a text or even be used within a self-written automatic code generator as this example.

A computer algebra system can be used to verify the results obtained from Boost.Math and Boost.Multiprecision. For example, the Wolfram Mathematica computer algebra system can obtain a similar table with the command:

    Table[N[Sin[Pi / (2^n)], 50], {n, 1, 31, 1}]

The Wolfram Alpha computational knowledge engine can also be used to generate this table. The same command can be pasted into the compute box.