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257 lines
9.4 KiB
C++
257 lines
9.4 KiB
C++
// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// Copyright Paul A. Bristow 2013.
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// Copyright Christopher Kormanyos 2012, 2013.
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// Copyright John Maddock 2013.
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// This file is written to be included from a Quickbook .qbk document.
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// It can be compiled by the C++ compiler, and run. Any output can
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// also be added here as comment or included or pasted in elsewhere.
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// Caution: this file contains Quickbook markup as well as code
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// and comments: don't change any of the special comment markups!
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#ifdef _MSC_VER
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# pragma warning (disable : 4996) // -D_SCL_SECURE_NO_WARNINGS.
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#endif
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//[fft_sines_table_example_1
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/*`[h5 Using Boost.Multiprecision to generate a high-precision array of sin coefficents for use with FFT.]
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The Boost.Multiprecision library can be used for computations requiring precision
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exceeding that of standard built-in types such as `float`, `double`
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and `long double`. For extended-precision calculations, Boost.Multiprecision
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supplies a template data type called `cpp_dec_float`. The number of decimal
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digits of precision is fixed at compile-time via template parameter.
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To use these floating-point types and constants, we need some includes:
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*/
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#include <boost/math/constants/constants.hpp>
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// using boost::math::constants::pi;
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#include <boost/multiprecision/cpp_dec_float.hpp>
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// using boost::multiprecision::cpp_dec_float
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#include <iostream>
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#include <limits>
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#include <vector>
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#include <algorithm>
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#include <iomanip>
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#include <iterator>
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#include <fstream>
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/*`Define a text string which is a C++ comment with the program licence, copyright etc.
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You could of course, tailor this to your needs, including your copyright claim.
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There are versions of `array` provided by Boost.Array in `boost::array` or
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the C++11 std::array, but since not all platforms provide C++11 support,
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this program provides the Boost version as fallback.
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*/
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static const char* prolog =
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{
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"// Use, modification and distribution are subject to the\n"
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"// Boost Software License, Version 1.0.\n"
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"// (See accompanying file LICENSE_1_0.txt\n"
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"// or copy at ""http://www.boost.org/LICENSE_1_0.txt)\n\n"
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"// Copyright ???? 2013.\n\n"
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"// Use boost/array if std::array (C++11 feature) is not available.\n"
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"#ifdef BOOST_NO_CXX11_HDR_ARRAY\n"
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"#include <boost/array/array.hpp>\n"
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"#else\n"
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"#include <array>\n"
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"#endif\n\n"
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};
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using boost::multiprecision::cpp_dec_float_50;
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using boost::math::constants::pi;
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// VS 2010 (wrongly) requires these at file scope, not local scope in `main`.
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// This program also requires `-std=c++11` option to compile using Clang and GCC.
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int main()
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{
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/*`One often needs to compute tables of numbers in mathematical software.
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A fast Fourier transform (FFT), for example, may use a table of the values of
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sin(([pi]/2[super n]) in its implementation details. In order to maximize the precision in
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the FFT implementation, the precision of the tabulated trigonometric values
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should exceed that of the built-in floating-point type used in the FFT.
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The sample below computes a table of the values of sin([pi]/2[super n])
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in the range 1 <= n <= 31.
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This program makes use of, among other program elements, the data type
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`boost::multiprecision::cpp_dec_float_50`
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for a precision of 50 decimal digits from Boost.Multiprecision,
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the value of constant [pi] retrieved from Boost.Math,
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guaranteed to be initialized with the very last bit of precision for the type,
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here `cpp_dec_float_50`,
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and a C++11 lambda function combined with `std::for_each()`.
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*/
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/*`define the number of values in the array.
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*/
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std::size_t size = 32U;
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cpp_dec_float_50 p = pi<cpp_dec_float_50>();
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cpp_dec_float_50 p2 = boost::math::constants::pi<cpp_dec_float_50>();
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std::vector <cpp_dec_float_50> sin_values (size);
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unsigned n = 1U;
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// Generate the sine values.
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std::for_each
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(
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sin_values.begin (),
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sin_values.end (),
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[&n](cpp_dec_float_50& y)
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{
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y = sin( pi<cpp_dec_float_50>() / pow(cpp_dec_float_50 (2), n));
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++n;
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}
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);
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/*`Define the floating-point type for the generated file, either built-in
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`double, `float, or `long double`, or a user defined type like `cpp_dec_float_50`.
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*/
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std::string fp_type = "double";
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std::cout << "Generating an `std::array` or `boost::array` for floating-point type: "
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<< fp_type << ". " << std::endl;
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/*`By default, output would only show the standard 6 decimal digits,
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so set precision to show enough significant digits for the chosen floating-point type.
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For `cpp_dec_float_50` is 50. (50 decimal digits should be ample for most applications).
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*/
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std::streamsize precision = std::numeric_limits<cpp_dec_float_50>::digits10;
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// std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
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std::cout << precision << " decimal digits precision. " << std::endl;
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/*`Of course, one could also choose less, for example, 36 would be sufficient
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for the most precise current `long double` implementations using 128-bit.
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In general, it should be a couple of decimal digits more (guard digits) than
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`std::numeric_limits<RealType>::max_digits10` for the target system floating-point type.
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If the implementation does not provide `max_digits10`, the the Kahan formula
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`std::numeric_limits<RealType>::digits * 3010/10000 + 2` can be used instead.
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The compiler will read these values as decimal digits strings and
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use the nearest representation for the floating-point type.
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Now output all the sine table, to a file of your chosen name.
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*/
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const char sines_name[] = "sines.hpp"; // In same directory as .exe
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std::ofstream fout(sines_name, std::ios_base::out); // Creates if no file exists,
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// & uses default overwrite/ ios::replace.
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if (fout.is_open() == false)
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{ // failed to open OK!
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std::cout << "Open file " << sines_name << " failed!" << std::endl;
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return EXIT_FAILURE;
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}
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else
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{
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std::cout << "Open file " << sines_name << " for output OK." << std::endl;
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fout << prolog << "// Table of " << sin_values.size() << " values with "
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<< precision << " decimal digits precision,\n"
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"// generated by program fft_sines_table.cpp.\n" << std::endl;
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fout <<
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"#ifdef BOOST_NO_CXX11_HDR_ARRAY""\n"
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" static const boost::array<double, " << size << "> sines =\n"
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"#else""\n"
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" static const std::array<double, " << size << "> sines =\n"
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"#endif""\n"
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"{{\n"; // 2nd { needed for some GCC compiler versions.
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fout.precision(precision);
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for (unsigned int i = 0U; ;)
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{
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fout << " " << sin_values[i];
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if (i == sin_values.size()-1)
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{ // next is last value.
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fout << "\n}};\n"; // 2nd } needed for some earlier GCC compiler versions.
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break;
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}
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else
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{
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fout << ",\n";
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i++;
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}
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}
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fout.close();
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std::cout << "Close file " << sines_name << " for output OK." << std::endl;
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}
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//`The output file generated can be seen at [@../../example/sines.hpp]
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//] [/fft_sines_table_example_1]
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return EXIT_SUCCESS;
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} // int main()
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/*
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//[fft_sines_table_example_output
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The printed table is:
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1
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0.70710678118654752440084436210484903928483593768847
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0.38268343236508977172845998403039886676134456248563
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0.19509032201612826784828486847702224092769161775195
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0.098017140329560601994195563888641845861136673167501
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0.049067674327418014254954976942682658314745363025753
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0.024541228522912288031734529459282925065466119239451
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0.012271538285719926079408261951003212140372319591769
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0.0061358846491544753596402345903725809170578863173913
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0.003067956762965976270145365490919842518944610213452
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0.0015339801862847656123036971502640790799548645752374
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0.00076699031874270452693856835794857664314091945206328
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0.00038349518757139558907246168118138126339502603496474
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0.00019174759731070330743990956198900093346887403385916
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9.5873799095977345870517210976476351187065612851145e-05
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4.7936899603066884549003990494658872746866687685767e-05
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2.3968449808418218729186577165021820094761474895673e-05
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1.1984224905069706421521561596988984804731977538387e-05
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5.9921124526424278428797118088908617299871778780951e-06
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2.9960562263346607504548128083570598118251878683408e-06
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1.4980281131690112288542788461553611206917585861527e-06
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7.4901405658471572113049856673065563715595930217207e-07
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3.7450702829238412390316917908463317739740476297248e-07
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1.8725351414619534486882457659356361712045272098287e-07
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9.3626757073098082799067286680885620193236507169473e-08
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4.681337853654909269511551813854009695950362701667e-08
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2.3406689268274552759505493419034844037886207223779e-08
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1.1703344634137277181246213503238103798093456639976e-08
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5.8516723170686386908097901008341396943900085051757e-09
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2.9258361585343193579282304690689559020175857150074e-09
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1.4629180792671596805295321618659637103742615227834e-09
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*/
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//] [/fft_sines_table_example_output]
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//[fft_sines_table_example_check
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/*`
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The output can be copied as text and readily integrated into a given source
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code. Alternatively, the output can be written to a text or even be used
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within a self-written automatic code generator as this example.
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A computer algebra system can be used to verify the results obtained from
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Boost.Math and Boost.Multiprecision. For example, the __Mathematica
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computer algebra system can obtain a similar table with the command:
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Table[N[Sin[Pi / (2^n)], 50], {n, 1, 31, 1}]
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The __WolframAlpha computational knowledge engine can also be used to generate
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this table. The same command can be pasted into the compute box.
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*/
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//] [/fft_sines_table_example_check]
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