ceph/src/boost/libs/math/example/root_n_finding_algorithms.cpp

   1 // Copyright Paul A. Bristow 2015
   2
   3 // Use, modification and distribution are subject to the
   4 // Boost Software License, Version 1.0.
   5 // (See accompanying file LICENSE_1_0.txt
   6 // or copy at http://www.boost.org/LICENSE_1_0.txt)
   7
   8 // Comparison of finding roots using TOMS748, Newton-Raphson, Halley & Schroder algorithms.
   9 // root_n_finding_algorithms.cpp  Generalised for nth root version.
  10
  11 // http://en.wikipedia.org/wiki/Cube_root
  12
  13 // Note that this file contains Quickbook mark-up as well as code
  14 // and comments, don't change any of the special comment mark-ups!
  15 // This program also writes files in Quickbook tables mark-up format.
  16
  17 #include <boost/cstdlib.hpp>
  18 #include <boost/config.hpp>
  19 #include <boost/array.hpp>
  20 #include <boost/type_traits/is_floating_point.hpp>
  21 #include <boost/math/concepts/real_concept.hpp>
  22 #include <boost/math/tools/roots.hpp>
  23
  24 //using boost::math::policies::policy;
  25 //using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits.
  26 //using boost::math::tools::bracket_and_solve_root;
  27 //using boost::math::tools::toms748_solve;
  28 //using boost::math::tools::halley_iterate;
  29 //using boost::math::tools::newton_raphson_iterate;
  30 //using boost::math::tools::schroder_iterate;
  31
  32 #include <boost/math/special_functions/next.hpp> // For float_distance.
  33 #include <boost/math/special_functions/pow.hpp> // For pow<N>.
  34 #include <boost/math/tools/tuple.hpp> // for tuple and make_tuple.
  35
  36 #include <boost/multiprecision/cpp_bin_float.hpp> // is binary.
  37 using boost::multiprecision::cpp_bin_float_100;
  38 using boost::multiprecision::cpp_bin_float_50;
  39
  40 #include <boost/timer/timer.hpp>
  41 #include <boost/system/error_code.hpp>
  42 #include <boost/preprocessor/stringize.hpp>
  43
  44 // STL
  45 #include <iostream>
  46 #include <iomanip>
  47 #include <string>
  48 #include <vector>
  49 #include <limits>
  50 #include <fstream> // std::ofstream
  51 #include <cmath>
  52 #include <typeinfo> // for type name using typid(thingy).name();
  53
  54 #ifdef __FILE__
  55   std::string sourcefilename = __FILE__;
  56 #else
  57   std::string sourcefilename("");
  58 #endif
  59
  60   std::string chop_last(std::string s)
  61   {
  62      std::string::size_type pos = s.find_last_of("\\/");
  63      if(pos != std::string::npos)
  64         s.erase(pos);
  65      else if(s.empty())
  66         abort();
  67      else
  68         s.erase();
  69      return s;
  70   }
  71
  72   std::string make_root()
  73   {
  74      std::string result;
  75      if(sourcefilename.find_first_of(":") != std::string::npos)
  76      {
  77         result = chop_last(sourcefilename); // lose filename part
  78         result = chop_last(result);   // lose /example/
  79         result = chop_last(result);   // lose /math/
  80         result = chop_last(result);   // lose /libs/
  81      }
  82      else
  83      {
  84         result = chop_last(sourcefilename); // lose filename part
  85         if(result.empty())
  86            result = ".";
  87         result += "/../../..";
  88      }
  89      return result;
  90   }
  91
  92   std::string short_file_name(std::string s)
  93   {
  94      std::string::size_type pos = s.find_last_of("\\/");
  95      if(pos != std::string::npos)
  96         s.erase(0, pos + 1);
  97      return s;
  98   }
  99
 100   std::string boost_root = make_root();
 101
 102
 103 std::string fp_hardware; // Any hardware features like SEE or AVX
 104
 105 const std::string roots_name = "libs/math/doc/roots/";
 106
 107 const std::string full_roots_name(boost_root + "/libs/math/doc/roots/");
 108
 109 const std::size_t nooftypes = 4;
 110 const std::size_t noofalgos = 4;
 111
 112 double digits_accuracy = 1.0; // 1 == maximum possible accuracy.
 113
 114 std::stringstream ss;
 115
 116 std::ofstream fout;
 117
 118 std::vector<std::string> algo_names =
 119 {
 120   "TOMS748", "Newton", "Halley", "Schr'''&#xf6;'''der"
 121 };
 122
 123 std::vector<std::string> names =
 124 {
 125   "float", "double", "long double", "cpp_bin_float50"
 126 };
 127
 128 uintmax_t iters; // Global as value of iterations is not returned.
 129
 130 struct root_info
 131 { // for a floating-point type, float, double ...
 132   std::size_t max_digits10; // for type.
 133   std::string full_typename; // for type from type_id.name().
 134   std::string short_typename; // for type "float", "double", "cpp_bin_float_50" ....
 135   std::size_t bin_digits;  // binary in floating-point type numeric_limits<T>::digits;
 136   int get_digits; // fraction of maximum possible accuracy required.
 137   // = digits * digits_accuracy
 138   // Vector of values (4) for each algorithm, TOMS748, Newton, Halley & Schroder.
 139   //std::vector< boost::int_least64_t> times;  converted to int.
 140   std::vector<int> times; // arbirary units (ticks).
 141   //boost::int_least64_t min_time = std::numeric_limits<boost::int_least64_t>::max(); // Used to normalize times (as int).
 142   std::vector<double> normed_times;
 143   int min_time = (std::numeric_limits<int>::max)(); // Used to normalize times.
 144   std::vector<uintmax_t> iterations;
 145   std::vector<long int> distances;
 146   std::vector<cpp_bin_float_100> full_results;
 147 }; // struct root_info
 148
 149 std::vector<root_info> root_infos;  // One element for each floating-point type used.
 150
 151 inline std::string build_test_name(const char* type_name, const char* test_name)
 152 {
 153   std::string result(BOOST_COMPILER);
 154   result += "|";
 155   result += BOOST_STDLIB;
 156   result += "|";
 157   result += BOOST_PLATFORM;
 158   result += "|";
 159   result += type_name;
 160   result += "|";
 161   result += test_name;
 162 #if defined(_DEBUG) || !defined(NDEBUG)
 163   result += "|";
 164   result += " debug";
 165 #else
 166   result += "|";
 167   result += " release";
 168 #endif
 169   result += "|";
 170   return result;
 171 } // std::string build_test_name
 172
 173 // Algorithms //////////////////////////////////////////////
 174
 175 // No derivatives - using TOMS748 internally.
 176
 177 template <int N, typename T = double>
 178 struct nth_root_functor_noderiv
 179 { //  Nth root of x using only function - no derivatives.
 180   nth_root_functor_noderiv(T const& to_find_root_of) : a(to_find_root_of)
 181   { // Constructor just stores value a to find root of.
 182   }
 183   T operator()(T const& x)
 184   {
 185     using boost::math::pow;
 186     T fx = pow<N>(x) -a; // Difference (estimate x^n - a).
 187     return fx;
 188   }
 189 private:
 190   T a; // to be 'cube_rooted'.
 191 }; // template <int N, class T> struct nth_root_functor_noderiv
 192
 193 template <int N, class T = double>
 194 T nth_root_noderiv(T x)
 195 { // return Nth root of x using bracket_and_solve (using NO derivatives).
 196   using namespace std;  // Help ADL of std functions.
 197   using namespace boost::math::tools; // For bracket_and_solve_root.
 198
 199   typedef double guess_type;
 200
 201   int exponent;
 202   frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
 203   T guess = static_cast<T>(ldexp(static_cast<guess_type>(1.), exponent / N)); // Rough guess is to divide the exponent by n.
 204   //T min = static_cast<T>(ldexp(static_cast<guess_type>(1.) / 2, exponent / N)); // Minimum possible value is half our guess.
 205   //T max = static_cast<T>(ldexp(static_cast<guess_type>(2.), exponent / N)); // Maximum possible value is twice our guess.
 206
 207   T factor = 2; // How big steps to take when searching.
 208
 209   const boost::uintmax_t maxit = 50; // Limit to maximum iterations.
 210   boost::uintmax_t it = maxit; // Initally our chosen max iterations, but updated with actual.
 211   bool is_rising = true; // So if result if guess^3 is too low, then try increasing guess.
 212   // Some fraction of digits is used to control how accurate to try to make the result.
 213   int get_digits = std::numeric_limits<T>::digits - 2;
 214   eps_tolerance<T> tol(get_digits); // Set the tolerance.
 215   std::pair<T, T> r;
 216   r =  bracket_and_solve_root(nth_root_functor_noderiv<N, T>(x), guess, factor, is_rising, tol, it);
 217   iters = it;
 218   T result = r.first + (r.second - r.first) / 2;  // Midway between brackets.
 219   return result;
 220 } // template <class T> T nth_root_noderiv(T x)
 221
 222 // Using 1st derivative only Newton-Raphson
 223
 224 template <int N, class T = double>
 225 struct nth_root_functor_1deriv
 226 { // Functor also returning 1st derviative.
 227   BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
 228   BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
 229
 230   nth_root_functor_1deriv(T const& to_find_root_of) : a(to_find_root_of)
 231   { // Constructor stores value a to find root of, for example:
 232   }
 233   std::pair<T, T> operator()(T const& x)
 234   { // Return both f(x) and f'(x).
 235     using boost::math::pow; // // Compile-time integral power.
 236     T p = pow<N - 1>(x);
 237     return std::make_pair(p * x - a, N * p); // 'return' both fx and dx.
 238   }
 239
 240 private:
 241   T a; // to be 'nth_rooted'.
 242 }; // struct nthroot__functor_1deriv
 243
 244 template <int N, class T = double>
 245 T nth_root_1deriv(T x)
 246 { // return nth root of x using 1st derivative and Newton_Raphson.
 247   using namespace std;  // Help ADL of std functions.
 248   using namespace boost::math::tools; // For newton_raphson_iterate.
 249
 250   BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
 251   BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
 252   BOOST_STATIC_ASSERT_MSG((N > 1000) == false, "root N is too big!");
 253
 254   typedef double guess_type;
 255
 256   int exponent;
 257   frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
 258   T guess = static_cast<T>(ldexp(static_cast<guess_type>(1.), exponent / N)); // Rough guess is to divide the exponent by n.
 259   T min = static_cast<T>(ldexp(static_cast<guess_type>(1.) / 2, exponent / N)); // Minimum possible value is half our guess.
 260   T max = static_cast<T>(ldexp(static_cast<guess_type>(2.), exponent / N)); // Maximum possible value is twice our guess.
 261
 262   int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
 263   int get_digits = static_cast<int>(digits * 0.6);
 264   const boost::uintmax_t maxit = 20;
 265   boost::uintmax_t it = maxit;
 266   T result = newton_raphson_iterate(nth_root_functor_1deriv<N, T>(x), guess, min, max, get_digits, it);
 267   iters = it;
 268   return result;
 269 } // T nth_root_1_deriv  Newton-Raphson
 270
 271 // Using 1st and 2nd derivatives with Halley algorithm.
 272
 273 template <int N, class T = double>
 274 struct nth_root_functor_2deriv
 275 { // Functor returning both 1st and 2nd derivatives.
 276   BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
 277   BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
 278
 279   nth_root_functor_2deriv(T const& to_find_root_of) : a(to_find_root_of)
 280   { // Constructor stores value a to find root of, for example:
 281   }
 282
 283   // using boost::math::tuple; // to return three values.
 284   std::tuple<T, T, T> operator()(T const& x)
 285   { // Return f(x), f'(x) and f''(x).
 286     using boost::math::pow; // Compile-time integral power.
 287     T p = pow<N - 2>(x);
 288
 289     return std::make_tuple(p * x * x - a, p * x * N, p * N * (N - 1)); // 'return' fx, dx and d2x.
 290   }
 291 private:
 292   T a; // to be 'nth_rooted'.
 293 };
 294
 295 template <int N, class T = double>
 296 T nth_root_2deriv(T x)
 297 { // return nth root of x using 1st and 2nd derivatives and Halley.
 298
 299   using namespace std;  // Help ADL of std functions.
 300   using namespace boost::math::tools; // For halley_iterate.
 301
 302   BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
 303   BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
 304   BOOST_STATIC_ASSERT_MSG((N > 1000) == false, "root N is too big!");
 305
 306   typedef double guess_type;
 307
 308   int exponent;
 309   frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
 310   T guess = static_cast<T>(ldexp(static_cast<guess_type>(1.), exponent / N)); // Rough guess is to divide the exponent by n.
 311   T min = static_cast<T>(ldexp(static_cast<guess_type>(1.) / 2, exponent / N)); // Minimum possible value is half our guess.
 312   T max = static_cast<T>(ldexp(static_cast<guess_type>(2.), exponent / N)); // Maximum possible value is twice our guess.
 313
 314   int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
 315   int get_digits = static_cast<int>(digits * 0.4);
 316   const boost::uintmax_t maxit = 20;
 317   boost::uintmax_t it = maxit;
 318   T result = halley_iterate(nth_root_functor_2deriv<N, T>(x), guess, min, max, get_digits, it);
 319   iters = it;
 320
 321   return result;
 322 } // nth_2deriv Halley
 323
 324 template <int N, class T = double>
 325 T nth_root_2deriv_s(T x)
 326 { // return nth root of x using 1st and 2nd derivatives and Schroder.
 327
 328   using namespace std;  // Help ADL of std functions.
 329   using namespace boost::math::tools; // For schroder_iterate.
 330
 331   BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
 332   BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
 333   BOOST_STATIC_ASSERT_MSG((N > 1000) == false, "root N is too big!");
 334
 335   typedef double guess_type;
 336
 337   int exponent;
 338   frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
 339   T guess = static_cast<T>(ldexp(static_cast<guess_type>(1.), exponent / N)); // Rough guess is to divide the exponent by n.
 340   T min = static_cast<T>(ldexp(static_cast<guess_type>(1.) / 2, exponent / N)); // Minimum possible value is half our guess.
 341   T max = static_cast<T>(ldexp(static_cast<guess_type>(2.), exponent / N)); // Maximum possible value is twice our guess.
 342
 343   int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.4);
 344   const boost::uintmax_t maxit = 20;
 345   boost::uintmax_t it = maxit;
 346   T result = schroder_iterate(nth_root_functor_2deriv<N, T>(x), guess, min, max, get_digits, it);
 347   iters = it;
 348
 349   return result;
 350 } // T nth_root_2deriv_s Schroder
 351
 352 //////////////////////////////////////////////////////// end of algorithms - perhaps in a separate .hpp?
 353
 354 //! Print 4 floating-point types info: max_digits10, digits and required accuracy digits as a Quickbook table.
 355 int table_type_info(double digits_accuracy)
 356 {
 357   std::string qbk_name = full_roots_name; // Prefix by boost_root file.
 358
 359   qbk_name += "type_info_table";
 360   std::stringstream ss;
 361   ss.precision(3);
 362   ss << "_" << digits_accuracy * 100;
 363   qbk_name += ss.str();
 364
 365 #ifdef _MSC_VER
 366   qbk_name += "_msvc.qbk";
 367 #else // assume GCC
 368   qbk_name += "_gcc.qbk";
 369 #endif
 370
 371   // Example: type_info_table_100_msvc.qbk
 372   fout.open(qbk_name, std::ios_base::out);
 373
 374   if (fout.is_open())
 375   {
 376     std::cout << "Output type table to " << qbk_name << std::endl;
 377   }
 378   else
 379   { // Failed to open.
 380     std::cout << " Open file " << qbk_name << " for output failed!" << std::endl;
 381     std::cout << "errno " << errno << std::endl;
 382     return errno;
 383   }
 384
 385   fout <<
 386     "[/"
 387     << qbk_name
 388     << "\n"
 389     "Copyright 2015 Paul A. Bristow.""\n"
 390     "Copyright 2015 John Maddock.""\n"
 391     "Distributed under the Boost Software License, Version 1.0.""\n"
 392     "(See accompanying file LICENSE_1_0.txt or copy at""\n"
 393     "http://www.boost.org/LICENSE_1_0.txt).""\n"
 394     "]""\n"
 395     << std::endl;
 396
 397   fout << "[h6 Fraction of maximum possible bits of accuracy required is " << digits_accuracy << ".]\n" << std::endl;
 398
 399   std::string table_id("type_info");
 400   table_id += ss.str(); // Fraction digits accuracy.
 401
 402 #ifdef _MSC_VER
 403   table_id += "_msvc";
 404 #else // assume GCC
 405   table_id += "_gcc";
 406 #endif
 407
 408   fout << "[table:" << table_id << " Digits for float, double, long double and cpp_bin_float_50\n"
 409     << "[[type name] [max_digits10] [binary digits] [required digits]]\n";// header.
 410
 411   // For all fout types:
 412
 413   fout  << "[[" << "float" << "]"
 414     << "[" << std::numeric_limits<float>::max_digits10 << "]"  // max_digits10
 415     << "[" << std::numeric_limits<float>::digits << "]"// < "Binary digits
 416     << "[" << static_cast<int>(std::numeric_limits<float>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
 417
 418   fout << "[[" << "float" << "]"
 419     << "[" << std::numeric_limits<double>::max_digits10 << "]"  // max_digits10
 420     << "[" << std::numeric_limits<double>::digits << "]"// < "Binary digits
 421     << "[" << static_cast<int>(std::numeric_limits<double>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
 422
 423   fout << "[[" << "long double" << "]"
 424     << "[" << std::numeric_limits<long double>::max_digits10 << "]"  // max_digits10
 425     << "[" << std::numeric_limits<long double>::digits << "]"// < "Binary digits
 426     << "[" << static_cast<int>(std::numeric_limits<long double>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
 427
 428   fout << "[[" << "cpp_bin_float_50" << "]"
 429     << "[" << std::numeric_limits<cpp_bin_float_50>::max_digits10 << "]"  // max_digits10
 430     << "[" << std::numeric_limits<cpp_bin_float_50>::digits << "]"// < "Binary digits
 431     << "[" << static_cast<int>(std::numeric_limits<cpp_bin_float_50>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
 432
 433   fout << "] [/table table_id_msvc] \n" << std::endl; // End of table.
 434
 435   fout.close();
 436   return 0;
 437 } // type_table
 438
 439 //! Evaluate root N timing for each algorithm, and for one floating-point type T.
 440 template <int N, typename T>
 441 int test_root(cpp_bin_float_100 big_value, cpp_bin_float_100 answer, const char* type_name, std::size_t type_no)
 442 {
 443   std::size_t max_digits = 2 + std::numeric_limits<T>::digits * 3010 / 10000;
 444   // For new versions use max_digits10
 445   // std::cout.precision(std::numeric_limits<T>::max_digits10);
 446   std::cout.precision(max_digits);
 447   std::cout << std::showpoint << std::endl; // Show trailing zeros too.
 448
 449   root_infos.push_back(root_info());
 450
 451   root_infos[type_no].max_digits10 = max_digits;
 452   root_infos[type_no].full_typename = typeid(T).name(); // Full typename.
 453   root_infos[type_no].short_typename = type_name; // Short typename.
 454   root_infos[type_no].bin_digits = std::numeric_limits<T>::digits;
 455   root_infos[type_no].get_digits = static_cast<int>(std::numeric_limits<T>::digits * digits_accuracy);
 456
 457   T to_root = static_cast<T>(big_value);
 458
 459   T result; // root
 460   T sum = 0;
 461   T ans = static_cast<T>(answer);
 462
 463   using boost::timer::nanosecond_type;
 464   using boost::timer::cpu_times;
 465   using boost::timer::cpu_timer;
 466
 467   int eval_count = boost::is_floating_point<T>::value ? 10000000 : 100000; // To give a sufficiently stable timing for the fast built-in types,
 468   //int eval_count = 1000000; // To give a sufficiently stable timing for the fast built-in types,
 469   // This takes an inconveniently long time for multiprecision cpp_bin_float_50 etc  types.
 470
 471   cpu_times now; // Holds wall, user and system times.
 472
 473   { // Evaluate times etc for each algorithm.
 474     //algorithm_names.push_back("TOMS748"); //
 475     cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
 476     ti.start();
 477     for (long i = 0; i < eval_count; ++i)
 478     {
 479       result = nth_root_noderiv<N, T>(to_root); //
 480       sum += result;
 481     }
 482     now = ti.elapsed();
 483     int time = static_cast<int>(now.user / eval_count);
 484     root_infos[type_no].times.push_back(time); // CPU time taken.
 485     if (time < root_infos[type_no].min_time)
 486     {
 487       root_infos[type_no].min_time = time;
 488     }
 489     ti.stop();
 490     long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
 491     root_infos[type_no].distances.push_back(distance);
 492     root_infos[type_no].iterations.push_back(iters); //
 493     root_infos[type_no].full_results.push_back(result);
 494   }
 495   {
 496     // algorithm_names.push_back("Newton"); // algorithm
 497     cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
 498     ti.start();
 499     for (long i = 0; i < eval_count; ++i)
 500     {
 501       result = nth_root_1deriv<N, T>(to_root); //
 502       sum += result;
 503     }
 504     now = ti.elapsed();
 505     int time = static_cast<int>(now.user / eval_count);
 506     root_infos[type_no].times.push_back(time); // CPU time taken.
 507     if (time < root_infos[type_no].min_time)
 508     {
 509       root_infos[type_no].min_time = time;
 510     }
 511
 512     ti.stop();
 513     long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
 514     root_infos[type_no].distances.push_back(distance);
 515     root_infos[type_no].iterations.push_back(iters); //
 516     root_infos[type_no].full_results.push_back(result);
 517   }
 518   {
 519     //algorithm_names.push_back("Halley"); // algorithm
 520     cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
 521     ti.start();
 522     for (long i = 0; i < eval_count; ++i)
 523     {
 524       result = nth_root_2deriv<N>(to_root); //
 525       sum += result;
 526     }
 527     now = ti.elapsed();
 528     int time = static_cast<int>(now.user / eval_count);
 529     root_infos[type_no].times.push_back(time); // CPU time taken.
 530     ti.stop();
 531     if (time < root_infos[type_no].min_time)
 532     {
 533       root_infos[type_no].min_time = time;
 534     }
 535     long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
 536     root_infos[type_no].distances.push_back(distance);
 537     root_infos[type_no].iterations.push_back(iters); //
 538     root_infos[type_no].full_results.push_back(result);
 539   }
 540   {
 541     // algorithm_names.push_back("Schroder"); // algorithm
 542     cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
 543     ti.start();
 544     for (long i = 0; i < eval_count; ++i)
 545     {
 546       result = nth_root_2deriv_s<N>(to_root); //
 547       sum += result;
 548     }
 549     now = ti.elapsed();
 550     int time = static_cast<int>(now.user / eval_count);
 551     root_infos[type_no].times.push_back(time); // CPU time taken.
 552     if (time < root_infos[type_no].min_time)
 553     {
 554       root_infos[type_no].min_time = time;
 555     }
 556     ti.stop();
 557     long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
 558     root_infos[type_no].distances.push_back(distance);
 559     root_infos[type_no].iterations.push_back(iters); //
 560     root_infos[type_no].full_results.push_back(result);
 561   }
 562   for (size_t i = 0; i != root_infos[type_no].times.size(); i++) // For each time.
 563   { // Normalize times.
 564     root_infos[type_no].normed_times.push_back(static_cast<double>(root_infos[type_no].times[i]) / root_infos[type_no].min_time);
 565   }
 566
 567   std::cout << "Accumulated result was: " << sum << std::endl;
 568
 569   return 4;  // eval_count of how many algorithms used.
 570 } // test_root
 571
 572 /*! Fill array of times, interations, etc for Nth root for all 4 types,
 573  and write a table of results in Quickbook format.
 574  */
 575 template <int N>
 576 void table_root_info(cpp_bin_float_100 full_value)
 577 {
 578    using std::abs;
 579   std::cout << nooftypes << " floating-point types tested:" << std::endl;
 580 #if defined(_DEBUG) || !defined(NDEBUG)
 581   std::cout << "Compiled in debug mode." << std::endl;
 582 #else
 583   std::cout << "Compiled in optimise mode." << std::endl;
 584 #endif
 585   std::cout << "FP hardware " << fp_hardware << std::endl;
 586   // Compute the 'right' answer for root N at 100 decimal digits.
 587   cpp_bin_float_100 full_answer = nth_root_noderiv<N, cpp_bin_float_100>(full_value);
 588
 589   root_infos.clear(); // Erase any previous data.
 590   // Fill the elements of the array for each floating-point type.
 591
 592   test_root<N, float>(full_value, full_answer, "float", 0);
 593   test_root<N, double>(full_value, full_answer, "double", 1);
 594   test_root<N, long double>(full_value, full_answer, "long double", 2);
 595   test_root<N, cpp_bin_float_50>(full_value, full_answer, "cpp_bin_float_50", 3);
 596
 597   // Use info from 4 floating point types to
 598
 599   // Prepare Quickbook table for a single root
 600   // with columns of times, iterations, distances repeated for various floating-point types,
 601   // and 4 rows for each algorithm.
 602
 603   std::stringstream table_info;
 604   table_info.precision(3);
 605   table_info << "[table:root_" << N << " " << N << "th root(" << static_cast<float>(full_value) << ") for float, double, long double and cpp_bin_float_50 types";
 606   if (fp_hardware != "")
 607   {
 608     table_info << ", using " << fp_hardware;
 609   }
 610   table_info << std::endl;
 611
 612   fout << table_info.str()
 613     << "[[][float][][][] [][double][][][] [][long d][][][] [][cpp50][][]]\n"
 614     << "[[Algo     ]";
 615   for (size_t tp = 0; tp != nooftypes; tp++)
 616   { // For all types:
 617     fout << "[Its]" << "[Times]" << "[Norm]" << "[Dis]" << "[ ]";
 618   }
 619   fout << "]" << std::endl;
 620
 621   // Row for all algorithms.
 622   for (std::size_t algo = 0; algo != noofalgos; algo++)
 623   {
 624     fout << "[[" << std::left << std::setw(9) << algo_names[algo] << "]";
 625     for (size_t tp = 0; tp != nooftypes; tp++)
 626     { // For all types:
 627       fout
 628         << "[" << std::right << std::showpoint
 629         << std::setw(3) << std::setprecision(2) << root_infos[tp].iterations[algo] << "]["
 630         << std::setw(5) << std::setprecision(5) << root_infos[tp].times[algo] << "][";
 631       fout << std::setw(3) << std::setprecision(3);
 632         double normed_time = root_infos[tp].normed_times[algo];
 633         if (abs(normed_time - 1.00) <= 0.05)
 634         { // At or near the best time, so show as blue.
 635           fout << "[role blue " << normed_time << "]";
 636         }
 637         else if (abs(normed_time) > 4.)
 638         { // markedly poor so show as red.
 639           fout << "[role red " << normed_time << "]";
 640         }
 641         else
 642         { // Not the best, so normal black.
 643           fout << normed_time;
 644         }
 645         fout << "]["
 646         << std::setw(3) << std::setprecision(2) << root_infos[tp].distances[algo] << "][ ]";
 647     } // tp
 648     fout << "]" << std::endl;
 649   } // for algo
 650   fout << "] [/end of table root]\n";
 651 } // void table_root_info
 652
 653 /*! Output program header, table of type info, and tables for 4 algorithms and 4 floating-point types,
 654  for Nth root required digits_accuracy.
 655  */
 656
 657 int roots_tables(cpp_bin_float_100 full_value, double digits_accuracy)
 658 {
 659   ::digits_accuracy = digits_accuracy;
 660   // Save globally so that it is available to root-finding algorithms. Ugly :-(
 661
 662 #if defined(_DEBUG) || !defined(NDEBUG)
 663   std::string debug_or_optimize("Compiled in debug mode.");
 664 #else
 665      std::string debug_or_optimize("Compiled in optimise mode.");
 666 #endif
 667
 668   // Create filename for roots_table
 669   std::string qbk_name = full_roots_name;
 670   qbk_name += "roots_table";
 671
 672   std::stringstream ss;
 673   ss.precision(3);
 674   // ss << "_" << N // now put all the tables in one .qbk file?
 675     ss << "_" << digits_accuracy * 100
 676     << std::flush;
 677   // Assume only save optimize mode runs, so don't add any  _DEBUG info.
 678   qbk_name += ss.str();
 679
 680 #ifdef _MSC_VER
 681   qbk_name += "_msvc";
 682 #else // assume GCC
 683   qbk_name += "_gcc";
 684 #endif
 685   if (fp_hardware != "")
 686   {
 687     qbk_name += fp_hardware;
 688   }
 689   qbk_name += ".qbk";
 690
 691   fout.open(qbk_name, std::ios_base::out);
 692
 693   if (fout.is_open())
 694   {
 695     std::cout << "Output root table to " << qbk_name << std::endl;
 696   }
 697   else
 698   { // Failed to open.
 699     std::cout << " Open file " << qbk_name << " for output failed!" << std::endl;
 700     std::cout << "errno " << errno << std::endl;
 701     return errno;
 702   }
 703
 704   fout <<
 705     "[/"
 706     << qbk_name
 707     << "\n"
 708     "Copyright 2015 Paul A. Bristow.""\n"
 709     "Copyright 2015 John Maddock.""\n"
 710     "Distributed under the Boost Software License, Version 1.0.""\n"
 711     "(See accompanying file LICENSE_1_0.txt or copy at""\n"
 712     "http://www.boost.org/LICENSE_1_0.txt).""\n"
 713     "]""\n"
 714     << std::endl;
 715
 716   // Print out the program/compiler/stdlib/platform names as a Quickbook comment:
 717   fout << "\n[h6 Program " << sourcefilename << ",\n "
 718     << BOOST_COMPILER << ", "
 719     << BOOST_STDLIB << ", "
 720     << BOOST_PLATFORM << "\n"
 721     << debug_or_optimize
 722     << ((fp_hardware != "") ? ", " + fp_hardware : "")
 723     << "]" // [h6 close].
 724     << std::endl;
 725
 726   fout << "Fraction of full accuracy " << digits_accuracy << std::endl;
 727
 728   table_root_info<5>(full_value);
 729   table_root_info<7>(full_value);
 730   table_root_info<11>(full_value);
 731
 732   fout.close();
 733
 734   //   table_type_info(digits_accuracy);
 735
 736   return 0;
 737 } // roots_tables
 738
 739
 740 int main()
 741 {
 742   using namespace boost::multiprecision;
 743   using namespace boost::math;
 744
 745
 746   try
 747   {
 748     std::cout << "Tests run with " << BOOST_COMPILER << ", "
 749       << BOOST_STDLIB << ", " << BOOST_PLATFORM << ", ";
 750
 751 // How to: Configure Visual C++ Projects to Target 64-Bit Platforms
 752 // https://msdn.microsoft.com/en-us/library/9yb4317s.aspx
 753
 754 #ifdef _M_X64 // Defined for compilations that target x64 processors.
 755     std::cout << "X64 " << std::endl;
 756     fp_hardware += "_X64";
 757 #else
 758 #  ifdef _M_IX86
 759      std::cout << "X32 " << std::endl;
 760      fp_hardware += "_X86";
 761 #  endif
 762 #endif
 763
 764 #ifdef _M_AMD64
 765     std::cout << "AMD64 " << std::endl;
 766  //   fp_hardware += "_AMD64";
 767 #endif
 768
 769 // https://msdn.microsoft.com/en-us/library/7t5yh4fd.aspx
 770 // /arch (x86) options /arch:[IA32|SSE|SSE2|AVX|AVX2]
 771 // default is to use SSE and SSE2 instructions by default.
 772 // https://msdn.microsoft.com/en-us/library/jj620901.aspx
 773 // /arch (x64) options /arch:AVX and /arch:AVX2
 774
 775 // MSVC doesn't bother to set these SSE macros!
 776 // http://stackoverflow.com/questions/18563978/sse-sse2-is-enabled-control-in-visual-studio
 777 // https://msdn.microsoft.com/en-us/library/b0084kay.aspx  predefined macros.
 778
 779 // But some of these macros are *not* defined by MSVC,
 780 // unlike AVX (but *are* defined by GCC and Clang).
 781 // So the macro code above does define them.
 782 #if (defined(_M_AMD64) || defined (_M_X64))
 783 #ifndef _M_X64
 784 #  define _M_X64
 785 #endif
 786 #ifndef __SSE2__
 787 #  define __SSE2__
 788 #endif
 789 #else
 790 #  ifdef _M_IX86_FP // Expands to an integer literal value indicating which /arch compiler option was used:
 791     std::cout << "Floating-point _M_IX86_FP = " << _M_IX86_FP << std::endl;
 792 #  if (_M_IX86_FP == 2) // 2 if /arch:SSE2, /arch:AVX or /arch:AVX2
 793 #    define __SSE2__ // x32
 794 #  elif (_M_IX86_FP == 1) // 1 if /arch:SSE was used.
 795 #    define __SSE__ // x32
 796 #  elif (_M_IX86_FP == 0) // 0 if /arch:IA32 was used.
 797 #    define _X32 // No special FP instructions.
 798 #  endif
 799 # endif
 800 #endif
 801 // Set the fp_hardware that is used in the .qbk filename.
 802 #ifdef __AVX2__
 803     std::cout << "Floating-point AVX2 " << std::endl;
 804     fp_hardware += "_AVX2";
 805 #  else
 806 #    ifdef __AVX__
 807     std::cout << "Floating-point AVX " << std::endl;
 808     fp_hardware += "_AVX";
 809 #    else
 810 #      ifdef __SSE2__
 811     std::cout << "Floating-point SSE2 " << std::endl;
 812     fp_hardware += "_SSE2";
 813 #      else
 814 #        ifdef __SSE__
 815     std::cout << "Floating-point SSE " << std::endl;
 816     fp_hardware += "_SSE";
 817 #        endif
 818 #      endif
 819 #   endif
 820 # endif
 821
 822 #ifdef _M_IX86
 823     std::cout << "Floating-point X86 _M_IX86 = " << _M_IX86 << std::endl;
 824     // https://msdn.microsoft.com/en-us/library/aa273918%28v=vs.60%29.aspx#_predir_table_1..3
 825     // 600 = Pentium Pro
 826 #endif
 827
 828 #ifdef _MSC_FULL_VER
 829     std::cout << "Floating-point _MSC_FULL_VER " << _MSC_FULL_VER << std::endl;
 830 #endif
 831
 832 #ifdef __MSVC_RUNTIME_CHECKS
 833     std::cout << "Runtime __MSVC_RUNTIME_CHECKS " << std::endl;
 834 #endif
 835
 836     BOOST_MATH_CONTROL_FP;
 837
 838     cpp_bin_float_100 full_value("28.");
 839     // Compute full answer to more than precision of tests.
 840     //T value = 28.; // integer (exactly representable as floating-point)
 841     // whose cube root is *not* exactly representable.
 842     // Wolfram Alpha command N[28 ^ (1 / 3), 100] computes cube root to 100 decimal digits.
 843     // 3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895
 844
 845     std::cout.precision(100);
 846     std::cout << "value " << full_value << std::endl;
 847    // std::cout << ",\n""answer = " << full_answer << std::endl;
 848     std::cout.precision(6);
 849    // cbrt cpp_bin_float_100 full_answer("3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895");
 850
 851     // Output the table of types, maxdigits10 and digits and required digits for some accuracies.
 852
 853     // Output tables for some roots at full accuracy.
 854     roots_tables(full_value, 1.);
 855
 856     // Output tables for some roots at less accuracy.
 857     //roots_tables(full_value, 0.75);
 858
 859     return boost::exit_success;
 860   }
 861   catch (std::exception const& ex)
 862   {
 863     std::cout << "exception thrown: " << ex.what() << std::endl;
 864     return boost::exit_failure;
 865   }
 866 } // int main()
 867
 868 /*
 869
 870 */