text: // Copyright Paul A. Bristow 2015
  
  // 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)
  
  // Comparison of finding roots using TOMS748, Newton-Raphson, Halley & Schroder algorithms.
  // root_n_finding_algorithms.cpp  Generalised for nth root version.
  
  // http://en.wikipedia.org/wiki/Cube_root
  
  // Note that this file contains Quickbook mark-up as well as code
  // and comments, don't change any of the special comment mark-ups!
  // This program also writes files in Quickbook tables mark-up format.
  
  #include <boost/cstdlib.hpp>
  #include <boost/config.hpp>
  #include <boost/array.hpp>
  #include <boost/type_traits/is_floating_point.hpp>
  #include <boost/math/tools/roots.hpp>
  #include <boost/math/special_functions/ellint_1.hpp>
  #include <boost/math/special_functions/ellint_2.hpp>
  
  //using boost::math::policies::policy;
  //using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits.
  //using boost::math::tools::bracket_and_solve_root;
  //using boost::math::tools::toms748_solve;
  //using boost::math::tools::halley_iterate; 
  //using boost::math::tools::newton_raphson_iterate;
  //using boost::math::tools::schroder_iterate;
  
  #include <boost/math/special_functions/next.hpp> // For float_distance.
  
  #include <boost/multiprecision/cpp_bin_float.hpp> // is binary.
  using boost::multiprecision::cpp_bin_float_100;
  using boost::multiprecision::cpp_bin_float_50;
  
  #include <boost/timer/timer.hpp>
  #include <boost/system/error_code.hpp>
  #include <boost/preprocessor/stringize.hpp>
  
  // STL
  #include <iostream>
  #include <iomanip>
  #include <string>
  #include <vector>
  #include <limits>
  #include <fstream> // std::ofstream
  #include <cmath>
  #include <typeinfo> // for type name using typid(thingy).name();
  
  #ifdef __FILE__
    std::string sourcefilename = __FILE__;
  #else
    std::string sourcefilename("");
  #endif
  
    std::string chop_last(std::string s)
    {
       std::string::size_type pos = s.find_last_of("\\/");
       if(pos != std::string::npos)
          s.erase(pos);
       else if(s.empty())
          abort();
       else
          s.erase();
       return s;
    }
  
    std::string make_root()
    {
       std::string result;
       if(sourcefilename.find_first_of(":") != std::string::npos)
       {
          result = chop_last(sourcefilename); // lose filename part
          result = chop_last(result);   // lose /example/
          result = chop_last(result);   // lose /math/
          result = chop_last(result);   // lose /libs/
       }
       else
       {
          result = chop_last(sourcefilename); // lose filename part
          if(result.empty())
             result = ".";
          result += "/../../..";
       }
       return result;
    }
  
    std::string short_file_name(std::string s)
    {
       std::string::size_type pos = s.find_last_of("\\/");
       if(pos != std::string::npos)
          s.erase(0, pos + 1);
       return s;
    }
  
    std::string boost_root = make_root();
  
  
  std::string fp_hardware; // Any hardware features like SEE or AVX
  
  const std::string roots_name = "libs/math/doc/roots/";
  
  const std::string full_roots_name(boost_root + "/libs/math/doc/roots/");
  
  const std::size_t nooftypes = 4;
  const std::size_t noofalgos = 4;
  const std::size_t noofroots = 3;
  
  double digits_accuracy = 1.0; // 1 == maximum possible accuracy.
  
  std::stringstream ss;
  
  std::ofstream fout;
  
  std::vector<std::string> algo_names =
  {
    "TOMS748", "Newton", "Halley", "Schr'''&#xf6;'''der"
  };
  
  std::vector<std::string> names =
  {
    "float", "double", "long double", "cpp_bin_float50"
  };
  
  uintmax_t iters; // Global as value of iterations is not returned.
  
  struct root_info
  { // for a floating-point type, float, double ...
    std::size_t max_digits10; // for type.
    std::string full_typename; // for type from type_id.name().
    std::string short_typename; // for type "float", "double", "cpp_bin_float_50" ....
    std::size_t bin_digits;  // binary in floating-point type numeric_limits<T>::digits;  
    int get_digits; // fraction of maximum possible accuracy required.
    // = digits * digits_accuracy
    // Vector of values (4) for each algorithm, TOMS748, Newton, Halley & Schroder.
    //std::vector< boost::int_least64_t> times;  converted to int.
    std::vector<int> times; // arbirary units (ticks).
    //boost::int_least64_t min_time = std::numeric_limits<boost::int_least64_t>::max(); // Used to normalize times (as int).
    std::vector<double> normed_times;
    int min_time = (std::numeric_limits<int>::max)(); // Used to normalize times.
    std::vector<uintmax_t> iterations;
    std::vector<long int> distances;
    std::vector<cpp_bin_float_100> full_results;
  }; // struct root_info
  
  std::vector<root_info> root_infos;  // One element for each floating-point type used.
  
  inline std::string build_test_name(const char* type_name, const char* test_name)
  {
    std::string result(BOOST_COMPILER);
    result += "|";
    result += BOOST_STDLIB;
    result += "|";
    result += BOOST_PLATFORM;
    result += "|";
    result += type_name;
    result += "|";
    result += test_name;
  #if defined(_DEBUG) || !defined(NDEBUG)
    result += "|";
    result += " debug";
  #else
    result += "|";
    result += " release";
  #endif
    result += "|";
    return result;
  } // std::string build_test_name
  
  // Algorithms //////////////////////////////////////////////
  
  // No derivatives - using TOMS748 internally.
  //[elliptic_noderv_func
  template <typename T = double>
  struct elliptic_root_functor_noderiv
  { //  Nth root of x using only function - no derivatives.
     elliptic_root_functor_noderiv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius)
     { // Constructor just stores value a to find root of.
     }
     T operator()(T const& x)
     {
        using std::sqrt;
        // return the difference between required arc-length, and the calculated arc-length for an
        // ellipse with radii m_radius and x:
        T a = (std::max)(m_radius, x);
        T b = (std::min)(m_radius, x);
        T k = sqrt(1 - b * b / (a * a));
        return 4 * a * boost::math::ellint_2(k) - m_arc;
     }
  private:
     T m_arc;     // length of arc.
     T m_radius;  // one of the two radii of the ellipse
  }; // template <class T> struct elliptic_root_functor_noderiv
  //]
  //[elliptic_root_noderiv
  template <class T = double>
  T elliptic_root_noderiv(T radius, T arc)
  { // return the other radius of an ellipse, given one radii and the arc-length
     using namespace std;  // Help ADL of std functions.
     using namespace boost::math::tools; // For bracket_and_solve_root.
  
     T guess = sqrt(arc * arc / 16 - radius * radius);
     T factor = 1.2;                     // How big steps to take when searching.
  
     const boost::uintmax_t maxit = 50;  // Limit to maximum iterations.
     boost::uintmax_t it = maxit;        // Initally our chosen max iterations, but updated with actual.
     bool is_rising = true;              // arc-length increases if one radii increases, so function is rising
     // Define a termination condition, stop when nearly all digits are correct, but allow for
     // the fact that we are returning a range, and must have some inaccuracy in the elliptic integral:
     eps_tolerance<T> tol(std::numeric_limits<T>::digits - 2);
     // Call bracket_and_solve_root to find the solution, note that this is a rising function:
     std::pair<T, T> r = bracket_and_solve_root(elliptic_root_functor_noderiv<T>(arc, radius), guess, factor, is_rising, tol, it);
     //<-
     iters = it;
     //->
     // Result is midway between the endpoints of the range:
     return r.first + (r.second - r.first) / 2;
  } // template <class T> T elliptic_root_noderiv(T x)
  //]
  // Using 1st derivative only Newton-Raphson
  //[elliptic_1deriv_func
  template <class T = double>
  struct elliptic_root_functor_1deriv
  { // Functor also returning 1st derviative.
     BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
  
     elliptic_root_functor_1deriv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius)
     { // Constructor just stores value a to find root of.
     }
     std::pair<T, T> operator()(T const& x)
     {
        using std::sqrt;
        // Return the difference between required arc-length, and the calculated arc-length for an
        // ellipse with radii m_radius and x, plus it's derivative.
        // See http://www.wolframalpha.com/input/?i=d%2Fda+[4+*+a+*+EllipticE%281+-+b^2%2Fa^2%29]
        // We require two elliptic integral calls, but from these we can calculate both
        // the function and it's derivative:
        T a = (std::max)(m_radius, x);
        T b = (std::min)(m_radius, x);
        T a2 = a * a;
        T b2 = b * b;
        T k = sqrt(1 - b2 / a2);
        T Ek = boost::math::ellint_2(k);
        T Kk = boost::math::ellint_1(k);
        T fx = 4 * a * Ek - m_arc;
        T dfx = 4 * (a2 * Ek - b2 * Kk) / (a2 - b2);
        return std::make_pair(fx, dfx);
     }
  private:
     T m_arc;     // length of arc.
     T m_radius;  // one of the two radii of the ellipse
  };  // struct elliptic_root__functor_1deriv
  //]
  //[elliptic_1deriv
  template <class T = double>
  T elliptic_root_1deriv(T radius, T arc)
  { 
     using namespace std;  // Help ADL of std functions.
     using namespace boost::math::tools; // For newton_raphson_iterate.
  
     BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
  
     T guess = sqrt(arc * arc / 16 - radius * radius);
     T min = 0;   // Minimum possible value is zero.
     T max = arc; // Maximum possible value is the arc length.
  
     // Accuracy doubles at each step, so stop when just over half of the digits are
     // correct, and rely on that step to polish off the remainder:
     int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.6);
     const boost::uintmax_t maxit = 20;
     boost::uintmax_t it = maxit;
     T result = newton_raphson_iterate(elliptic_root_functor_1deriv<T>(arc, radius), guess, min, max, get_digits, it);
     //<-
     iters = it;
     //->
     return result;
  } // T elliptic_root_1_deriv  Newton-Raphson
  //]
  
  // Using 1st and 2nd derivatives with Halley algorithm.
  //[elliptic_2deriv_func
  template <class T = double>
  struct elliptic_root_functor_2deriv
  { // Functor returning both 1st and 2nd derivatives.
     BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
  
     elliptic_root_functor_2deriv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius) {}
     std::tuple<T, T, T> operator()(T const& x)
     {
        using std::sqrt;
        // Return the difference between required arc-length, and the calculated arc-length for an
        // ellipse with radii m_radius and x, plus it's derivative.
        // See http://www.wolframalpha.com/input/?i=d^2%2Fda^2+[4+*+a+*+EllipticE%281+-+b^2%2Fa^2%29]
        // for the second derivative.
        T a = (std::max)(m_radius, x);
        T b = (std::min)(m_radius, x);
        T a2 = a * a;
        T b2 = b * b;
        T k = sqrt(1 - b2 / a2);
        T Ek = boost::math::ellint_2(k);
        T Kk = boost::math::ellint_1(k);
        T fx = 4 * a * Ek - m_arc;
        T dfx = 4 * (a2 * Ek - b2 * Kk) / (a2 - b2);
        T dfx2 = 4 * b2 * ((a2 + b2) * Kk - 2 * a2 * Ek) / (a * (a2 - b2) * (a2 - b2));
        return std::make_tuple(fx, dfx, dfx2);
     }
  private:
     T m_arc;     // length of arc.
     T m_radius;  // one of the two radii of the ellipse
  };
  //]
  //[elliptic_2deriv
  template <class T = double>
  T elliptic_root_2deriv(T radius, T arc)
  {
     using namespace std;                // Help ADL of std functions.
     using namespace boost::math::tools; // For halley_iterate.
  
     BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
  
     T guess = sqrt(arc * arc / 16 - radius * radius);
     T min = 0;                                   // Minimum possible value is zero.
     T max = arc;                                 // radius can't be larger than the arc length.
  
     // Accuracy triples at each step, so stop when just over one-third of the digits
     // are correct, and the last iteration will polish off the remaining digits:
     int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.4);
     const boost::uintmax_t maxit = 20;
     boost::uintmax_t it = maxit;
     T result = halley_iterate(elliptic_root_functor_2deriv<T>(arc, radius), guess, min, max, get_digits, it);
     //<-
     iters = it;
     //->
     return result;
  } // nth_2deriv Halley
  //]
  // Using 1st and 2nd derivatives using Schroder algorithm.
  
  template <class T = double>
  T elliptic_root_2deriv_s(T arc, T radius)
  { // return nth root of x using 1st and 2nd derivatives and Schroder.
  
    using namespace std;  // Help ADL of std functions.
    using namespace boost::math::tools; // For schroder_iterate.
  
    BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
  
    T guess = sqrt(arc * arc / 16 - radius * radius);
    T min = 0; // Minimum possible value is zero.
    T max = arc; // radius can't be larger than the arc length.
  
    int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
    int get_digits = static_cast<int>(digits * digits_accuracy);
    const boost::uintmax_t maxit = 20;
    boost::uintmax_t it = maxit;
    T result = schroder_iterate(elliptic_root_functor_2deriv<T>(arc, radius), guess, min, max, get_digits, it);
    iters = it;
  
    return result;
  } // T elliptic_root_2deriv_s Schroder
  
  //////////////////////////////////////////////////////// end of algorithms - perhaps in a separate .hpp?
  
  //! Print 4 floating-point types info: max_digits10, digits and required accuracy digits as a Quickbook table.
  int table_type_info(double digits_accuracy)
  {
    std::string qbk_name = full_roots_name; // Prefix by boost_root file.
  
    qbk_name += "type_info_table";
    std::stringstream ss;
    ss.precision(3);
    ss << "_" << digits_accuracy * 100;
    qbk_name += ss.str();
  
  #ifdef _MSC_VER
    qbk_name += "_msvc.qbk";
  #else // assume GCC
    qbk_name += "_gcc.qbk";
  #endif
  
    // Example: type_info_table_100_msvc.qbk
    fout.open(qbk_name, std::ios_base::out);
  
    if (fout.is_open())
    {
      std::cout << "Output type table to " << qbk_name << std::endl;
    }
    else
    { // Failed to open.
      std::cout << " Open file " << qbk_name << " for output failed!" << std::endl;
      std::cout << "errno " << errno << std::endl;
      return errno;
    }
  
    fout <<
      "[/"
      << qbk_name
      << "\n"
      "Copyright 2015 Paul A. Bristow.""\n"
      "Copyright 2015 John Maddock.""\n"
      "Distributed under the Boost Software License, Version 1.0.""\n"
      "(See accompanying file LICENSE_1_0.txt or copy at""\n"
      "http://www.boost.org/LICENSE_1_0.txt).""\n"
      "]""\n"
      << std::endl;
  
    fout << "[h6 Fraction of maximum possible bits of accuracy required is " << digits_accuracy << ".]\n" << std::endl;
  
    std::string table_id("type_info");
    table_id += ss.str(); // Fraction digits accuracy.
  
  #ifdef _MSC_VER
    table_id += "_msvc";
  #else // assume GCC
    table_id += "_gcc";
  #endif
  
    fout << "[table:" << table_id << " Digits for float, double, long double and cpp_bin_float_50\n"
      << "[[type name] [max_digits10] [binary digits] [required digits]]\n";// header.
  
    // For all fout types:
  
    fout  << "[[" << "float" << "]"
      << "[" << std::numeric_limits<float>::max_digits10 << "]"  // max_digits10
      << "[" << std::numeric_limits<float>::digits << "]"// < "Binary digits 
      << "[" << static_cast<int>(std::numeric_limits<float>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
  
    fout << "[[" << "float" << "]"
      << "[" << std::numeric_limits<double>::max_digits10 << "]"  // max_digits10
      << "[" << std::numeric_limits<double>::digits << "]"// < "Binary digits 
      << "[" << static_cast<int>(std::numeric_limits<double>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
  
    fout << "[[" << "long double" << "]"
      << "[" << std::numeric_limits<long double>::max_digits10 << "]"  // max_digits10
      << "[" << std::numeric_limits<long double>::digits << "]"// < "Binary digits 
      << "[" << static_cast<int>(std::numeric_limits<long double>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
  
    fout << "[[" << "cpp_bin_float_50" << "]"
      << "[" << std::numeric_limits<cpp_bin_float_50>::max_digits10 << "]"  // max_digits10
      << "[" << std::numeric_limits<cpp_bin_float_50>::digits << "]"// < "Binary digits 
      << "[" << static_cast<int>(std::numeric_limits<cpp_bin_float_50>::digits * digits_accuracy) << "]]\n"; // Accuracy digits.
  
    fout << "] [/table table_id_msvc] \n" << std::endl; // End of table.
  
    fout.close();
    return 0;
  } // type_table
  
  //! Evaluate root N timing for each algorithm, and for one floating-point type T. 
  template <typename T>
  int test_root(cpp_bin_float_100 big_radius, cpp_bin_float_100 big_arc, cpp_bin_float_100 answer, const char* type_name, std::size_t type_no)
  {
    std::size_t max_digits = 2 + std::numeric_limits<T>::digits * 3010 / 10000;
    // For new versions use max_digits10
    // std::cout.precision(std::numeric_limits<T>::max_digits10);
    std::cout.precision(max_digits);
    std::cout << std::showpoint << std::endl; // Show trailing zeros too.
  
    root_infos.push_back(root_info()); 
  
    root_infos[type_no].max_digits10 = max_digits;
    root_infos[type_no].full_typename = typeid(T).name(); // Full typename.
    root_infos[type_no].short_typename = type_name; // Short typename.
    root_infos[type_no].bin_digits = std::numeric_limits<T>::digits;
    root_infos[type_no].get_digits = static_cast<int>(std::numeric_limits<T>::digits * digits_accuracy);
  
    T radius = static_cast<T>(big_radius);
    T arc = static_cast<T>(big_arc);
  
    T result; // root
    T sum = 0;
    T ans = static_cast<T>(answer);
  
    using boost::timer::nanosecond_type;
    using boost::timer::cpu_times;
    using boost::timer::cpu_timer;
  
    long eval_count = boost::is_floating_point<T>::value ? 1000000 : 10000; // To give a sufficiently stable timing for the fast built-in types,
    // This takes an inconveniently long time for multiprecision cpp_bin_float_50 etc  types.
  
    cpu_times now; // Holds wall, user and system times.
  
    { // Evaluate times etc for each algorithm.
      //algorithm_names.push_back("TOMS748"); // 
      cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
      ti.start();
      for(long i = eval_count; i >= 0; --i)
      {
        result = elliptic_root_noderiv(radius, arc); // 
        sum += result;
      }
      now = ti.elapsed();
      int time = static_cast<int>(now.user / eval_count);
      root_infos[type_no].times.push_back(time); // CPU time taken.
      if (time < root_infos[type_no].min_time)
      {
        root_infos[type_no].min_time = time;
      }
      ti.stop();
      long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
      root_infos[type_no].distances.push_back(distance);
      root_infos[type_no].iterations.push_back(iters); // 
      root_infos[type_no].full_results.push_back(result);
    }
    {
      // algorithm_names.push_back("Newton"); // algorithm
      cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
      ti.start();
      for(long i = eval_count; i >= 0; --i)
      {
        result = elliptic_root_1deriv(radius, arc); // 
        sum += result;
      }
      now = ti.elapsed();
      int time = static_cast<int>(now.user / eval_count);
      root_infos[type_no].times.push_back(time); // CPU time taken.
      if (time < root_infos[type_no].min_time)
      {
        root_infos[type_no].min_time = time;
      }
  
      ti.stop();
      long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
      root_infos[type_no].distances.push_back(distance);
      root_infos[type_no].iterations.push_back(iters); //
      root_infos[type_no].full_results.push_back(result);
    }
    {
      //algorithm_names.push_back("Halley"); // algorithm
      cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
      ti.start();
      for(long i = eval_count; i >= 0; --i)
      {
        result = elliptic_root_2deriv(radius, arc); // 
        sum += result;
      }
      now = ti.elapsed();
      int time = static_cast<int>(now.user / eval_count);
      root_infos[type_no].times.push_back(time); // CPU time taken.
      ti.stop();
      if (time < root_infos[type_no].min_time)
      {
        root_infos[type_no].min_time = time;
      }
      long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
      root_infos[type_no].distances.push_back(distance);
      root_infos[type_no].iterations.push_back(iters); // 
      root_infos[type_no].full_results.push_back(result);
    }
    {
      // algorithm_names.push_back("Schr'''&#xf6;'''der"); // algorithm
      cpu_timer ti; // Can start, pause, resume and stop, and read elapsed.
      ti.start();
      for(long i = eval_count; i >= 0; --i)
      {
        result = elliptic_root_2deriv_s(arc, radius); // 
        sum += result;
      }
      now = ti.elapsed();
      int time = static_cast<int>(now.user / eval_count);
      root_infos[type_no].times.push_back(time); // CPU time taken.
      if (time < root_infos[type_no].min_time)
      {
        root_infos[type_no].min_time = time;
      }
      ti.stop();
      long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans));
      root_infos[type_no].distances.push_back(distance);
      root_infos[type_no].iterations.push_back(iters); // 
      root_infos[type_no].full_results.push_back(result);
    }
    for (size_t i = 0; i != root_infos[type_no].times.size(); i++) // For each time.
    { // Normalize times.
      root_infos[type_no].normed_times.push_back(static_cast<double>(root_infos[type_no].times[i]) / root_infos[type_no].min_time);
    }
  
    std::cout << "Accumulated result was: " << sum << std::endl;
  
    return 4;  // eval_count of how many algorithms used.
  } // test_root
  
  /*! Fill array of times, interations, etc for Nth root for all 4 types,
   and write a table of results in Quickbook format.
   */
  void table_root_info(cpp_bin_float_100 radius, cpp_bin_float_100 arc)
  {
     using std::abs;
  
    std::cout << nooftypes << " floating-point types tested:" << std::endl;
  #if defined(_DEBUG) || !defined(NDEBUG)
    std::cout << "Compiled in debug mode." << std::endl;
  #else
    std::cout << "Compiled in optimise mode." << std::endl;
  #endif
    std::cout << "FP hardware " << fp_hardware << std::endl;
    // Compute the 'right' answer for root N at 100 decimal digits.
    cpp_bin_float_100 full_answer = elliptic_root_noderiv(radius, arc);
  
    int type_count = 0;
    root_infos.clear(); // Erase any previous data.
    // Fill the elements of the array for each floating-point type.
  
    type_count = test_root<float>(radius, arc, full_answer, "float", 0);
    type_count = test_root<double>(radius, arc, full_answer,  "double", 1);
    type_count = test_root<long double>(radius, arc, full_answer, "long double", 2);
    type_count = test_root<cpp_bin_float_50>(radius, arc, full_answer, "cpp_bin_float_50", 3);
  
    // Use info from 4 floating point types to
  
    // Prepare Quickbook table for a single root
    // with columns of times, iterations, distances repeated for various floating-point types,
    // and 4 rows for each algorithm.
  
    std::stringstream table_info;
    table_info.precision(3);
    table_info << "[table:elliptic root with radius " << radius << " and arc length " << arc << ") for float, double, long double and cpp_bin_float_50 types";
    if (fp_hardware != "")
    {
      table_info << ", using " << fp_hardware;
    }
    table_info << std::endl;
  
    fout << table_info.str()
      << "[[][float][][][] [][double][][][] [][long d][][][] [][cpp50][][]]\n"
      << "[[Algo     ]";
    for (size_t tp = 0; tp != nooftypes; tp++)
    { // For all types:
      fout << "[Its]" << "[Times]" << "[Norm]" << "[Dis]" << "[ ]";
    }
    fout << "]" << std::endl;
  
    // Row for all algorithms.
    for (std::size_t algo = 0; algo != noofalgos; algo++)
    {
      fout << "[[" << std::left << std::setw(9) << algo_names[algo] << "]";
      for (size_t tp = 0; tp != nooftypes; tp++)
      { // For all types:
        fout
          << "[" << std::right << std::showpoint
          << std::setw(3) << std::setprecision(2) << root_infos[tp].iterations[algo] << "]["
          << std::setw(5) << std::setprecision(5) << root_infos[tp].times[algo] << "][";
        fout << std::setw(3) << std::setprecision(3);
          double normed_time = root_infos[tp].normed_times[algo];
          if (abs(normed_time - 1.00) <= 0.05)
          { // At or near the best time, so show as blue.
            fout << "[role blue " << normed_time << "]";
          }
          else if (abs(normed_time) > 4.)
          { // markedly poor so show as red.
            fout << "[role red " << normed_time << "]";
          }
          else
          { // Not the best, so normal black.
            fout << normed_time;
          }
          fout << "]["
          << std::setw(3) << std::setprecision(2) << root_infos[tp].distances[algo] << "][ ]";
      } // tp
      fout << "]" << std::endl;
    } // for algo
    fout << "] [/end of table root]\n";
  } // void table_root_info
  
  /*! Output program header, table of type info, and tables for 4 algorithms and 4 floating-point types,
   for Nth root required digits_accuracy.
   */
  
  int roots_tables(cpp_bin_float_100 radius, cpp_bin_float_100 arc, double digits_accuracy)
  {
    ::digits_accuracy = digits_accuracy;
    // Save globally so that it is available to root-finding algorithms. Ugly :-(
  
  #if defined(_DEBUG) || !defined(NDEBUG)
    std::string debug_or_optimize("Compiled in debug mode.");
  #else
       std::string debug_or_optimize("Compiled in optimise mode.");
  #endif
  
    // Create filename for roots_table
    std::string qbk_name = full_roots_name;
    qbk_name += "elliptic_table";
  
    std::stringstream ss;
    ss.precision(3);
    // ss << "_" << N // now put all the tables in one .qbk file?
      ss << "_" << digits_accuracy * 100
      << std::flush;
    // Assume only save optimize mode runs, so don't add any  _DEBUG info.
    qbk_name += ss.str();
  
  #ifdef _MSC_VER
    qbk_name += "_msvc";
  #else // assume GCC
    qbk_name += "_gcc";
  #endif 
    if (fp_hardware != "")
    {
      qbk_name += fp_hardware;
    }
    qbk_name += ".qbk";
  
    fout.open(qbk_name, std::ios_base::out);
  
    if (fout.is_open())
    {
      std::cout << "Output root table to " << qbk_name << std::endl;
    }
    else
    { // Failed to open.
      std::cout << " Open file " << qbk_name << " for output failed!" << std::endl;
      std::cout << "errno " << errno << std::endl;
      return errno;
    }
  
    fout <<
      "[/"
      << qbk_name
      << "\n"
      "Copyright 2015 Paul A. Bristow.""\n"
      "Copyright 2015 John Maddock.""\n"
      "Distributed under the Boost Software License, Version 1.0.""\n"
      "(See accompanying file LICENSE_1_0.txt or copy at""\n"
      "http://www.boost.org/LICENSE_1_0.txt).""\n"
      "]""\n"
      << std::endl;
  
    // Print out the program/compiler/stdlib/platform names as a Quickbook comment:
    fout << "\n[h6 Program [@../../example/" << short_file_name(sourcefilename) << " " << short_file_name(sourcefilename) << "],\n "
      << BOOST_COMPILER << ", "
      << BOOST_STDLIB << ", "
      << BOOST_PLATFORM << "\n"
      << debug_or_optimize 
      << ((fp_hardware != "") ? ", " + fp_hardware : "")
      << "]" // [h6 close].
      << std::endl;
  
    //fout << "Fraction of full accuracy " << digits_accuracy << std::endl;
  
    table_root_info(radius, arc);
  
    fout.close();
  
    //   table_type_info(digits_accuracy);
  
    return 0;
  } // roots_tables
  
  
  int main()
  {
    using namespace boost::multiprecision;
    using namespace boost::math;
  
  
    try
    {
      std::cout << "Tests run with " << BOOST_COMPILER << ", "
        << BOOST_STDLIB << ", " << BOOST_PLATFORM << ", ";
  
  // How to: Configure Visual C++ Projects to Target 64-Bit Platforms
  // https://msdn.microsoft.com/en-us/library/9yb4317s.aspx
  
  #ifdef _M_X64 // Defined for compilations that target x64 processors.
      std::cout << "X64 " << std::endl;
      fp_hardware += "_X64";
  #else
  #  ifdef _M_IX86
       std::cout << "X32 " << std::endl;
       fp_hardware += "_X86";
  #  endif
  #endif
  
  #ifdef _M_AMD64
      std::cout << "AMD64 " << std::endl;
   //   fp_hardware += "_AMD64";
  #endif
  
  // https://msdn.microsoft.com/en-us/library/7t5yh4fd.aspx  
  // /arch (x86) options /arch:[IA32|SSE|SSE2|AVX|AVX2]
  // default is to use SSE and SSE2 instructions by default.
  // https://msdn.microsoft.com/en-us/library/jj620901.aspx
  // /arch (x64) options /arch:AVX and /arch:AVX2
  
  // MSVC doesn't bother to set these SSE macros!
  // http://stackoverflow.com/questions/18563978/sse-sse2-is-enabled-control-in-visual-studio
  // https://msdn.microsoft.com/en-us/library/b0084kay.aspx  predefined macros.
  
  // But some of these macros are *not* defined by MSVC, 
  // unlike AVX (but *are* defined by GCC and Clang). 
  // So the macro code above does define them.
  #if (defined(_M_AMD64) || defined (_M_X64))
  #  define _M_X64 
  #  define __SSE2__
  #else
  #  ifdef _M_IX86_FP // Expands to an integer literal value indicating which /arch compiler option was used:
      std::cout << "Floating-point _M_IX86_FP = " << _M_IX86_FP << std::endl;
  #  if (_M_IX86_FP == 2) // 2 if /arch:SSE2, /arch:AVX or /arch:AVX2 
  #    define __SSE2__ // x32
  #  elif (_M_IX86_FP == 1) // 1 if /arch:SSE was used.
  #    define __SSE__ // x32
  #  elif (_M_IX86_FP == 0) // 0 if /arch:IA32 was used.
  #    define _X32 // No special FP instructions.
  #  endif
  # endif
  #endif
  // Set the fp_hardware that is used in the .qbk filename.
  #ifdef __AVX2__
      std::cout << "Floating-point AVX2 " << std::endl;
      fp_hardware += "_AVX2";
  #  else 
  #    ifdef __AVX__
      std::cout << "Floating-point AVX " << std::endl;
      fp_hardware += "_AVX";
  #    else
  #      ifdef __SSE2__
      std::cout << "Floating-point SSE2 " << std::endl;
      fp_hardware += "_SSE2";
  #      else
  #        ifdef __SSE__
      std::cout << "Floating-point SSE " << std::endl;
      fp_hardware += "_SSE";
  #        endif
  #      endif
  #   endif
  # endif
  
  #ifdef _M_IX86
      std::cout << "Floating-point X86 _M_IX86 = " << _M_IX86 << std::endl;
      // https://msdn.microsoft.com/en-us/library/aa273918%28v=vs.60%29.aspx#_predir_table_1..3
      // 600 = Pentium Pro
  #endif
  
  #ifdef _MSC_FULL_VER
      std::cout << "Floating-point _MSC_FULL_VER " << _MSC_FULL_VER << std::endl;
  #endif
  
  #ifdef __MSVC_RUNTIME_CHECKS
      std::cout << "Runtime __MSVC_RUNTIME_CHECKS " << std::endl;
  #endif
  
      BOOST_MATH_CONTROL_FP;
  
      cpp_bin_float_100 radius("28.");
      cpp_bin_float_100 arc("300.");
      // Compute full answer to more than precision of tests.
      //T value = 28.; // integer (exactly representable as floating-point)
      // whose cube root is *not* exactly representable.
      // Wolfram Alpha command N[28 ^ (1 / 3), 100] computes cube root to 100 decimal digits.
      // 3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895
  
      std::cout.precision(100);
      std::cout << "radius 1" << radius << std::endl;
      std::cout << "arc length" << arc << std::endl;
      // std::cout << ",\n""answer = " << full_answer << std::endl;
      std::cout.precision(6);
     // cbrt cpp_bin_float_100 full_answer("3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895");
  
      // Output the table of types, maxdigits10 and digits and required digits for some accuracies.
  
      // Output tables for some roots at full accuracy.
      roots_tables(radius, arc, 1.);
  
      // Output tables for some roots at less accuracy.
      //roots_tables(full_value, 0.75);
  
      return boost::exit_success;
    }
    catch (std::exception ex)
    {
      std::cout << "exception thrown: " << ex.what() << std::endl;
      return boost::exit_failure;
    }
  } // int main()
  
  /*
  
  */
  