Adding upstream version 16.2.11+ds.upstream/16.2.11+dsupstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

4 files changed, 1324 insertions, 0 deletions

diff --git a/src/boost/libs/math/minimax/Jamfile.v2 b/src/boost/libs/math/minimax/Jamfile.v2
new file mode 100644
index 000000000..81790f9a6
--- /dev/null
+++ b/src/boost/libs/math/minimax/Jamfile.v2
@@ -0,0 +1,46 @@
+# Copyright John Maddock 2010
+# Copyright Paul A. Bristow 2018
+# Distributed under 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.
+# \math_toolkit\libs\math\minimax\jamfile.v2
+# Runs minimax using multiprecision, (rather than gmp and mpfr)
+
+# bring in the rules for testing.
+import modules ;
+import path ;
+
+project
+    : requirements
+      <toolset>gcc:<cxxflags>-Wno-missing-braces
+      <toolset>darwin:<cxxflags>-Wno-missing-braces
+      <toolset>acc:<cxxflags>+W2068,2461,2236,4070,4069
+      <toolset>intel-win:<cxxflags>-nologo
+      <toolset>intel-win:<linkflags>-nologo
+      <toolset>msvc:<warnings>all
+      <toolset>msvc:<asynch-exceptions>on
+      <toolset>msvc:<cxxflags>/wd4996
+      <toolset>msvc:<cxxflags>/wd4512
+      <toolset>msvc:<cxxflags>/wd4610
+      <toolset>msvc:<cxxflags>/wd4510
+      <toolset>msvc:<cxxflags>/wd4127
+      <toolset>msvc:<cxxflags>/wd4701 # needed for lexical cast - temporary.
+      <link>static
+      <toolset>borland:<runtime-link>static
+      <include>../../..
+      <define>BOOST_ALL_NO_LIB=1
+      <define>BOOST_UBLAS_UNSUPPORTED_COMPILER=0
+      <include>.
+      <include>../include_private
+      #<include>$(ntl-path)/include
+    ;
+
+#lib mpfr : gmp : <name>mpfr ;
+
+#lib gmp : : <name>gmp ;
+
+# exe minimax : f.cpp main.cpp gmp mpfr ;
+exe minimax : f.cpp main.cpp ;
+
+install bin : minimax ;
+
diff --git a/src/boost/libs/math/minimax/f.cpp b/src/boost/libs/math/minimax/f.cpp
new file mode 100644
index 000000000..6ea405800
--- /dev/null
+++ b/src/boost/libs/math/minimax/f.cpp
@@ -0,0 +1,404 @@
+//  (C) Copyright John Maddock 2006.
+//  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)
+
+#define L22
+//#include "../tools/ntl_rr_lanczos.hpp"
+//#include "../tools/ntl_rr_digamma.hpp"
+#include "multiprecision.hpp"
+#include <boost/math/tools/polynomial.hpp>
+#include <boost/math/special_functions.hpp>
+#include <boost/math/special_functions/zeta.hpp>
+#include <boost/math/special_functions/expint.hpp>
+#include <boost/math/special_functions/lambert_w.hpp>
+
+#include <cmath>
+
+
+mp_type f(const mp_type& x, int variant)
+{
+   static const mp_type tiny = boost::math::tools::min_value<mp_type>() * 64;
+   switch(variant)
+   {
+   case 0:
+      {
+      mp_type x_ = sqrt(x == 0 ? 1e-80 : x);
+      return boost::math::erf(x_) / x_;
+      }
+   case 1:
+      {
+      mp_type x_ = 1 / x;
+      return boost::math::erfc(x_) * x_ / exp(-x_ * x_);
+      }
+   case 2:
+      {
+      return boost::math::erfc(x) * x / exp(-x * x);
+      }
+   case 3:
+      {
+         mp_type y(x);
+         if(y == 0) 
+            y += tiny;
+         return boost::math::lgamma(y+2) / y - 0.5;
+      }
+   case 4:
+      //
+      // lgamma in the range [2,3], use:
+      //
+      // lgamma(x) = (x-2) * (x + 1) * (c + R(x - 2))
+      //
+      // Works well at 80-bit long double precision, but doesn't
+      // stretch to 128-bit precision.
+      //
+      if(x == 0)
+      {
+         return boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008") / 3;
+      }
+      return boost::math::lgamma(x+2) / (x * (x+3));
+   case 5:
+      {
+         //
+         // lgamma in the range [1,2], use:
+         //
+         // lgamma(x) = (x - 1) * (x - 2) * (c + R(x - 1))
+         //
+         // works well over [1, 1.5] but not near 2 :-(
+         //
+         mp_type r1 = boost::lexical_cast<mp_type>("0.57721566490153286060651209008240243104215933593992");
+         mp_type r2 = boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008");
+         if(x == 0)
+         {
+            return r1;
+         }
+         if(x == 1)
+         {
+            return r2;
+         }
+         return boost::math::lgamma(x+1) / (x * (x - 1));
+      }
+   case 6:
+      {
+         //
+         // lgamma in the range [1.5,2], use:
+         //
+         // lgamma(x) = (2 - x) * (1 - x) * (c + R(2 - x))
+         //
+         // works well over [1.5, 2] but not near 1 :-(
+         //
+         mp_type r1 = boost::lexical_cast<mp_type>("0.57721566490153286060651209008240243104215933593992");
+         mp_type r2 = boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008");
+         if(x == 0)
+         {
+            return r2;
+         }
+         if(x == 1)
+         {
+            return r1;
+         }
+         return boost::math::lgamma(2-x) / (x * (x - 1));
+      }
+   case 7:
+      {
+         //
+         // erf_inv in range [0, 0.5]
+         //
+         mp_type y = x;
+         if(y == 0)
+            y = boost::math::tools::epsilon<mp_type>() / 64;
+         return boost::math::erf_inv(y) / (y * (y+10));
+      }
+   case 8:
+      {
+         // 
+         // erfc_inv in range [0.25, 0.5]
+         // Use an y-offset of 0.25, and range [0, 0.25]
+         // abs error, auto y-offset.
+         //
+         mp_type y = x;
+         if(y == 0)
+            y = boost::lexical_cast<mp_type>("1e-5000");
+         return sqrt(-2 * log(y)) / boost::math::erfc_inv(y);
+      }
+   case 9:
+      {
+         mp_type x2 = x;
+         if(x2 == 0)
+            x2 = boost::lexical_cast<mp_type>("1e-5000");
+         mp_type y = exp(-x2*x2); // sqrt(-log(x2)) - 5;
+         return boost::math::erfc_inv(y) / x2;
+      }
+   case 10:
+      {
+         //
+         // Digamma over the interval [1,2], set x-offset to 1
+         // and optimise for absolute error over [0,1].
+         //
+         int current_precision = get_working_precision();
+         if(current_precision < 1000)
+            set_working_precision(1000);
+         //
+         // This value for the root of digamma is calculated using our
+         // differentiated lanczos approximation.  It agrees with Cody
+         // to ~ 25 digits and to Morris to 35 digits.  See:
+         // TOMS ALGORITHM 708 (Didonato and Morris).
+         // and Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher.
+         //
+         //mp_type root = boost::lexical_cast<mp_type>("1.4616321449683623412626595423257213234331845807102825466429633351908372838889871");
+         //
+         // Actually better to calculate the root on the fly, it appears to be more
+         // accurate: convergence is easier with the 1000-bit value, the approximation
+         // produced agrees with functions.mathworld.com values to 35 digits even quite
+         // near the root.
+         //
+         static boost::math::tools::eps_tolerance<mp_type> tol(1000);
+         static boost::uintmax_t max_iter = 1000;
+         mp_type (*pdg)(mp_type) = &boost::math::digamma;
+         static const mp_type root = boost::math::tools::bracket_and_solve_root(pdg, mp_type(1.4), mp_type(1.5), true, tol, max_iter).first;
+
+         mp_type x2 = x;
+         double lim = 1e-65;
+         if(fabs(x2 - root) < lim)
+         {
+            //
+            // This is a problem area:
+            // x2-root suffers cancellation error, so does digamma.
+            // That gets compounded again when Remez calculates the error
+            // function.  This cludge seems to stop the worst of the problems:
+            //
+            static const mp_type a = boost::math::digamma(root - lim) / -lim;
+            static const mp_type b = boost::math::digamma(root + lim) / lim;
+            mp_type fract = (x2 - root + lim) / (2*lim);
+            mp_type r = (1-fract) * a + fract * b;
+            std::cout << "In root area: " << r;
+            return r;
+         }
+         mp_type result =  boost::math::digamma(x2) / (x2 - root);
+         if(current_precision < 1000)
+            set_working_precision(current_precision);
+         return result;
+      }
+   case 11:
+      // expm1:
+      if(x == 0)
+      {
+         static mp_type lim = 1e-80;
+         static mp_type a = boost::math::expm1(-lim);
+         static mp_type b = boost::math::expm1(lim);
+         static mp_type l = (b-a) / (2 * lim);
+         return l;
+      }
+      return boost::math::expm1(x) / x;
+   case 12:
+      // demo, and test case:
+      return exp(x);
+   case 13:
+      // K(k):
+      {
+      return boost::math::ellint_1(x);
+      }
+   case 14:
+      // K(k)
+      {
+      return boost::math::ellint_1(1-x) / log(x);
+   }
+   case 15:
+      // E(k)
+      {
+         // x = 1-k^2
+         mp_type z = 1 - x * log(x);
+         return boost::math::ellint_2(sqrt(1-x)) / z;
+      }
+   case 16:
+      // Bessel I0(x) over [0,16]:
+      {
+         return boost::math::cyl_bessel_i(0, sqrt(x));
+      }
+   case 17:
+      // Bessel I0(x) over [16,INF]
+      {
+         mp_type z = 1 / (mp_type(1)/16 - x);
+         return boost::math::cyl_bessel_i(0, z) * sqrt(z) / exp(z);
+      }
+   case 18:
+      // Zeta over [0, 1]
+      {
+         return boost::math::zeta(1 - x) * x - x;
+      }
+   case 19:
+      // Zeta over [1, n]
+      {
+         return boost::math::zeta(x) - 1 / (x - 1);
+      }
+   case 20:
+      // Zeta over [a, b] : a >> 1
+      {
+         return log(boost::math::zeta(x) - 1);
+      }
+   case 21:
+      // expint[1] over [0,1]:
+      {
+         mp_type tiny = boost::lexical_cast<mp_type>("1e-5000");
+         mp_type z = (x <= tiny) ? tiny : x;
+         return boost::math::expint(1, z) - z + log(z);
+      }
+   case 22:
+      // expint[1] over [1,N],
+      // Note that x varies from [0,1]:
+      {
+         mp_type z = 1 / x;
+         return boost::math::expint(1, z) * exp(z) * z;
+      }
+   case 23:
+      // expin Ei over [0,R]
+      {
+         static const mp_type root = 
+            boost::lexical_cast<mp_type>("0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392");
+         mp_type z = x < (std::numeric_limits<long double>::min)() ? (std::numeric_limits<long double>::min)() : x;
+         return (boost::math::expint(z) - log(z / root)) / (z - root);
+      }
+   case 24:
+      // Expint Ei for large x:
+      {
+         static const mp_type root = 
+            boost::lexical_cast<mp_type>("0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392");
+         mp_type z = x < (std::numeric_limits<long double>::min)() ? (std::numeric_limits<long double>::max)() : mp_type(1 / x);
+         return (boost::math::expint(z) - z) * z * exp(-z);
+         //return (boost::math::expint(z) - log(z)) * z * exp(-z);
+      }
+   case 25:
+      // Expint Ei for large x:
+      {
+         return (boost::math::expint(x) - x) * x * exp(-x);
+      }
+   case 26:
+      {
+         //
+         // erf_inv in range [0, 0.5]
+         //
+         mp_type y = x;
+         if(y == 0)
+            y = boost::math::tools::epsilon<mp_type>() / 64;
+         y = sqrt(y);
+         return boost::math::erf_inv(y) / (y);
+      }
+   case 28:
+      {
+         // log1p over [-0.5,0.5]
+         mp_type y = x;
+         if(fabs(y) < 1e-100)
+            y = (y == 0) ? 1e-100 : boost::math::sign(y) * 1e-100;
+         return (boost::math::log1p(y) - y + y * y / 2) / (y);
+      }
+   case 29:
+      {
+         // cbrt over [0.5, 1]
+         return boost::math::cbrt(x);
+      }
+   case 30:
+   {
+      // trigamma over [x,y]
+      mp_type y = x;
+      y = sqrt(y);
+      return boost::math::trigamma(x) * (x * x);
+   }
+   case 31:
+   {
+      // trigamma over [x, INF]
+      if(x == 0) return 1;
+      mp_type y = (x == 0) ? (std::numeric_limits<double>::max)() / 2 : mp_type(1/x);
+      return boost::math::trigamma(y) * y;
+   }
+   case 32:
+   {
+      // I0 over [N, INF]
+      // Don't need to go past x = 1/1000 = 1e-3 for double, or
+      // 1/15000 = 0.0006 for long double, start at 1/7.75=0.13
+      mp_type arg = 1 / x;
+      return sqrt(arg) * exp(-arg) * boost::math::cyl_bessel_i(0, arg);
+   }
+   case 33:
+   {
+      // I0 over [0, N]
+      mp_type xx = sqrt(x) * 2;
+      return (boost::math::cyl_bessel_i(0, xx) - 1) / x;
+   }
+   case 34:
+   {
+      // I1 over [0, N]
+      mp_type xx = sqrt(x) * 2;
+      return (boost::math::cyl_bessel_i(1, xx) * 2 / xx - 1 - x / 2) / (x * x);
+   }
+   case 35:
+   {
+      // I1 over [N, INF]
+      mp_type xx = 1 / x;
+      return boost::math::cyl_bessel_i(1, xx) * sqrt(xx) * exp(-xx);
+   }
+   case 36:
+   {
+      // K0 over [0, 1]
+      mp_type xx = sqrt(x);
+      return boost::math::cyl_bessel_k(0, xx) + log(xx) * boost::math::cyl_bessel_i(0, xx);
+   }
+   case 37:
+   {
+      // K0 over [1, INF]
+      mp_type xx = 1 / x;
+      return boost::math::cyl_bessel_k(0, xx) * exp(xx) * sqrt(xx);
+   }
+   case 38:
+   {
+      // K1 over [0, 1]
+      mp_type xx = sqrt(x);
+      return (boost::math::cyl_bessel_k(1, xx) - log(xx) * boost::math::cyl_bessel_i(1, xx) - 1 / xx) / xx;
+   }
+   case 39:
+   {
+      // K1 over [1, INF]
+      mp_type xx = 1 / x;
+      return boost::math::cyl_bessel_k(1, xx) * sqrt(xx) * exp(xx);
+   }
+   // Lambert W0
+   case 40:
+      return boost::math::lambert_w0(x);
+   case 41:
+   {
+      if (x == 0)
+         return 1;
+      return boost::math::lambert_w0(x) / x;
+   }
+   case 42:
+   {
+      static const mp_type e1 = exp(mp_type(-1));
+      return x / -boost::math::lambert_w0(-e1 + x);
+   }
+   case 43:
+   {
+      mp_type xx = 1 / x;
+      return 1 / boost::math::lambert_w0(xx);
+   }
+   case 44:
+   {
+      mp_type ex = exp(x);
+      return boost::math::lambert_w0(ex) - x;
+   }
+   }
+   return 0;
+}
+
+void show_extra(
+   const boost::math::tools::polynomial<mp_type>& n, 
+   const boost::math::tools::polynomial<mp_type>& d, 
+   const mp_type& x_offset, 
+   const mp_type& y_offset, 
+   int variant)
+{
+   switch(variant)
+   {
+   default:
+      // do nothing here...
+      ;
+   }
+}
+
diff --git a/src/boost/libs/math/minimax/main.cpp b/src/boost/libs/math/minimax/main.cpp
new file mode 100644
index 000000000..6ff018762
--- /dev/null
+++ b/src/boost/libs/math/minimax/main.cpp
@@ -0,0 +1,650 @@
+//  (C) Copyright John Maddock 2006.
+//  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)
+
+#define BOOST_TEST_MODULE foobar
+#define BOOST_UBLAS_TYPE_CHECK_EPSILON (type_traits<real_type>::type_sqrt (boost::math::tools::epsilon <real_type>()))
+#define BOOST_UBLAS_TYPE_CHECK_MIN (type_traits<real_type>::type_sqrt ( boost::math::tools::min_value<real_type>()))
+#define BOOST_UBLAS_NDEBUG
+
+#include "multiprecision.hpp"
+
+#include <boost/math/tools/remez.hpp>
+#include <boost/math/tools/test.hpp>
+#include <boost/math/special_functions/binomial.hpp>
+#include <boost/spirit/include/classic_core.hpp>
+#include <boost/spirit/include/classic_actor.hpp>
+#include <boost/lexical_cast.hpp>
+#include <iostream>
+#include <iomanip>
+#include <string>
+#include <boost/test/included/unit_test.hpp> // for test_main
+#include <boost/multiprecision/cpp_bin_float.hpp>
+
+
+extern mp_type f(const mp_type& x, int variant);
+extern void show_extra(
+   const boost::math::tools::polynomial<mp_type>& n, 
+   const boost::math::tools::polynomial<mp_type>& d, 
+   const mp_type& x_offset, 
+   const mp_type& y_offset, 
+   int variant);
+
+using namespace boost::spirit::classic;
+
+mp_type a(0), b(1);   // range to optimise over
+bool rel_error(true);
+bool pin(false);
+int orderN(3);
+int orderD(1);
+int target_precision = boost::math::tools::digits<long double>();
+int working_precision = target_precision * 2;
+bool started(false);
+int variant(0);
+int skew(0);
+int brake(50);
+mp_type x_offset(0), y_offset(0), x_scale(1);
+bool auto_offset_y;
+
+boost::shared_ptr<boost::math::tools::remez_minimax<mp_type> > p_remez;
+
+mp_type the_function(const mp_type& val)
+{
+   return f(x_scale * (val + x_offset), variant) + y_offset;
+}
+
+void step_some(unsigned count)
+{
+   try{
+      set_working_precision(working_precision);
+      if(!started)
+      {
+         //
+         // If we have an automatic y-offset calculate it now:
+         //
+         if(auto_offset_y)
+         {
+            mp_type fa, fb, fm;
+            fa = f(x_scale * (a + x_offset), variant);
+            fb = f(x_scale * (b + x_offset), variant);
+            fm = f(x_scale * ((a+b)/2 + x_offset), variant);
+            y_offset = -(fa + fb + fm) / 3;
+            set_output_precision(5);
+            std::cout << "Setting auto-y-offset to " << y_offset << std::endl;
+         }
+         //
+         // Truncate offsets to float precision:
+         //
+         x_offset = round_to_precision(x_offset, 20);
+         y_offset = round_to_precision(y_offset, 20);
+         //
+         // Construct new Remez state machine:
+         //
+         p_remez.reset(new boost::math::tools::remez_minimax<mp_type>(
+            &the_function, 
+            orderN, orderD, 
+            a, b, 
+            pin, 
+            rel_error, 
+            skew, 
+            working_precision));
+         std::cout << "Max error in interpolated form: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(p_remez->max_error()) << std::endl;
+         //
+         // Signal that we've started:
+         //
+         started = true;
+      }
+      unsigned i;
+      for(i = 0; i < count; ++i)
+      {
+         std::cout << "Stepping..." << std::endl;
+         p_remez->set_brake(brake);
+         mp_type r = p_remez->iterate();
+         set_output_precision(3);
+         std::cout 
+            << "Maximum Deviation Found:                     " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(p_remez->max_error()) << std::endl
+            << "Expected Error Term:                         " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(p_remez->error_term()) << std::endl
+            << "Maximum Relative Change in Control Points:   " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast<double>(r) << std::endl;
+      }
+   }
+   catch(const std::exception& e)
+   {
+      std::cout << "Step failed with exception: " << e.what() << std::endl;
+   }
+}
+
+void step(const char*, const char*)
+{
+   step_some(1);
+}
+
+void show(const char*, const char*)
+{
+   set_working_precision(working_precision);
+   if(started)
+   {
+      boost::math::tools::polynomial<mp_type> n = p_remez->numerator();
+      boost::math::tools::polynomial<mp_type> d = p_remez->denominator();
+      std::vector<mp_type> cn = n.chebyshev();
+      std::vector<mp_type> cd = d.chebyshev();
+      int prec = 2 + (target_precision * 3010LL)/10000;
+      std::cout << std::scientific << std::setprecision(prec);
+      set_output_precision(prec);
+      boost::numeric::ublas::vector<mp_type> v = p_remez->zero_points();
+      
+      std::cout << "  Zeros = {\n";
+      unsigned i;
+      for(i = 0; i < v.size(); ++i)
+      {
+         std::cout << "    " << v[i] << std::endl;
+      }
+      std::cout << "  }\n";
+
+      v = p_remez->chebyshev_points();
+      std::cout << "  Chebeshev Control Points = {\n";
+      for(i = 0; i < v.size(); ++i)
+      {
+         std::cout << "    " << v[i] << std::endl;
+      }
+      std::cout << "  }\n";
+
+      std::cout << "X offset: " << x_offset << std::endl;
+      std::cout << "X scale:  " << x_scale << std::endl;
+      std::cout << "Y offset: " << y_offset << std::endl;
+
+      std::cout << "P = {";
+      for(i = 0; i < n.size(); ++i)
+      {
+         std::cout << "    " << n[i] << "L," << std::endl;
+      }
+      std::cout << "  }\n";
+
+      std::cout << "Q = {";
+      for(i = 0; i < d.size(); ++i)
+      {
+         std::cout << "    " << d[i] << "L," << std::endl;
+      }
+      std::cout << "  }\n";
+
+      std::cout << "CP = {";
+      for(i = 0; i < cn.size(); ++i)
+      {
+         std::cout << "    " << cn[i] << "L," << std::endl;
+      }
+      std::cout << "  }\n";
+
+      std::cout << "CQ = {";
+      for(i = 0; i < cd.size(); ++i)
+      {
+         std::cout << "    " << cd[i] << "L," << std::endl;
+      }
+      std::cout << "  }\n";
+
+      show_extra(n, d, x_offset, y_offset, variant);
+   }
+   else
+   {
+      std::cerr << "Nothing to display" << std::endl;
+   }
+}
+
+void do_graph(unsigned points)
+{
+   set_working_precision(working_precision);
+   mp_type step = (b - a) / (points - 1);
+   mp_type x = a;
+   while(points > 1)
+   {
+      set_output_precision(10);
+      std::cout << std::setprecision(10) << std::setw(30) << std::left 
+         << boost::lexical_cast<std::string>(x) << the_function(x) << std::endl;
+      --points;
+      x += step;
+   }
+   std::cout << std::setprecision(10) << std::setw(30) << std::left 
+      << boost::lexical_cast<std::string>(b) << the_function(b) << std::endl;
+}
+
+void graph(const char*, const char*)
+{
+   do_graph(3);
+}
+
+template <class T>
+mp_type convert_to_rr(const T& val)
+{
+   return val;
+}
+template <class Backend, boost::multiprecision::expression_template_option ET>
+mp_type convert_to_rr(const boost::multiprecision::number<Backend, ET>& val)
+{
+   return boost::lexical_cast<mp_type>(val.str());
+}
+
+template <class T>
+void do_test(T, const char* name)
+{
+   set_working_precision(working_precision);
+   if(started)
+   {
+      //
+      // We want to test the approximation at fixed precision:
+      // either float, double or long double.  Begin by getting the
+      // polynomials:
+      //
+      boost::math::tools::polynomial<T> n, d;
+      boost::math::tools::polynomial<mp_type> nr, dr;
+      nr = p_remez->numerator();
+      dr = p_remez->denominator();
+      n = nr;
+      d = dr;
+
+      std::vector<mp_type> cn1, cd1;
+      cn1 = nr.chebyshev();
+      cd1 = dr.chebyshev();
+      std::vector<T> cn, cd;
+      for(unsigned i = 0; i < cn1.size(); ++i)
+      {
+         cn.push_back(boost::math::tools::real_cast<T>(cn1[i]));
+      }
+      for(unsigned i = 0; i < cd1.size(); ++i)
+      {
+         cd.push_back(boost::math::tools::real_cast<T>(cd1[i]));
+      }
+      //
+      // We'll test at the Chebeshev control points which is where
+      // (in theory) the largest deviation should occur.  For good
+      // measure we'll test at the zeros as well:
+      //
+      boost::numeric::ublas::vector<mp_type> 
+         zeros(p_remez->zero_points()),
+         cheb(p_remez->chebyshev_points());
+
+      mp_type max_error(0), cheb_max_error(0);
+
+      //
+      // Do the tests at the zeros:
+      //
+      std::cout << "Starting tests at " << name << " precision...\n";
+      std::cout << "Absissa        Error (Poly)   Error (Cheb)\n";
+      for(unsigned i = 0; i < zeros.size(); ++i)
+      {
+         mp_type true_result = the_function(zeros[i]);
+         T absissa = boost::math::tools::real_cast<T>(zeros[i]);
+         mp_type test_result = convert_to_rr(n.evaluate(absissa) / d.evaluate(absissa));
+         mp_type cheb_result = convert_to_rr(boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa));
+         mp_type err, cheb_err;
+         if(rel_error)
+         {
+            err = boost::math::tools::relative_error(test_result, true_result);
+            cheb_err = boost::math::tools::relative_error(cheb_result, true_result);
+         }
+         else
+         {
+            err = fabs(test_result - true_result);
+            cheb_err = fabs(cheb_result - true_result);
+         }
+         if(err > max_error)
+            max_error = err;
+         if(cheb_err > cheb_max_error)
+            cheb_max_error = cheb_err;
+         std::cout << std::setprecision(6) << std::setw(15) << std::left << absissa
+            << std::setw(15) << std::left << boost::math::tools::real_cast<T>(err) << boost::math::tools::real_cast<T>(cheb_err) << std::endl;
+      }
+      //
+      // Do the tests at the Chebeshev control points:
+      //
+      for(unsigned i = 0; i < cheb.size(); ++i)
+      {
+         mp_type true_result = the_function(cheb[i]);
+         T absissa = boost::math::tools::real_cast<T>(cheb[i]);
+         mp_type test_result = convert_to_rr(n.evaluate(absissa) / d.evaluate(absissa));
+         mp_type cheb_result = convert_to_rr(boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa));
+         mp_type err, cheb_err;
+         if(rel_error)
+         {
+            err = boost::math::tools::relative_error(test_result, true_result);
+            cheb_err = boost::math::tools::relative_error(cheb_result, true_result);
+         }
+         else
+         {
+            err = fabs(test_result - true_result);
+            cheb_err = fabs(cheb_result - true_result);
+         }
+         if(err > max_error)
+            max_error = err;
+         std::cout << std::setprecision(6) << std::setw(15) << std::left << absissa
+            << std::setw(15) << std::left << boost::math::tools::real_cast<T>(err) << 
+            boost::math::tools::real_cast<T>(cheb_err) << std::endl;
+      }
+      std::string msg = "Max Error found at ";
+      msg += name;
+      msg += " precision = ";
+      msg.append(62 - 17 - msg.size(), ' ');
+      std::cout << msg << std::setprecision(6) << "Poly: " << std::setw(20) << std::left
+         << boost::math::tools::real_cast<T>(max_error) << "Cheb: " << boost::math::tools::real_cast<T>(cheb_max_error) << std::endl;
+   }
+   else
+   {
+      std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl;
+   }
+}
+
+void test_float(const char*, const char*)
+{
+   do_test(float(0), "float");
+}
+
+void test_double(const char*, const char*)
+{
+   do_test(double(0), "double");
+}
+
+void test_long(const char*, const char*)
+{
+   do_test((long double)(0), "long double");
+}
+
+void test_float80(const char*, const char*)
+{
+   do_test((boost::multiprecision::cpp_bin_float_double_extended)(0), "float80");
+}
+
+void test_float128(const char*, const char*)
+{
+   do_test((boost::multiprecision::cpp_bin_float_quad)(0), "float128");
+}
+
+void test_all(const char*, const char*)
+{
+   do_test(float(0), "float");
+   do_test(double(0), "double");
+   do_test((long double)(0), "long double");
+}
+
+template <class T>
+void do_test_n(T, const char* name, unsigned count)
+{
+   set_working_precision(working_precision);
+   if(started)
+   {
+      //
+      // We want to test the approximation at fixed precision:
+      // either float, double or long double.  Begin by getting the
+      // polynomials:
+      //
+      boost::math::tools::polynomial<T> n, d;
+      boost::math::tools::polynomial<mp_type> nr, dr;
+      nr = p_remez->numerator();
+      dr = p_remez->denominator();
+      n = nr;
+      d = dr;
+
+      std::vector<mp_type> cn1, cd1;
+      cn1 = nr.chebyshev();
+      cd1 = dr.chebyshev();
+      std::vector<T> cn, cd;
+      for(unsigned i = 0; i < cn1.size(); ++i)
+      {
+         cn.push_back(boost::math::tools::real_cast<T>(cn1[i]));
+      }
+      for(unsigned i = 0; i < cd1.size(); ++i)
+      {
+         cd.push_back(boost::math::tools::real_cast<T>(cd1[i]));
+      }
+
+      mp_type max_error(0), max_cheb_error(0);
+      mp_type step = (b - a) / count;
+
+      //
+      // Do the tests at the zeros:
+      //
+      std::cout << "Starting tests at " << name << " precision...\n";
+      std::cout << "Absissa        Error (poly)   Error (Cheb)\n";
+      for(mp_type x = a; x <= b; x += step)
+      {
+         mp_type true_result = the_function(x);
+         //std::cout << true_result << std::endl;
+         T absissa = boost::math::tools::real_cast<T>(x);
+         mp_type test_result = convert_to_rr(n.evaluate(absissa) / d.evaluate(absissa));
+         //std::cout << test_result << std::endl;
+         mp_type cheb_result = convert_to_rr(boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa));
+         //std::cout << cheb_result << std::endl;
+         mp_type err, cheb_err;
+         if(rel_error)
+         {
+            err = boost::math::tools::relative_error(test_result, true_result);
+            cheb_err = boost::math::tools::relative_error(cheb_result, true_result);
+         }
+         else
+         {
+            err = fabs(test_result - true_result);
+            cheb_err = fabs(cheb_result - true_result);
+         }
+         if(err > max_error)
+            max_error = err;
+         if(cheb_err > max_cheb_error)
+            max_cheb_error = cheb_err;
+         std::cout << std::setprecision(6) << std::setw(15) << std::left << boost::math::tools::real_cast<double>(absissa)
+            << (test_result < true_result ? "-" : "") << std::setw(20) << std::left 
+            << boost::math::tools::real_cast<double>(err) 
+            << boost::math::tools::real_cast<double>(cheb_err) << std::endl;
+      }
+      std::string msg = "Max Error found at ";
+      msg += name;
+      msg += " precision = ";
+      //msg.append(62 - 17 - msg.size(), ' ');
+      std::cout << msg << "Poly: " << std::setprecision(6) 
+         //<< std::setw(15) << std::left 
+         << boost::math::tools::real_cast<T>(max_error) 
+         << " Cheb: " << boost::math::tools::real_cast<T>(max_cheb_error) << std::endl;
+   }
+   else
+   {
+      std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl;
+   }
+}
+
+void test_n(unsigned n)
+{
+   do_test_n(mp_type(), "mp_type", n);
+}
+
+void test_float_n(unsigned n)
+{
+   do_test_n(float(0), "float", n);
+}
+
+void test_double_n(unsigned n)
+{
+   do_test_n(double(0), "double", n);
+}
+
+void test_long_n(unsigned n)
+{
+   do_test_n((long double)(0), "long double", n);
+}
+
+void test_float80_n(unsigned n)
+{
+   do_test_n((boost::multiprecision::cpp_bin_float_double_extended)(0), "float80", n);
+}
+
+void test_float128_n(unsigned n)
+{
+   do_test_n((boost::multiprecision::cpp_bin_float_quad)(0), "float128", n);
+}
+
+void rotate(const char*, const char*)
+{
+   if(p_remez)
+   {
+      p_remez->rotate();
+   }
+   else
+   {
+      std::cerr << "Nothing to rotate" << std::endl;
+   }
+}
+
+void rescale(const char*, const char*)
+{
+   if(p_remez)
+   {
+      p_remez->rescale(a, b);
+   }
+   else
+   {
+      std::cerr << "Nothing to rescale" << std::endl;
+   }
+}
+
+void graph_poly(const char*, const char*)
+{
+   int i = 50;
+   set_working_precision(working_precision);
+   if(started)
+   {
+      //
+      // We want to test the approximation at fixed precision:
+      // either float, double or long double.  Begin by getting the
+      // polynomials:
+      //
+      boost::math::tools::polynomial<mp_type> n, d;
+      n = p_remez->numerator();
+      d = p_remez->denominator();
+
+      mp_type max_error(0);
+      mp_type step = (b - a) / i;
+
+      std::cout << "Evaluating Numerator...\n";
+      mp_type val;
+      for(val = a; val <= b; val += step)
+         std::cout << n.evaluate(val) << std::endl;
+      std::cout << "Evaluating Denominator...\n";
+      for(val = a; val <= b; val += step)
+         std::cout << d.evaluate(val) << std::endl;
+   }
+   else
+   {
+      std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl;
+   }
+}
+
+BOOST_AUTO_TEST_CASE( test_main )
+{
+   std::string line;
+   real_parser<long double/*mp_type*/ > const rr_p;
+   while(std::getline(std::cin, line))
+   {
+      if(parse(line.c_str(), str_p("quit"), space_p).full)
+         return;
+      if(false == parse(line.c_str(), 
+         (
+
+            str_p("range")[assign_a(started, false)] && real_p[assign_a(a)] && real_p[assign_a(b)]
+      ||
+            str_p("relative")[assign_a(started, false)][assign_a(rel_error, true)]
+      ||
+            str_p("absolute")[assign_a(started, false)][assign_a(rel_error, false)]
+      ||
+            str_p("pin")[assign_a(started, false)] && str_p("true")[assign_a(pin, true)]
+      ||
+            str_p("pin")[assign_a(started, false)] && str_p("false")[assign_a(pin, false)]
+      ||
+            str_p("pin")[assign_a(started, false)] && str_p("1")[assign_a(pin, true)]
+      ||
+            str_p("pin")[assign_a(started, false)] && str_p("0")[assign_a(pin, false)]
+      ||
+            str_p("pin")[assign_a(started, false)][assign_a(pin, true)]
+      ||
+            str_p("order")[assign_a(started, false)] && uint_p[assign_a(orderN)] && uint_p[assign_a(orderD)]
+      ||
+            str_p("order")[assign_a(started, false)] && uint_p[assign_a(orderN)]
+      ||
+            str_p("target-precision") && uint_p[assign_a(target_precision)]
+      ||
+            str_p("working-precision")[assign_a(started, false)] && uint_p[assign_a(working_precision)]
+      ||
+            str_p("variant")[assign_a(started, false)] && int_p[assign_a(variant)]
+      ||
+            str_p("skew")[assign_a(started, false)] && int_p[assign_a(skew)]
+      ||
+            str_p("brake") && int_p[assign_a(brake)]
+      ||
+            str_p("step") && int_p[&step_some]
+      ||
+            str_p("step")[&step]
+      ||
+            str_p("poly")[&graph_poly]
+      ||
+            str_p("info")[&show]
+      ||
+            str_p("graph") && uint_p[&do_graph]
+      ||
+            str_p("graph")[&graph]
+      ||
+            str_p("x-offset") && real_p[assign_a(x_offset)]
+      ||
+            str_p("x-scale") && real_p[assign_a(x_scale)]
+      ||
+            str_p("y-offset") && str_p("auto")[assign_a(auto_offset_y, true)]
+      ||
+            str_p("y-offset") && real_p[assign_a(y_offset)][assign_a(auto_offset_y, false)]
+      ||
+            str_p("test") && str_p("float80") && uint_p[&test_float80_n]
+      ||
+            str_p("test") && str_p("float80")[&test_float80]
+      ||
+            str_p("test") && str_p("float128") && uint_p[&test_float128_n]
+      ||
+            str_p("test") && str_p("float128")[&test_float128]
+      ||
+            str_p("test") && str_p("float") && uint_p[&test_float_n]
+      ||
+            str_p("test") && str_p("float")[&test_float]
+      ||
+            str_p("test") && str_p("double") && uint_p[&test_double_n]
+      ||
+            str_p("test") && str_p("double")[&test_double]
+      ||
+            str_p("test") && str_p("long") && uint_p[&test_long_n]
+      ||
+            str_p("test") && str_p("long")[&test_long]
+      ||
+            str_p("test") && str_p("all")[&test_all]
+      ||
+            str_p("test") && uint_p[&test_n]
+      ||
+            str_p("rotate")[&rotate]
+      ||
+            str_p("rescale") && real_p[assign_a(a)] && real_p[assign_a(b)] && epsilon_p[&rescale]
+
+         ), space_p).full)
+      {
+         std::cout << "Unable to parse directive: \"" << line << "\"" << std::endl;
+      }
+      else
+      {
+         std::cout << "Variant              = " << variant << std::endl;
+         std::cout << "range                = [" << a << "," << b << "]" << std::endl;
+         std::cout << "Relative Error       = " << rel_error << std::endl;
+         std::cout << "Pin to Origin        = " << pin << std::endl;
+         std::cout << "Order (Num/Denom)    = " << orderN << "/" << orderD << std::endl;
+         std::cout << "Target Precision     = " << target_precision << std::endl;
+         std::cout << "Working Precision    = " << working_precision << std::endl;
+         std::cout << "Skew                 = " << skew << std::endl;
+         std::cout << "Brake                = " << brake << std::endl;
+         std::cout << "X Offset             = " << x_offset << std::endl;
+         std::cout << "X scale              = " << x_scale << std::endl;
+         std::cout << "Y Offset             = ";
+         if(auto_offset_y)
+            std::cout << "Auto (";
+         std::cout << y_offset;
+         if(auto_offset_y)
+            std::cout << ")";
+         std::cout << std::endl;
+     }
+   }
+}
diff --git a/src/boost/libs/math/minimax/multiprecision.hpp b/src/boost/libs/math/minimax/multiprecision.hpp
new file mode 100644
index 000000000..2f44ac07b
--- /dev/null
+++ b/src/boost/libs/math/minimax/multiprecision.hpp
@@ -0,0 +1,224 @@
+//  (C) Copyright John Maddock 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)
+
+#ifndef BOOST_REMEZ_MULTIPRECISION_HPP
+#define BOOST_REMEZ_MULTIPRECISION_HPP
+
+#include <boost/multiprecision/cpp_bin_float.hpp>
+
+#ifdef USE_NTL
+#include <boost/math/bindings/rr.hpp>
+namespace std {
+   using boost::math::ntl::pow;
+} // workaround for spirit parser.
+
+typedef boost::math::ntl::RR mp_type;
+
+inline void set_working_precision(int n)
+{
+   NTL::RR::SetPrecision(working_precision);
+}
+
+inline int get_working_precision()
+{
+   return mp_type::precision(working_precision);
+}
+
+inline void set_output_precision(int n)
+{
+   NTL::RR::SetOutputPrecision(n);
+}
+
+inline mp_type round_to_precision(mp_type m, int bits)
+{
+   return NTL::RoundToPrecision(m.value(), bits);
+}
+
+
+namespace boost {
+   namespace math {
+      namespace tools {
+
+         template <>
+         inline boost::multiprecision::cpp_bin_float_double_extended real_cast<boost::multiprecision::cpp_bin_float_double_extended, mp_type>(mp_type val)
+         {
+            unsigned p = NTL::RR::OutputPrecision();
+            NTL::RR::SetOutputPrecision(20);
+            boost::multiprecision::cpp_bin_float_double_extended r = boost::lexical_cast<boost::multiprecision::cpp_bin_float_double_extended>(val);
+            NTL::RR::SetOutputPrecision(p);
+            return r;
+         }
+         template <>
+         inline boost::multiprecision::cpp_bin_float_quad real_cast<boost::multiprecision::cpp_bin_float_quad, mp_type>(mp_type val)
+         {
+            unsigned p = NTL::RR::OutputPrecision();
+            NTL::RR::SetOutputPrecision(35);
+            boost::multiprecision::cpp_bin_float_quad r = boost::lexical_cast<boost::multiprecision::cpp_bin_float_quad>(val);
+            NTL::RR::SetOutputPrecision(p);
+            return r;
+         }
+
+      }
+   }
+}
+
+#elif defined(USE_CPP_BIN_FLOAT_100)
+
+#include <boost/multiprecision/cpp_bin_float.hpp>
+
+typedef boost::multiprecision::cpp_bin_float_100 mp_type;
+
+inline void set_working_precision(int n)
+{
+}
+
+inline void set_output_precision(int n)
+{
+   std::cout << std::setprecision(n);
+   std::cerr << std::setprecision(n);
+}
+
+inline mp_type round_to_precision(mp_type m, int bits)
+{
+   int i;
+   mp_type f = frexp(m, &i);
+   f = ldexp(f, bits);
+   i -= bits;
+   f = floor(f);
+   return ldexp(f, i);
+}
+
+inline int get_working_precision()
+{
+   return std::numeric_limits<mp_type>::digits;
+}
+
+namespace boost {
+   namespace math {
+      namespace tools {
+
+         template <>
+         inline boost::multiprecision::cpp_bin_float_double_extended real_cast<boost::multiprecision::cpp_bin_float_double_extended, mp_type>(mp_type val)
+         {
+            return boost::multiprecision::cpp_bin_float_double_extended(val);
+         }
+         template <>
+         inline boost::multiprecision::cpp_bin_float_quad real_cast<boost::multiprecision::cpp_bin_float_quad, mp_type>(mp_type val)
+         {
+            return boost::multiprecision::cpp_bin_float_quad(val);
+         }
+
+      }
+   }
+}
+
+
+#elif defined(USE_MPFR_100)
+
+#include <boost/multiprecision/mpfr.hpp>
+
+typedef boost::multiprecision::mpfr_float_100 mp_type;
+
+inline void set_working_precision(int n)
+{
+}
+
+inline void set_output_precision(int n)
+{
+   std::cout << std::setprecision(n);
+   std::cerr << std::setprecision(n);
+}
+
+inline mp_type round_to_precision(mp_type m, int bits)
+{
+   mpfr_prec_round(m.backend().data(), bits, MPFR_RNDN);
+   return m;
+}
+
+inline int get_working_precision()
+{
+   return std::numeric_limits<mp_type>::digits;
+}
+
+namespace boost {
+   namespace math {
+      namespace tools {
+
+         template <>
+         inline boost::multiprecision::cpp_bin_float_double_extended real_cast<boost::multiprecision::cpp_bin_float_double_extended, mp_type>(mp_type val)
+         {
+            return boost::multiprecision::cpp_bin_float_double_extended(val);
+         }
+         template <>
+         inline boost::multiprecision::cpp_bin_float_quad real_cast<boost::multiprecision::cpp_bin_float_quad, mp_type>(mp_type val)
+         {
+            return boost::multiprecision::cpp_bin_float_quad(val);
+         }
+
+      }
+   }
+}
+
+
+#else
+
+#include <boost/multiprecision/mpfr.hpp>
+
+typedef boost::multiprecision::mpfr_float mp_type;
+
+inline void set_working_precision(int n)
+{
+   boost::multiprecision::mpfr_float::default_precision(boost::multiprecision::detail::digits2_2_10(n));
+}
+
+inline void set_output_precision(int n)
+{
+   std::cout << std::setprecision(n);
+   std::cerr << std::setprecision(n);
+}
+
+inline mp_type round_to_precision(mp_type m, int bits)
+{
+   mpfr_prec_round(m.backend().data(), bits, MPFR_RNDN);
+   return m;
+}
+
+inline int get_working_precision()
+{
+   return mp_type::default_precision();
+}
+
+namespace boost {
+   namespace math {
+      namespace tools {
+
+         template <>
+         inline boost::multiprecision::cpp_bin_float_double_extended real_cast<boost::multiprecision::cpp_bin_float_double_extended, mp_type>(mp_type val)
+         {
+            return boost::multiprecision::cpp_bin_float_double_extended(val);
+         }
+         template <>
+         inline boost::multiprecision::cpp_bin_float_quad real_cast<boost::multiprecision::cpp_bin_float_quad, mp_type>(mp_type val)
+         {
+            return boost::multiprecision::cpp_bin_float_quad(val);
+         }
+
+      }
+   }
+}
+
+
+
+#endif
+
+
+
+
+#endif // BOOST_REMEZ_MULTIPRECISION_HPP
+
+
+
+
+