complex_test.cpp

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//  (C) Copyright John Maddock 2005.
//  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)

#include <boost/test/test_tools.hpp>
#include <boost/test/included/test_exec_monitor.hpp>
#include <boost/test/floating_point_comparison.hpp>
#include <boost/type_traits/is_same.hpp>
#include <boost/type_traits/is_floating_point.hpp>
#include <boost/mpl/if.hpp>
#include <boost/static_assert.hpp>
#include <boost/math/complex.hpp>

#include <iostream>
#include <iomanip>
#include <cmath>
#include <typeinfo>

#ifdef BOOST_NO_STDC_NAMESPACE
namespace std{ using ::sqrt; using ::tan; using ::tanh; }
#endif

#ifndef VERBOSE
#undef BOOST_MESSAGE
#define BOOST_MESSAGE(x)
#endif

//
// check_complex:
// Verifies that expected value "a" and found value "b" have a relative error
// less than "max_error" epsilons.  Note that relative error is calculated for
// the complex number as a whole; this means that the error in the real or 
// imaginary parts alone can be much higher than max_error when the real and 
// imaginary parts are of very different magnitudes.  This is important, because
// the Hull et al analysis of the acos and asin algorithms requires that very small
// real/imaginary components can be safely ignored if they are negligible compared
// to the other component.
//
template <class T>
bool check_complex(const std::complex<T>& a, const std::complex<T>& b, int max_error)
{
   //
   // a is the expected value, b is what was actually found,
   // compute | (a-b)/b | and compare with max_error which is the 
   // multiple of E to permit:
   //
   bool result = true;
   static const std::complex<T> zero(0);
   static const T eps = std::pow(static_cast<T>(std::numeric_limits<T>::radix), 1 - std::numeric_limits<T>::digits);
   if(a == zero)
   {
      if(b != zero)
      {
         if(boost::math::fabs(b) > eps)
         {
            result = false;
            BOOST_ERROR("Expected {0,0} but got: " << b);
         }
         else
         {
            BOOST_MESSAGE("Expected {0,0} but got: " << b);
         }
      }
      return result;
   }
   else if(b == zero)
   {
      if(boost::math::fabs(a) > eps)
      {
         BOOST_ERROR("Found {0,0} but expected: " << a);
         return false;;
      }
      else
      {
         BOOST_MESSAGE("Found {0,0} but expected: " << a);
      }
   }

   T rel = boost::math::fabs((b-a)/b) / eps;
   if( rel > max_error)
   {
      result = false;
      BOOST_ERROR("Error in result exceeded permitted limit of " << max_error << " (actual relative error was " << rel << "e).  Found " << b << " expected " << a);
   }
   return result;
}

//
// test_inverse_trig:
// This is nothing more than a sanity check, computes trig(atrig(z)) 
// and compare the result to z.  Note that:
//
// atrig(trig(z)) != z
//
// for certain z because the inverse trig functions are multi-valued, this 
// essentially rules this out as a testing method.  On the other hand:
//
// trig(atrig(z))
//
// can vary compare to z by an arbitrarily large amount.  For one thing we 
// have no control over the implementation of the trig functions, for another
// even if both functions were accurate to 1ulp (as accurate as transcendental
// number can get, thanks to the "table makers dilemma"), the errors can still
// be arbitrarily large - often the inverse trig functions will map a very large
// part of the complex domain into a small output domain, so you can never get
// back exactly where you started from.  Consequently these tests are no more than
// sanity checks (just verifies that signs are correct and so on).
//
template <class T>
void test_inverse_trig(T)
{
   using namespace std;

   static const T interval = static_cast<T>(2.0L/128.0L);

   T x, y;

   std::cout << std::setprecision(std::numeric_limits<T>::digits10+2);

   for(x = -1; x <= 1; x += interval)
   {
      for(y = -1; y <= 1; y += interval)
      {
         // acos:
         std::complex<T> val(x, y), inter, result;
         inter = boost::math::acos(val);
         result = cos(inter);
         if(!check_complex(val, result, 50))
         {
            std::cout << "Error in testing inverse complex cos for type " << typeid(T).name() << std::endl;
            std::cout << "   val=             " << val << std::endl;
            std::cout << "   acos(val) =      " << inter << std::endl;
            std::cout << "   cos(acos(val)) = " << result << std::endl;
         }
         // asin:
         inter = boost::math::asin(val);
         result = sin(inter);
         if(!check_complex(val, result, 5))
         {
            std::cout << "Error in testing inverse complex sin for type " << typeid(T).name() << std::endl;
            std::cout << "   val=             " << val << std::endl;
            std::cout << "   asin(val) =      " << inter << std::endl;
            std::cout << "   sin(asin(val)) = " << result << std::endl;
         }
      }
   }

   static const T interval2 = static_cast<T>(3.0L/256.0L);
   for(x = -3; x <= 3; x += interval2)
   {
      for(y = -3; y <= 3; y += interval2)
      {
         // asinh:
         std::complex<T> val(x, y), inter, result;
         inter = boost::math::asinh(val);
         result = sinh(inter);
         if(!check_complex(val, result, 5))
         {
            std::cout << "Error in testing inverse complex sinh for type " << typeid(T).name() << std::endl;
            std::cout << "   val=               " << val << std::endl;
            std::cout << "   asinh(val) =       " << inter << std::endl;
            std::cout << "   sinh(asinh(val)) = " << result << std::endl;
         }
         // acosh:
         if(!((y == 0) && (x <= 1))) // can't test along the branch cut
         {
            inter = boost::math::acosh(val);
            result = cosh(inter);
            if(!check_complex(val, result, 60))
            {
               std::cout << "Error in testing inverse complex cosh for type " << typeid(T).name() << std::endl;
               std::cout << "   val=               " << val << std::endl;
               std::cout << "   acosh(val) =       " << inter << std::endl;
               std::cout << "   cosh(acosh(val)) = " << result << std::endl;
            }
         }
         //
         // There is a problem in testing atan and atanh:
         // The inverse functions map a large input range to a much
         // smaller output range, so at the extremes too rather different
         // inputs may map to the same output value once rounded to N places.
         // Consequently tan(atan(z)) can suffer from arbitrarily large errors
         // even if individually they each have a small error bound.  On the other
         // hand we can't test atan(tan(z)) either because atan is multi-valued, so
         // round-tripping in this direction isn't always possible.
         // The following heuristic is designed to make the best of a bad job,
         // using atan(tan(z)) where possible and tan(atan(z)) when it's not.
         //
         static const int tanh_error = 20;
         if((0 != x) && (0 != y) && ((std::fabs(y) < 1) || (std::fabs(x) < 1)))
         {
            // atanh:
            val = boost::math::atanh(val);
            inter = tanh(val);
            result = boost::math::atanh(inter);
            if(!check_complex(val, result, tanh_error))
            {
               std::cout << "Error in testing inverse complex tanh for type " << typeid(T).name() << std::endl;
               std::cout << "   val=               " << val << std::endl;
               std::cout << "   tanh(val) =        " << inter << std::endl;
               std::cout << "   atanh(tanh(val)) = " << result << std::endl;
            }
            // atan:
            if(!((x == 0) && (std::fabs(y) == 1))) // we can't test infinities here
            {
               val = std::complex<T>(x, y);
               val = boost::math::atan(val);
               inter = tan(val);
               result = boost::math::atan(inter);
               if(!check_complex(val, result, tanh_error))
               {
                  std::cout << "Error in testing inverse complex tan for type " << typeid(T).name() << std::endl;
                  std::cout << "   val=               " << val << std::endl;
                  std::cout << "   tan(val) =         " << inter << std::endl;
                  std::cout << "   atan(tan(val)) =   " << result << std::endl;
               }
            }
         }
         else
         {
            // atanh:
            inter = boost::math::atanh(val);
            result = tanh(inter);
            if(!check_complex(val, result, tanh_error))
            {
               std::cout << "Error in testing inverse complex atanh for type " << typeid(T).name() << std::endl;
               std::cout << "   val=                 " << val << std::endl;
               std::cout << "   atanh(val) =         " << inter << std::endl;
               std::cout << "   tanh(atanh(val)) =   " << result << std::endl;
            }
            // atan:
            if(!((x == 0) && (std::fabs(y) == 1))) // we can't test infinities here
            {
               inter = boost::math::atan(val);
               result = tan(inter);
               if(!check_complex(val, result, tanh_error))
               {
                  std::cout << "Error in testing inverse complex atan for type " << typeid(T).name() << std::endl;
                  std::cout << "   val=                 " << val << std::endl;
                  std::cout << "   atan(val) =          " << inter << std::endl;
                  std::cout << "   tan(atan(val)) =     " << result << std::endl;
               }
            }
         }
      }
   }
}

//
// check_spots:
// Various spot values, mostly the C99 special cases (infinites and NAN's).
// TODO: add spot checks for the Wolfram spot values.
//
template <class T>
void check_spots(const T&)
{
   typedef std::complex<T> ct;
   ct result;
   static const T two = 2.0;
   T eps = std::pow(two, 1-std::numeric_limits<T>::digits); // numeric_limits<>::epsilon way too small to be useful on Darwin.
   static const T zero = 0;
   static const T mzero = -zero;
   static const T one = 1;
   static const T pi = static_cast<T>(3.141592653589793238462643383279502884197L);
   static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L);
   static const T quarter_pi = static_cast<T>(0.78539816339744830961566084581987572L);
   static const T three_quarter_pi = static_cast<T>(2.35619449019234492884698253745962716L);
   //static const T log_two = static_cast<T>(0.69314718055994530941723212145817657L);
   T infinity = std::numeric_limits<T>::infinity();
   bool test_infinity = std::numeric_limits<T>::has_infinity;
   T nan = 0;
   bool test_nan = false;
#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x564))
   // numeric_limits reports that a quiet NaN is present
   // but an attempt to access it will terminate the program!!!!
   if(std::numeric_limits<T>::has_quiet_NaN)
      nan = std::numeric_limits<T>::quiet_NaN();
   if(boost::math::detail::test_is_nan(nan))
      test_nan = true;
#endif
#if defined(__DECCXX) && !defined(_IEEE_FP)
   // Tru64 cxx traps infinities unless the -ieee option is used:
   test_infinity = false;
#endif

   //
   // C99 spot tests for acos:
   //
   result = boost::math::acos(ct(zero));