text: // Boost.Geometry
  
  // Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
  
  // Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.
  
  // Use, modification and distribution is 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)
  
  // This file is converted from GeographicLib, https://geographiclib.sourceforge.io
  // GeographicLib is originally written by Charles Karney.
  
  // Author: Charles Karney (2008-2017)
  
  // Last updated version of GeographicLib: 1.49
  
  // Original copyright notice:
  
  // Copyright (c) Charles Karney (2008-2017) <charles@karney.com> and licensed
  // under the MIT/X11 License. For more information, see
  // https://geographiclib.sourceforge.io
  
  #ifndef BOOST_GEOMETRY_UTIL_SERIES_EXPANSION_HPP
  #define BOOST_GEOMETRY_UTIL_SERIES_EXPANSION_HPP
  
  #include <boost/geometry/core/assert.hpp>
  #include <boost/geometry/util/math.hpp>
  
  namespace boost { namespace geometry { namespace series_expansion {
  
      /*
       Generate and evaluate the series expansion of the following integral
  
       I1 = integrate( sqrt(1+k2*sin(sigma1)^2), sigma1, 0, sigma )
  
       which is valid for k2 small. We substitute k2 = 4 * eps / (1 - eps)^2
       and expand (1 - eps) * I1 retaining terms up to order eps^maxpow
       in A1 and C1[l].
  
       The resulting series is of the form
  
       A1 * ( sigma + sum(C1[l] * sin(2*l*sigma), l, 1, maxpow) ).
  
       The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
  
       The expansion above is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_A1 below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
  
       To replace each number x by CT(x) the following
       script can be used:
         sed -e 's/[0-9]\+/CT(&)/g; s/\[CT/\[/g; s/)\]/\]/g;
                 s/case\sCT(/case /g; s/):/:/g; s/epsCT(2)/eps2/g;'
      */
      template <size_t SeriesOrder, typename CT>
      inline CT evaluate_A1(CT eps)
      {
          CT eps2 = math::sqr(eps);
          CT t;
          switch (SeriesOrder/2) {
          case 0:
              t = CT(0);
              break;
          case 1:
              t = eps2/CT(4);
              break;
          case 2:
              t = eps2*(eps2+CT(16))/CT(64);
              break;
          case 3:
              t = eps2*(eps2*(eps2+CT(4))+CT(64))/CT(256);
              break;
          case 4:
              t = eps2*(eps2*(eps2*(CT(25)*eps2+CT(64))+CT(256))+CT(4096))/CT(16384);
              break;
          }
          return (t + eps) / (CT(1) - eps);
      }
  
      /*
       Generate and evaluate the series expansion of the following integral
  
       I2 = integrate( 1/sqrt(1+k2*sin(sigma1)^2), sigma1, 0, sigma )
  
       which is valid for k2 small. We substitute k2 = 4 * eps / (1 - eps)^2
       and expand (1 - eps) * I2 retaining terms up to order eps^maxpow
       in A2 and C2[l].
  
       The resulting series is of the form
  
       A2 * ( sigma + sum(C2[l] * sin(2*l*sigma), l, 1, maxpow) )
  
       The scale factor A2-1 = mean value of (d/dsigma)2 - 1
  
       The expansion above is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_A2 below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
      */
      template <size_t SeriesOrder, typename CT>
      inline CT evaluate_A2(CT const& eps)
      {
          CT const eps2 = math::sqr(eps);
          CT t;
          switch (SeriesOrder/2) {
          case 0:
              t = CT(0);
              break;
          case 1:
              t = -CT(3)*eps2/CT(4);
              break;
          case 2:
              t = (-CT(7)*eps2-CT(48))*eps2/CT(64);
              break;
          case 3:
              t = eps2*((-CT(11)*eps2-CT(28))*eps2-CT(192))/CT(256);
              break;
          default:
              t = eps2*(eps2*((-CT(375)*eps2-CT(704))*eps2-CT(1792))-CT(12288))/CT(16384);
              break;
          }
          return (t - eps) / (CT(1) + eps);
      }
  
      /*
       Express
  
          I3 = integrate( (2-f)/(1+(1-f)*sqrt(1+k2*sin(sigma1)^2)), sigma1, 0, sigma )
  
       as a series
  
          A3 * ( sigma + sum(C3[l] * sin(2*l*sigma), l, 1, maxpow-1) )
  
       valid for f and k2 small.  It is convenient to write k2 = 4 * eps / (1 -
       eps)^2 and f = 2*n/(1+n) and expand in eps and n.  This procedure leads
       to a series where the coefficients of eps^j are terminating series in n.
  
       The scale factor A3 = mean value of (d/dsigma)I3
  
       The expansion above is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_coeffs_A3 below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
      */
      template <typename Coeffs, typename CT>
      inline void evaluate_coeffs_A3(Coeffs &c, CT const& n)
      {
          switch (int(Coeffs::static_size)) {
          case 0:
              break;
          case 1:
              c[0] = CT(1);
              break;
          case 2:
              c[0] = CT(1);
              c[1] = -CT(1)/CT(2);
              break;
          case 3:
              c[0] = CT(1);
              c[1] = (n-CT(1))/CT(2);
              c[2] = -CT(1)/CT(4);
              break;
          case 4:
              c[0] = CT(1);
              c[1] = (n-CT(1))/CT(2);
              c[2] = (-n-CT(2))/CT(8);
              c[3] = -CT(1)/CT(16);
              break;
          case 5:
              c[0] = CT(1);
              c[1] = (n-CT(1))/CT(2);
              c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
              c[3] = (-CT(3)*n-CT(1))/CT(16);
              c[4] = -CT(3)/CT(64);
              break;
          case 6:
              c[0] = CT(1);
              c[1] = (n-CT(1))/CT(2);
              c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
              c[3] = ((-n-CT(3))*n-CT(1))/CT(16);
              c[4] = (-CT(2)*n-CT(3))/CT(64);
              c[5] = -CT(3)/CT(128);
              break;
          case 7:
              c[0] = CT(1);
              c[1] = (n-CT(1))/CT(2);
              c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
              c[3] = (n*(n*(CT(5)*n-CT(1))-CT(3))-CT(1))/CT(16);
              c[4] = ((-CT(10)*n-CT(2))*n-CT(3))/CT(64);
              c[5] = (-CT(5)*n-CT(3))/CT(128);
              c[6] = -CT(5)/CT(256);
              break;
          default:
              c[0] = CT(1);
              c[1] = (n-CT(1))/CT(2);
              c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
              c[3] = (n*(n*(CT(5)*n-CT(1))-CT(3))-CT(1))/CT(16);
              c[4] = (n*((-CT(5)*n-CT(20))*n-CT(4))-CT(6))/CT(128);
              c[5] = ((-CT(5)*n-CT(10))*n-CT(6))/CT(256);
              c[6] = (-CT(15)*n-CT(20))/CT(1024);
              c[7] = -CT(25)/CT(2048);
              break;
          }
      }
  
      /*
       The coefficients C1[l] in the Fourier expansion of B1.
  
       The expansion below is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_coeffs_C1 below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
      */
      template <typename Coeffs, typename CT>
      inline void evaluate_coeffs_C1(Coeffs &c, CT const& eps)
      {
          CT eps2 = math::sqr(eps);
          CT d = eps;
          switch (int(Coeffs::static_size) - 1) {
          case 0:
              break;
          case 1:
              c[1] = -d/CT(2);
              break;
          case 2:
              c[1] = -d/CT(2);
              d *= eps;
              c[2] = -d/CT(16);
              break;
          case 3:
              c[1] = d*(CT(3)*eps2-CT(8))/CT(16);
              d *= eps;
              c[2] = -d/CT(16);
              d *= eps;
              c[3] = -d/CT(48);
              break;
          case 4:
              c[1] = d*(CT(3)*eps2-CT(8))/CT(16);
              d *= eps;
              c[2] = d*(eps2-CT(2))/CT(32);
              d *= eps;
              c[3] = -d/CT(48);
              d *= eps;
              c[4] = -CT(5)*d/CT(512);
              break;
          case 5:
              c[1] = d*((CT(6)-eps2)*eps2-CT(16))/CT(32);
              d *= eps;
              c[2] = d*(eps2-CT(2))/CT(32);
              d *= eps;
              c[3] = d*(CT(9)*eps2-CT(16))/CT(768);
              d *= eps;
              c[4] = -CT(5)*d/CT(512);
              d *= eps;
              c[5] = -CT(7)*d/CT(1280);
              break;
          case 6:
              c[1] = d*((CT(6)-eps2)*eps2-CT(16))/CT(32);
              d *= eps;
              c[2] = d*((CT(64)-CT(9)*eps2)*eps2-CT(128))/CT(2048);
              d *= eps;
              c[3] = d*(CT(9)*eps2-CT(16))/CT(768);
              d *= eps;
              c[4] = d*(CT(3)*eps2-CT(5))/CT(512);
              d *= eps;
              c[5] = -CT(7)*d/CT(1280);
              d *= eps;
              c[6] = -CT(7)*d/CT(2048);
              break;
          case 7:
              c[1] = d*(eps2*(eps2*(CT(19)*eps2-CT(64))+CT(384))-CT(1024))/CT(2048);
              d *= eps;
              c[2] = d*((CT(64)-CT(9)*eps2)*eps2-CT(128))/CT(2048);
              d *= eps;
              c[3] = d*((CT(72)-CT(9)*eps2)*eps2-CT(128))/CT(6144);
              d *= eps;
              c[4] = d*(CT(3)*eps2-CT(5))/CT(512);
              d *= eps;
              c[5] = d*(CT(35)*eps2-CT(56))/CT(10240);
              d *= eps;
              c[6] = -CT(7)*d/CT(2048);
              d *= eps;
              c[7] = -CT(33)*d/CT(14336);
              break;
          default:
              c[1] = d*(eps2*(eps2*(CT(19)*eps2-CT(64))+CT(384))-CT(1024))/CT(2048);
              d *= eps;
              c[2] = d*(eps2*(eps2*(CT(7)*eps2-CT(18))+CT(128))-CT(256))/CT(4096);
              d *= eps;
              c[3] = d*((CT(72)-CT(9)*eps2)*eps2-CT(128))/CT(6144);
              d *= eps;
              c[4] = d*((CT(96)-CT(11)*eps2)*eps2-CT(160))/CT(16384);
              d *= eps;
              c[5] = d*(CT(35)*eps2-CT(56))/CT(10240);
              d *= eps;
              c[6] = d*(CT(9)*eps2-CT(14))/CT(4096);
              d *= eps;
              c[7] = -CT(33)*d/CT(14336);
              d *= eps;
              c[8] = -CT(429)*d/CT(262144);
              break;
          }
      }
  
      /*
       The coefficients C1p[l] in the Fourier expansion of B1p.
  
       The expansion below is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_coeffs_C1p below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
      */
      template <typename Coeffs, typename CT>
      inline void evaluate_coeffs_C1p(Coeffs& c, CT const& eps)
      {
          CT const eps2 = math::sqr(eps);
          CT d = eps;
          switch (int(Coeffs::static_size) - 1) {
          case 0:
              break;
          case 1:
              c[1] = d/CT(2);
              break;
          case 2:
              c[1] = d/CT(2);
              d *= eps;
              c[2] = CT(5)*d/CT(16);
              break;
          case 3:
              c[1] = d*(CT(16)-CT(9)*eps2)/CT(32);
              d *= eps;
              c[2] = CT(5)*d/CT(16);
              d *= eps;
              c[3] = CT(29)*d/CT(96);
              break;
          case 4:
              c[1] = d*(CT(16)-CT(9)*eps2)/CT(32);
              d *= eps;
              c[2] = d*(CT(30)-CT(37)*eps2)/CT(96);
              d *= eps;
              c[3] = CT(29)*d/CT(96);
              d *= eps;
              c[4] = CT(539)*d/CT(1536);
              break;
          case 5:
              c[1] = d*(eps2*(CT(205)*eps2-CT(432))+CT(768))/CT(1536);
              d *= eps;
              c[2] = d*(CT(30)-CT(37)*eps2)/CT(96);
              d *= eps;
              c[3] = d*(CT(116)-CT(225)*eps2)/CT(384);
              d *= eps;
              c[4] = CT(539)*d/CT(1536);
              d *= eps;
              c[5] = CT(3467)*d/CT(7680);
              break;
          case 6:
              c[1] = d*(eps2*(CT(205)*eps2-CT(432))+CT(768))/CT(1536);
              d *= eps;
              c[2] = d*(eps2*(CT(4005)*eps2-CT(4736))+CT(3840))/CT(12288);
              d *= eps;
              c[3] = d*(CT(116)-CT(225)*eps2)/CT(384);
              d *= eps;
              c[4] = d*(CT(2695)-CT(7173)*eps2)/CT(7680);
              d *= eps;
              c[5] = CT(3467)*d/CT(7680);
              d *= eps;
              c[6] = CT(38081)*d/CT(61440);
              break;
          case 7:
              c[1] = d*(eps2*((CT(9840)-CT(4879)*eps2)*eps2-CT(20736))+CT(36864))/CT(73728);
              d *= eps;
              c[2] = d*(eps2*(CT(4005)*eps2-CT(4736))+CT(3840))/CT(12288);
              d *= eps;
              c[3] = d*(eps2*(CT(8703)*eps2-CT(7200))+CT(3712))/CT(12288);
              d *= eps;
              c[4] = d*(CT(2695)-CT(7173)*eps2)/CT(7680);
              d *= eps;
              c[5] = d*(CT(41604)-CT(141115)*eps2)/CT(92160);
              d *= eps;
              c[6] = CT(38081)*d/CT(61440);
              d *= eps;
              c[7] = CT(459485)*d/CT(516096);
              break;
          default:
              c[1] = d*(eps2*((CT(9840)-CT(4879)*eps2)*eps2-CT(20736))+CT(36864))/CT(73728);
              d *= eps;
              c[2] = d*(eps2*((CT(120150)-CT(86171)*eps2)*eps2-CT(142080))+CT(115200))/CT(368640);
              d *= eps;
              c[3] = d*(eps2*(CT(8703)*eps2-CT(7200))+CT(3712))/CT(12288);
              d *= eps;
              c[4] = d*(eps2*(CT(1082857)*eps2-CT(688608))+CT(258720))/CT(737280);
              d *= eps;
              c[5] = d*(CT(41604)-CT(141115)*eps2)/CT(92160);
              d *= eps;
              c[6] = d*(CT(533134)-CT(2200311)*eps2)/CT(860160);
              d *= eps;
              c[7] = CT(459485)*d/CT(516096);
              d *= eps;
              c[8] = CT(109167851)*d/CT(82575360);
              break;
          }
      }
  
      /*
       The coefficients C2[l] in the Fourier expansion of B2.
  
       The expansion below is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_coeffs_C2 below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
      */
      template <typename Coeffs, typename CT>
      inline void evaluate_coeffs_C2(Coeffs& c, CT const& eps)
      {
          CT const eps2 = math::sqr(eps);
          CT d = eps;
          switch (int(Coeffs::static_size) - 1) {
          case 0:
              break;
          case 1:
              c[1] = d/CT(2);
              break;
          case 2:
              c[1] = d/CT(2);
              d *= eps;
              c[2] = CT(3)*d/CT(16);
              break;
          case 3:
              c[1] = d*(eps2+CT(8))/CT(16);
              d *= eps;
              c[2] = CT(3)*d/CT(16);
              d *= eps;
              c[3] = CT(5)*d/CT(48);
              break;
          case 4:
              c[1] = d*(eps2+CT(8))/CT(16);
              d *= eps;
              c[2] = d*(eps2+CT(6))/CT(32);
              d *= eps;
              c[3] = CT(5)*d/CT(48);
              d *= eps;
              c[4] = CT(35)*d/CT(512);
              break;
          case 5:
              c[1] = d*(eps2*(eps2+CT(2))+CT(16))/CT(32);
              d *= eps;
              c[2] = d*(eps2+CT(6))/CT(32);
              d *= eps;
              c[3] = d*(CT(15)*eps2+CT(80))/CT(768);
              d *= eps;
              c[4] = CT(35)*d/CT(512);
              d *= eps;
              c[5] = CT(63)*d/CT(1280);
              break;
          case 6:
              c[1] = d*(eps2*(eps2+CT(2))+CT(16))/CT(32);
              d *= eps;
              c[2] = d*(eps2*(CT(35)*eps2+CT(64))+CT(384))/CT(2048);
              d *= eps;
              c[3] = d*(CT(15)*eps2+CT(80))/CT(768);
              d *= eps;
              c[4] = d*(CT(7)*eps2+CT(35))/CT(512);
              d *= eps;
              c[5] = CT(63)*d/CT(1280);
              d *= eps;
              c[6] = CT(77)*d/CT(2048);
              break;
          case 7:
              c[1] = d*(eps2*(eps2*(CT(41)*eps2+CT(64))+CT(128))+CT(1024))/CT(2048);
              d *= eps;
              c[2] = d*(eps2*(CT(35)*eps2+CT(64))+CT(384))/CT(2048);
              d *= eps;
              c[3] = d*(eps2*(CT(69)*eps2+CT(120))+CT(640))/CT(6144);
              d *= eps;
              c[4] = d*(CT(7)*eps2+CT(35))/CT(512);
              d *= eps;
              c[5] = d*(CT(105)*eps2+CT(504))/CT(10240);
              d *= eps;
              c[6] = CT(77)*d/CT(2048);
              d *= eps;
              c[7] = CT(429)*d/CT(14336);
              break;
          default:
              c[1] = d*(eps2*(eps2*(CT(41)*eps2+CT(64))+CT(128))+CT(1024))/CT(2048);
              d *= eps;
              c[2] = d*(eps2*(eps2*(CT(47)*eps2+CT(70))+CT(128))+CT(768))/CT(4096);
              d *= eps;
              c[3] = d*(eps2*(CT(69)*eps2+CT(120))+CT(640))/CT(6144);
              d *= eps;
              c[4] = d*(eps2*(CT(133)*eps2+CT(224))+CT(1120))/CT(16384);
              d *= eps;
              c[5] = d*(CT(105)*eps2+CT(504))/CT(10240);
              d *= eps;
              c[6] = d*(CT(33)*eps2+CT(154))/CT(4096);
              d *= eps;
              c[7] = CT(429)*d/CT(14336);
              d *= eps;
              c[8] = CT(6435)*d/CT(262144);
              break;
          }
      }
  
      /*
       The coefficients C3[l] in the Fourier expansion of B3.
  
       The expansion below is performed in Maxima, a Computer Algebra System.
       The C++ code (that yields the function evaluate_coeffs_C3 below) is
       generated by the following Maxima script:
       geometry/doc/other/maxima/geod.mac
      */
      template <size_t SeriesOrder, typename Coeffs, typename CT>
      inline void evaluate_coeffs_C3x(Coeffs &c, CT const& n) {
          size_t const coeff_size = Coeffs::static_size;
          size_t const expected_size = (SeriesOrder * (SeriesOrder - 1)) / 2;
          BOOST_GEOMETRY_ASSERT((coeff_size == expected_size));
  
          const CT n2 = math::sqr(n);
          switch (SeriesOrder) {
          case 0:
              break;
          case 1:
              break;
          case 2:
              c[0] = (CT(1)-n)/CT(4);
              break;
          case 3:
              c[0] = (CT(1)-n)/CT(4);
              c[1] = (CT(1)-n2)/CT(8);
              c[2] = ((n-CT(3))*n+CT(2))/CT(32);
              break;
          case 4:
              c[0] = (CT(1)-n)/CT(4);
              c[1] = (CT(1)-n2)/CT(8);
              c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
              c[3] = ((n-CT(3))*n+CT(2))/CT(32);
              c[4] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
              c[5] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
              break;
          case 5:
              c[0] = (CT(1)-n)/CT(4);
              c[1] = (CT(1)-n2)/CT(8);
              c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
              c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
              c[4] = ((n-CT(3))*n+CT(2))/CT(32);
              c[5] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
              c[6] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
              c[7] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
              c[8] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
              c[9] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
              break;
          case 6:
              c[0] = (CT(1)-n)/CT(4);
              c[1] = (CT(1)-n2)/CT(8);
              c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
              c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
              c[4] = (n*(CT(3)*n+CT(11))+CT(12))/CT(512);
              c[5] = ((n-CT(3))*n+CT(2))/CT(32);
              c[6] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
              c[7] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
              c[8] = ((CT(1)-CT(2)*n)*n+CT(5))/CT(256);
              c[9] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
              c[10] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
              c[11] = ((-CT(77)*n-CT(8))*n+CT(42))/CT(3072);
              c[12] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
              c[13] = ((-CT(7)*n-CT(40))*n+CT(28))/CT(2048);
              c[14] = (n*(CT(75)*n-CT(90))+CT(42))/CT(5120);
              break;
          case 7:
              c[0] = (CT(1)-n)/CT(4);
              c[1] = (CT(1)-n2)/CT(8);
              c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
              c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
              c[4] = (n*(CT(3)*n+CT(11))+CT(12))/CT(512);
              c[5] = (CT(10)*n+CT(21))/CT(1024);
              c[6] = ((n-CT(3))*n+CT(2))/CT(32);
              c[7] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
              c[8] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
              c[9] = ((CT(1)-CT(2)*n)*n+CT(5))/CT(256);
              c[10] = (CT(69)*n+CT(108))/CT(8192);
              c[11] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
              c[12] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
              c[13] = ((-CT(77)*n-CT(8))*n+CT(42))/CT(3072);
              c[14] = (CT(12)-n)/CT(1024);
              c[15] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
              c[16] = ((-CT(7)*n-CT(40))*n+CT(28))/CT(2048);
              c[17] = (CT(72)-CT(43)*n)/CT(8192);
              c[18] = (n*(CT(75)*n-CT(90))+CT(42))/CT(5120);
              c[19] = (CT(9)-CT(15)*n)/CT(1024);
              c[20] = (CT(44)-CT(99)*n)/CT(8192);
              break;
          default:
              c[0] = (CT(1)-n)/CT(4);
              c[1] = (CT(1)-n2)/CT(8);
              c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
              c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
              c[4] = (n*(CT(3)*n+CT(11))+CT(12))/CT(512);
              c[5] = (CT(10)*n+CT(21))/CT(1024);
              c[6] = CT(243)/CT(16384);
              c[7] = ((n-CT(3))*n+CT(2))/CT(32);
              c[8] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
              c[9] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
              c[10] = ((CT(1)-CT(2)*n)*n+CT(5))/CT(256);
              c[11] = (CT(69)*n+CT(108))/CT(8192);
              c[12] = CT(187)/CT(16384);
              c[13] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
              c[14] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
              c[15] = ((-CT(77)*n-CT(8))*n+CT(42))/CT(3072);
              c[16] = (CT(12)-n)/CT(1024);
              c[17] = CT(139)/CT(16384);
              c[18] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
              c[19] = ((-CT(7)*n-CT(40))*n+CT(28))/CT(2048);
              c[20] = (CT(72)-CT(43)*n)/CT(8192);
              c[21] = CT(127)/CT(16384);
              c[22] = (n*(CT(75)*n-CT(90))+CT(42))/CT(5120);
              c[23] = (CT(9)-CT(15)*n)/CT(1024);
              c[24] = CT(99)/CT(16384);
              c[25] = (CT(44)-CT(99)*n)/CT(8192);
              c[26] = CT(99)/CT(16384);
              c[27] = CT(429)/CT(114688);
              break;
          }
      }
  
      /*
      \brief Given the set of coefficients coeffs2[] evaluate on
        C3 and return the set of coefficients coeffs1[].
  
        Elements coeffs1[1] through coeffs1[SeriesOrder - 1] are set.
      */
      template <typename Coeffs1, typename Coeffs2, typename CT>
      inline void evaluate_coeffs_C3(Coeffs1 &coeffs1, Coeffs2 &coeffs2, CT const& eps)
      {
          CT mult = 1;
          int offset = 0;
  
          // l is the index of C3[l].
          for (size_t l = 1; l < Coeffs1::static_size; ++l)
          {
              // Order of polynomial in eps.
              int m = Coeffs1::static_size - l;
              mult *= eps;
  
              coeffs1[l] = mult * math::horner_evaluate(eps, coeffs2.begin() + offset,
                                                             coeffs2.begin() + offset + m);
  
              offset += m;
          }
          // Post condition: offset == coeffs_C3_size
      }
  
      /*
      \brief Evaluate the following:
  
       y = sum(c[i] * sin(2*i * x), i, 1, n)
  
       using Clenshaw summation.
      */
      template <typename CT, typename Coeffs>
      inline CT sin_cos_series(CT const& sinx, CT const& cosx, Coeffs const& coeffs)
      {
          size_t n = Coeffs::static_size - 1;
          size_t index = 0;
  
          // Point to one beyond last element.
          index += (n + 1);
          CT ar = 2 * (cosx - sinx) * (cosx + sinx);
  
          // If n is odd, get the last element.
          CT k0 = n & 1 ? coeffs[--index] : 0;
          CT k1 = 0;
  
          // Make n even.
          n /= 2;
          while (n--) {
              // Unroll loop x 2, so accumulators return to their original role.
              k1 = ar * k0 - k1 + coeffs[--index];
              k0 = ar * k1 - k0 + coeffs[--index];
          }
  
          return 2 * sinx * cosx * k0;
      }
  
      /*
       The coefficient containers for the series expansions.
       These structs allow the caller to only know the series order.
      */
      template <size_t SeriesOrder, typename CT>
      struct coeffs_C1 : boost::array<CT, SeriesOrder + 1>
      {
          coeffs_C1(CT const& epsilon)
          {
              evaluate_coeffs_C1(*this, epsilon);
          }
      };
  
      template <size_t SeriesOrder, typename CT>
      struct coeffs_C1p : boost::array<CT, SeriesOrder + 1>
      {
          coeffs_C1p(CT const& epsilon)
          {
              evaluate_coeffs_C1p(*this, epsilon);
          }
      };
  
      template <size_t SeriesOrder, typename CT>
      struct coeffs_C2 : boost::array<CT, SeriesOrder + 1>
      {
          coeffs_C2(CT const& epsilon)
          {
              evaluate_coeffs_C2(*this, epsilon);
          }
      };
  
      template <size_t SeriesOrder, typename CT>
      struct coeffs_C3x : boost::array<CT, (SeriesOrder * (SeriesOrder - 1)) / 2>
      {
          coeffs_C3x(CT const& n)
          {
              evaluate_coeffs_C3x<SeriesOrder>(*this, n);
          }
      };
  
      template <size_t SeriesOrder, typename CT>
      struct coeffs_C3 : boost::array<CT, SeriesOrder>
      {
          coeffs_C3(CT const& n, CT const& epsilon)
          {
              coeffs_C3x<SeriesOrder, CT> coeffs_C3x(n);
  
              evaluate_coeffs_C3(*this, coeffs_C3x, epsilon);
          }
      };
  
      template <size_t SeriesOrder, typename CT>
      struct coeffs_A3 : boost::array<CT, SeriesOrder>
      {
          coeffs_A3(CT const& n)
          {
              evaluate_coeffs_A3(*this, n);
          }
      };
  
  }}} // namespace boost::geometry::series_expansion
  
  #endif // BOOST_GEOMETRY_UTIL_SERIES_EXPANSION_HPP
  