text: // David Eberly, Geometric Tools, Redmond WA 98052
  // Copyright (c) 1998-2020
  // Distributed under the Boost Software License, Version 1.0.
  // https://www.boost.org/LICENSE_1_0.txt
  // https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
  // Version: 4.0.2019.08.13
  
  #pragma once
  
  #include <Mathematics/Math.h>
  
  // Minimax polynomial approximations to sin(x).  The polynomial p(x) of
  // degree D has only odd-power terms, is required to have linear term x,
  // and p(pi/2) = sin(pi/2) = 1.  It minimizes the quantity
  // maximum{|sin(x) - p(x)| : x in [-pi/2,pi/2]} over all polynomials of
  // degree D subject to the constraints mentioned.
  
  namespace gte
  {
      template <typename Real>
      class SinEstimate
      {
      public:
          // The input constraint is x in [-pi/2,pi/2].  For example,
          //   float x; // in [-pi/2,pi/2]
          //   float result = SinEstimate<float>::Degree<3>(x);
          template <int D>
          inline static Real Degree(Real x)
          {
              return Evaluate(degree<D>(), x);
          }
  
          // The input x can be any real number.  Range reduction is used to
          // generate a value y in [-pi/2,pi/2] for which sin(y) = sin(x).
          // For example,
          //   float x;  // x any real number
          //   float result = SinEstimate<float>::DegreeRR<3>(x);
          template <int D>
          inline static Real DegreeRR(Real x)
          {
              return Degree<D>(Reduce(x));
          }
  
      private:
          // Metaprogramming and private implementation to allow specialization
          // of a template member function.
          template <int D> struct degree {};
  
          inline static Real Evaluate(degree<3>, Real x)
          {
              Real xsqr = x * x;
              Real poly;
              poly = (Real)GTE_C_SIN_DEG3_C1;
              poly = (Real)GTE_C_SIN_DEG3_C0 + poly * xsqr;
              poly = poly * x;
              return poly;
          }
  
          inline static Real Evaluate(degree<5>, Real x)
          {
              Real xsqr = x * x;
              Real poly;
              poly = (Real)GTE_C_SIN_DEG5_C2;
              poly = (Real)GTE_C_SIN_DEG5_C1 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG5_C0 + poly * xsqr;
              poly = poly * x;
              return poly;
          }
  
          inline static Real Evaluate(degree<7>, Real x)
          {
              Real xsqr = x * x;
              Real poly;
              poly = (Real)GTE_C_SIN_DEG7_C3;
              poly = (Real)GTE_C_SIN_DEG7_C2 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG7_C1 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG7_C0 + poly * xsqr;
              poly = poly * x;
              return poly;
          }
  
          inline static Real Evaluate(degree<9>, Real x)
          {
              Real xsqr = x * x;
              Real poly;
              poly = (Real)GTE_C_SIN_DEG9_C4;
              poly = (Real)GTE_C_SIN_DEG9_C3 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG9_C2 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG9_C1 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG9_C0 + poly * xsqr;
              poly = poly * x;
              return poly;
          }
  
          inline static Real Evaluate(degree<11>, Real x)
          {
              Real xsqr = x * x;
              Real poly;
              poly = (Real)GTE_C_SIN_DEG11_C5;
              poly = (Real)GTE_C_SIN_DEG11_C4 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG11_C3 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG11_C2 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG11_C1 + poly * xsqr;
              poly = (Real)GTE_C_SIN_DEG11_C0 + poly * xsqr;
              poly = poly * x;
              return poly;
          }
  
          // Support for range reduction.
          inline static Real Reduce(Real x)
          {
              // Map x to y in [-pi,pi], x = 2*pi*quotient + remainder.
              Real quotient = (Real)GTE_C_INV_TWO_PI * x;
              if (x >= (Real)0)
              {
                  quotient = (Real)((int)(quotient + (Real)0.5));
              }
              else
              {
                  quotient = (Real)((int)(quotient - (Real)0.5));
              }
              Real y = x - (Real)GTE_C_TWO_PI * quotient;
  
              // Map y to [-pi/2,pi/2] with sin(y) = sin(x).
              if (y > (Real)GTE_C_HALF_PI)
              {
                  y = (Real)GTE_C_PI - y;
              }
              else if (y < (Real)-GTE_C_HALF_PI)
              {
                  y = (Real)-GTE_C_PI - y;
              }
              return y;
          }
      };
  }
  