text: // David Eberly, Geometric Tools, Redmond WA 98052
  // Copyright (c) 1998-2019
  // Distributed under the Boost Software License, Version 1.0.
  // http://www.boost.org/LICENSE_1_0.txt
  // http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
  // File Version: 3.0.1 (2018/10/05)
  
  #pragma once
  
  #include <Mathematics/GteMath.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);
  
      // 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);
  
  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);
      inline static Real Evaluate(degree<5>, Real x);
      inline static Real Evaluate(degree<7>, Real x);
      inline static Real Evaluate(degree<9>, Real x);
      inline static Real Evaluate(degree<11>, Real x);
  
      // Support for range reduction.
      inline static Real Reduce(Real x);
  };
  
  
  template <typename Real>
  template <int D>
  inline Real SinEstimate<Real>::Degree(Real x)
  {
      return Evaluate(degree<D>(), x);
  }
  
  template <typename Real>
  template <int D>
  inline Real SinEstimate<Real>::DegreeRR(Real x)
  {
      return Degree<D>(Reduce(x));
  }
  
  template <typename Real>
  inline Real SinEstimate<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;
  }
  
  template <typename Real>
  inline Real SinEstimate<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;
  }
  
  template <typename Real>
  inline Real SinEstimate<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;
  }
  
  template <typename Real>
  inline Real SinEstimate<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;
  }
  
  template <typename Real>
  inline Real SinEstimate<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;
  }
  
  template <typename Real>
  inline Real SinEstimate<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;
  }
  
  
  }
  