GteCosEstimate.h

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// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2017
// 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: 3.0.0 (2016/06/19)

#pragma once

#include <Mathematics/GteConstants.h>

// Minimax polynomial approximations to cos(x).  The polynomial p(x) of
// degree D has only even-power terms, is required to have constant term 1,
// and p(pi/2) = cos(pi/2) = 0.  It minimizes the quantity
// maximum{|cos(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 CosEstimate
{
public:
    // The input constraint is x in [-pi/2,pi/2].  For example,
    //   float x; // in [-pi/2,pi/2]
    //   float result = CosEstimate<float>::Degree<4>(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] and a sign s for which
    // cos(y) = s*cos(x).  For example,
    //   float x;  // x any real number
    //   float result = CosEstimate<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<2>, Real x);
    inline static Real Evaluate(degree<4>, Real x);
    inline static Real Evaluate(degree<6>, Real x);
    inline static Real Evaluate(degree<8>, Real x);
    inline static Real Evaluate(degree<10>, Real x);

    // Support for range reduction.
    inline static void Reduce(Real x, Real& y, Real& sign);
};


template <typename Real>
template <int D>
inline Real CosEstimate<Real>::Degree(Real x)
{
    return Evaluate(degree<D>(), x);
}

template <typename Real>
template <int D>
inline Real CosEstimate<Real>::DegreeRR(Real x)
{
    Real y, sign;
    Reduce(x, y, sign);
    Real poly = sign * Degree<D>(y);
    return poly;
}

template <typename Real>
inline Real CosEstimate<Real>::Evaluate(degree<2>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_COS_DEG2_C1;
    poly = (Real)GTE_C_COS_DEG2_C0 + poly * xsqr;
    return poly;
}

template <typename Real>
inline Real CosEstimate<Real>::Evaluate(degree<4>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_COS_DEG4_C2;
    poly = (Real)GTE_C_COS_DEG4_C1 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG4_C0 + poly * xsqr;
    return poly;
}

template <typename Real>
inline Real CosEstimate<Real>::Evaluate(degree<6>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_COS_DEG6_C3;
    poly = (Real)GTE_C_COS_DEG6_C2 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG6_C1 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG6_C0 + poly * xsqr;
    return poly;
}

template <typename Real>
inline Real CosEstimate<Real>::Evaluate(degree<8>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_COS_DEG8_C4;
    poly = (Real)GTE_C_COS_DEG8_C3 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG8_C2 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG8_C1 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG8_C0 + poly * xsqr;
    return poly;
}

template <typename Real>
inline Real CosEstimate<Real>::Evaluate(degree<10>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_COS_DEG10_C5;
    poly = (Real)GTE_C_COS_DEG10_C4 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG10_C3 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG10_C2 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG10_C1 + poly * xsqr;
    poly = (Real)GTE_C_COS_DEG10_C0 + poly * xsqr;
    return poly;
}

template <typename Real>
inline void CosEstimate<Real>::Reduce(Real x, Real& y, Real& sign)
{
    // 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));
    }
    y = x - (Real)GTE_C_TWO_PI * quotient;

    // Map y to [-pi/2,pi/2] with cos(y) = sign*cos(x).
    if (y > (Real)GTE_C_HALF_PI)
    {
        y = (Real)GTE_C_PI - y;
        sign = (Real)-1;
    }
    else if (y < (Real)-GTE_C_HALF_PI)
    {
        y = (Real)-GTE_C_PI - y;
        sign = (Real)-1;
    }
    else
    {
        sign = (Real)1;
    }
}


}