GteATanEstimate.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>
#include <cmath>

// Minimax polynomial approximations to atan(x).  The polynomial p(x) of
// degree D has only odd-power terms, is required to have linear term x,
// and p(1) = atan(1) = pi/4.  It minimizes the quantity
// maximum{|atan(x) - p(x)| : x in [-1,1]} over all polynomials of
// degree D subject to the constraints mentioned.

namespace gte
{

template <typename Real>
class ATanEstimate
{
public:
    // The input constraint is x in [-1,1].  For example,
    //   float x; // in [-1,1]
    //   float result = ATanEstimate<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 via
    // the identities atan(x) = pi/2 - atan(1/x) for x > 0, and
    // atan(x) = -pi/2 - atan(1/x) for x < 0.  For example,
    //   float x;  // x any real number
    //   float result = ATanEstimate<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);
    inline static Real Evaluate(degree<13>, Real x);
};


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

template <typename Real>
template <int D>
inline Real ATanEstimate<Real>::DegreeRR(Real x)
{
    if (std::abs(x) <= (Real)1)
    {
        return Degree<D>(x);
    }
    else if (x > (Real)1)
    {
        return (Real)GTE_C_HALF_PI - Degree<D>((Real)1 / x);
    }
    else
    {
        return (Real)-GTE_C_HALF_PI - Degree<D>((Real)1 / x);
    }
}

template <typename Real>
inline Real ATanEstimate<Real>::Evaluate(degree<3>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_ATAN_DEG3_C1;
    poly = (Real)GTE_C_ATAN_DEG3_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real ATanEstimate<Real>::Evaluate(degree<5>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_ATAN_DEG5_C2;
    poly = (Real)GTE_C_ATAN_DEG5_C1 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG5_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real ATanEstimate<Real>::Evaluate(degree<7>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_ATAN_DEG7_C3;
    poly = (Real)GTE_C_ATAN_DEG7_C2 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG7_C1 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG7_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real ATanEstimate<Real>::Evaluate(degree<9>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_ATAN_DEG9_C4;
    poly = (Real)GTE_C_ATAN_DEG9_C3 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG9_C2 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG9_C1 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG9_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real ATanEstimate<Real>::Evaluate(degree<11>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_ATAN_DEG11_C5;
    poly = (Real)GTE_C_ATAN_DEG11_C4 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG11_C3 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG11_C2 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG11_C1 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG11_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real ATanEstimate<Real>::Evaluate(degree<13>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_ATAN_DEG13_C6;
    poly = (Real)GTE_C_ATAN_DEG13_C5 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG13_C4 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG13_C3 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG13_C2 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG13_C1 + poly * xsqr;
    poly = (Real)GTE_C_ATAN_DEG13_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}


}