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

namespace gte
{

template <typename Real>
class TanEstimate
{
public:
    // The input constraint is x in [-pi/4,pi/4].  For example,
    //   float x; // in [-pi/4,pi/4]
    //   float result = TanEstimate<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].  If |y| <= pi/4, then the
    // polynomial is evaluated.  If y in (pi/4,pi/2), set z = y - pi/4
    // and use the identity
    //   tan(y) = tan(z + pi/4) = [1 + tan(z)]/[1 - tan(z)]
    // Be careful when evaluating at y nearly pi/2, because tan(y)
    // becomes infinite.  For example,
    //   float x;  // x any real number
    //   float result = TanEstimate<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);

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


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

template <typename Real>
template <int D>
inline Real TanEstimate<Real>::DegreeRR(Real x)
{
    Real y;
    Reduce(x, y);
    if (std::abs(y) <= (Real)GTE_C_QUARTER_PI)
    {
        return Degree<D>(y);
    }
    else if (y > (Real)GTE_C_QUARTER_PI)
    {
        Real poly = Degree<D>(y - (Real)GTE_C_QUARTER_PI);
        return ((Real)1 + poly) / ((Real)1 - poly);
    }
    else
    {
        Real poly = Degree<D>(y + (Real)GTE_C_QUARTER_PI);
        return -((Real)1 - poly) / ((Real)1 + poly);
    }
}

template <typename Real>
inline Real TanEstimate<Real>::Evaluate(degree<3>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_TAN_DEG3_C1;
    poly = (Real)GTE_C_TAN_DEG3_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real TanEstimate<Real>::Evaluate(degree<5>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_TAN_DEG5_C2;
    poly = (Real)GTE_C_TAN_DEG5_C1 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG5_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real TanEstimate<Real>::Evaluate(degree<7>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_TAN_DEG7_C3;
    poly = (Real)GTE_C_TAN_DEG7_C2 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG7_C1 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG7_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real TanEstimate<Real>::Evaluate(degree<9>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_TAN_DEG9_C4;
    poly = (Real)GTE_C_TAN_DEG9_C3 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG9_C2 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG9_C1 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG9_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real TanEstimate<Real>::Evaluate(degree<11>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_TAN_DEG11_C5;
    poly = (Real)GTE_C_TAN_DEG11_C4 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG11_C3 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG11_C2 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG11_C1 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG11_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline Real TanEstimate<Real>::Evaluate(degree<13>, Real x)
{
    Real xsqr = x * x;
    Real poly;
    poly = (Real)GTE_C_TAN_DEG13_C6;
    poly = (Real)GTE_C_TAN_DEG13_C5 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG13_C4 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG13_C3 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG13_C2 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG13_C1 + poly * xsqr;
    poly = (Real)GTE_C_TAN_DEG13_C0 + poly * xsqr;
    poly = poly * x;
    return poly;
}

template <typename Real>
inline void TanEstimate<Real>::Reduce(Real x, Real& y)
{
    // Map x to y in [-pi,pi], x = pi*quotient + remainder.
    y = fmod(x, (Real)GTE_C_PI);

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


}