\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

atan2.cpp

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The AD atan2 Function: Example and Test

# include <cppad/cppad.hpp>
# define N_THETA 20

bool atan2(void)
{   bool ok = true;
    //
    using CppAD::AD;
    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
    double pi    = 2.0 * std::atan(1.0);
    //
    for(size_t k = 0; k < N_THETA; ++k)
    {   // theta
        double theta =  2.0 * pi * double(k+1) / double(N_THETA) - pi;
        //
        // radius
        double radius = 1.0 + double(k) / double(N_THETA);
        //
        // x, y
        double x = radius * std::cos(theta);
        double y = radius * std::sin(theta);
        //
        // au
        CPPAD_TESTVECTOR(AD<double>) au(2);
        au[0] = x;
        au[1] = y;
        CppAD::Independent(au);
        //
        // av
        CPPAD_TESTVECTOR(AD<double>) av(1);
        av[0] = CppAD::atan2(au[1], au[0]);
        //
        // f(x, y) = atan2(y, x)
        CppAD::ADFun<double> f(au, av);
        //
        // check value
        ok &= NearEqual(av[0] , theta, eps99, eps99);
        //
        // partial_x, partial_y
        // see https://en.wikipedia.org/wiki/Atan2#Derivative
        double partial_x = - y / (radius * radius);
        double partial_y =   x / (radius * radius);
        //
        // check forward mode
        CPPAD_TESTVECTOR(double) du(2), dv(1);
        du[0] = 1.0;
        du[1] = 0.0;
        dv    = f.Forward(1, du);
        ok   &= NearEqual(dv[0], partial_x, eps99, eps99);
        du[0] = 0.0;
        du[1] = 1.0;
        dv    = f.Forward(1, du);
        ok   &= NearEqual(dv[0], partial_y, eps99, eps99);
        //
        // check reverse mode
        CPPAD_TESTVECTOR(double)  w(1);
        CPPAD_TESTVECTOR(double) dw(2);
        w[0]  = 1.;
        dw    = f.Reverse(1, w);
        ok   &= NearEqual(dw[0], partial_x, eps99, eps99);
        ok   &= NearEqual(dw[1], partial_y, eps99, eps99);
        //
    }
    return ok;
}