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

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;
}