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

pow.cpp

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The AD Power Function: Example and Test

# include <cppad/cppad.hpp>
# include <cmath>

bool pow(void)
{  bool ok = true;

   using CppAD::AD;
   using CppAD::NearEqual;
   double eps = 10. * std::numeric_limits<double>::epsilon();

   // domain space vector
   size_t n  = 2;
   double x = 0.5;
   double y = 2.;
   CPPAD_TESTVECTOR(AD<double>) axy(n);
   axy[0]      = x;
   axy[1]      = y;

   // declare independent variables and start tape recording
   CppAD::Independent(axy);

   // range space vector
   size_t m = 3;
   CPPAD_TESTVECTOR(AD<double>) az(m);
   az[0] = CppAD::pow(axy[0], axy[1]); // pow(variable, variable)
   az[1] = CppAD::pow(axy[0], y);      // pow(variable, parameter)
   az[2] = CppAD::pow(x,     axy[1]);  // pow(parameter, variable)

   // create f: axy -> az and stop tape recording
   CppAD::ADFun<double> f(axy, az);

   // check value
   double check = std::pow(x, y);
   size_t i;
   for(i = 0; i < m; i++)
      ok &= NearEqual(az[i] , check,  eps, eps);

   // forward computation of first partial w.r.t. x
   CPPAD_TESTVECTOR(double) dxy(n);
   CPPAD_TESTVECTOR(double) dz(m);
   dxy[0] = 1.;
   dxy[1] = 0.;
   dz    = f.Forward(1, dxy);
   check = y * std::pow(x, y-1.);
   ok   &= NearEqual(dz[0], check, eps, eps);
   ok   &= NearEqual(dz[1], check, eps, eps);
   ok   &= NearEqual(dz[2],    0., eps, eps);

   // forward computation of first partial w.r.t. y
   dxy[0] = 0.;
   dxy[1] = 1.;
   dz    = f.Forward(1, dxy);
   check = std::log(x) * std::pow(x, y);
   ok   &= NearEqual(dz[0], check, eps, eps);
   ok   &= NearEqual(dz[1],    0., eps, eps);
   ok   &= NearEqual(dz[2], check, eps, eps);

   // reverse computation of derivative of z[0] + z[1] + z[2]
   CPPAD_TESTVECTOR(double)  w(m);
   CPPAD_TESTVECTOR(double) dw(n);
   w[0]  = 1.;
   w[1]  = 1.;
   w[2]  = 1.;
   dw    = f.Reverse(1, w);
   check = y * std::pow(x, y-1.);
   ok   &= NearEqual(dw[0], 2. * check, eps, eps);
   check = std::log(x) * std::pow(x, y);
   ok   &= NearEqual(dw[1], 2. * check, eps, eps);

   // use a VecAD<Base>::reference object with pow
   CppAD::VecAD<double> v(2);
   AD<double> zero(0);
   AD<double> one(1);
   v[zero]           = axy[0];
   v[one]            = axy[1];
   AD<double> result = CppAD::pow(v[zero], v[one]);
   ok               &= NearEqual(result, az[0], eps, eps);

   return ok;
}