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

sqrt.cpp

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The AD sqrt Function: Example and Test

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

bool Sqrt(void)
{  bool ok = true;

   using CppAD::AD;
   using CppAD::NearEqual;
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

   // domain space vector
   size_t n  = 1;
   double x0 = 0.5;
   CPPAD_TESTVECTOR(AD<double>) x(n);
   x[0]      = x0;

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

   // range space vector
   size_t m = 1;
   CPPAD_TESTVECTOR(AD<double>) y(m);
   y[0] = CppAD::sqrt(x[0]);

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

   // check value
   double check = std::sqrt(x0);
   ok &= NearEqual(y[0] , check, eps99, eps99);

   // forward computation of first partial w.r.t. x[0]
   CPPAD_TESTVECTOR(double) dx(n);
   CPPAD_TESTVECTOR(double) dy(m);
   dx[0] = 1.;
   dy    = f.Forward(1, dx);
   check = 1. / (2. * std::sqrt(x0) );
   ok   &= NearEqual(dy[0], check, eps99, eps99);

   // reverse computation of derivative of y[0]
   CPPAD_TESTVECTOR(double)  w(m);
   CPPAD_TESTVECTOR(double) dw(n);
   w[0]  = 1.;
   dw    = f.Reverse(1, w);
   ok   &= NearEqual(dw[0], check, eps99, eps99);

   // use a VecAD<Base>::reference object with sqrt
   CppAD::VecAD<double> v(1);
   AD<double> zero(0);
   v[zero]           = x0;
   AD<double> result = CppAD::sqrt(v[zero]);
   check = std::sqrt(x0);
   ok   &= NearEqual(result, check, eps99, eps99);

   return ok;
}