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@(@\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}} }@)@
Fadbad Speed: Second Derivative of a Polynomial

Specifications
See link_poly .

Implementation 
# include <cppad/utility/vector.hpp>
# include <cppad/utility/poly.hpp>
# include <cppad/speed/uniform_01.hpp>
# include <FADBAD++/tadiff.h>

// list of possible options
# include <map>
extern std::map<std::string, bool> global_option;

bool link_poly(
     size_t                     size     ,
     size_t                     repeat   ,
     CppAD::vector<double>     &a        ,  // coefficients of polynomial
     CppAD::vector<double>     &z        ,  // polynomial argument value
     CppAD::vector<double>     &ddp      )  // second derivative w.r.t z
{
     if( global_option["atomic"] )
          return false;
     if( global_option["memory"] || global_option["onetape"] || global_option["optimize"] )
          return false;
     // -----------------------------------------------------
     // setup
     size_t i;             // temporary index
     fadbad::T<double>  Z; // domain space AD value
     fadbad::T<double>  P; // range space AD value

     // choose the polynomial coefficients
     CppAD::uniform_01(size, a);

     // AD copy of the polynomial coefficients
     CppAD::vector< fadbad::T<double> > A(size);
     for(i = 0; i < size; i++)
          A[i] = a[i];

     // ------------------------------------------------------
     while(repeat--)
     {     // get the next argument value
          CppAD::uniform_01(1, z);

          // independent variable value
          Z    = z[0]; // argument value
          Z[1] = 1;    // argument first order Taylor coefficient

          // AD computation of the dependent variable
          P = CppAD::Poly(0, A, Z);

          // Taylor-expand P to degree one
          P.eval(2);

          // second derivative is twice second order Taylor coefficient
          ddp[0] = 2. * P[2];

          // Free DAG corresponding to P does not seem to improve speed.
          // Probably because it gets freed the next time P is assigned.
          // P.reset();
     }
     // ------------------------------------------------------
     return true;
}
Input File: speed/fadbad/poly.cpp