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

Specifications

Implementation
# include <cppad/utility/vector.hpp>

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

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

// choose the polynomial coefficients

// AD copy of the polynomial coefficients
for(i = 0; i < size; i++)
A[i] = a[i];

// ------------------------------------------------------
while(repeat--)
{     // get the next argument value

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

// AD computation of the dependent variable

// 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;
}