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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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
Second Order Reverse ModeExample and Test
# include <cppad/cppad.hpp>
namespace { // ----------------------------------------------------------
// define the template function reverse_two_cases<Vector> in empty namespace
template <class Vector>
bool reverse_two_cases(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 = 2;
    CPPAD_TESTVECTOR(AD<double>) X(n);
    X[0] = 0.;
    X[1] = 1.;

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

    // range space vector
    size_t m = 1;
    CPPAD_TESTVECTOR(AD<double>) Y(m);
    Y[0] = X[0] * X[0] * X[1];

    // create f : X -> Y and stop recording
    CppAD::ADFun<double> f(X, Y);

    // use zero order forward mode to evaluate y at x = (3, 4)
    // use the template parameter Vector for the vector type
    Vector x(n), y(m);
    x[0]  = 3.;
    x[1]  = 4.;
    y     = f.Forward(0, x);
    ok    &= NearEqual(y[0] , x[0]*x[0]*x[1], eps99, eps99);

    // use first order forward mode in x[0] direction
    // (all second order partials below involve x[0])
    Vector dx(n), dy(m);
    dx[0] = 1.;
    dx[1] = 1.;
    dy    = f.Forward(1, dx);
    double check = 2.*x[0]*x[1]*dx[0] + x[0]*x[0]*dx[1];
    ok   &= NearEqual(dy[0], check, eps99, eps99);

    // use second order reverse mode to evalaute second partials of y[0]
    // with respect to (x[0], x[0]) and with respect to (x[0], x[1])
    Vector w(m), dw( n * 2 );
    w[0]  = 1.;
    dw    = f.Reverse(2, w);

    // check derivative of f
    ok   &= NearEqual(dw[0*2+0] , 2.*x[0]*x[1], eps99, eps99);
    ok   &= NearEqual(dw[1*2+0] ,    x[0]*x[0], eps99, eps99);

    // check derivative of f^{(1)} (x) * dx
    check = 2.*x[1]*dx[1] + 2.*x[0]*dx[1];
    ok   &= NearEqual(dw[0*2+1] , check, eps99, eps99);
    check = 2.*x[0]*dx[1];
    ok   &= NearEqual(dw[1*2+1] , check, eps99, eps99);

    return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool reverse_two(void)
{   bool ok = true;
    ok &= reverse_two_cases< CppAD::vector  <double> >();
    ok &= reverse_two_cases< std::vector    <double> >();
    ok &= reverse_two_cases< std::valarray  <double> >();
    return ok;
}
Input File: example/general/reverse_two.cpp