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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 .
Forward Mode: Example and Test of Multiple Orders
# include <limits>
# include <cppad/cppad.hpp>
bool forward_order(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;
    CPPAD_TESTVECTOR(AD<double>) ax(n);
    ax[0] = 0.;
    ax[1] = 1.;

    // declare independent variables and starting recording
    CppAD::Independent(ax);

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

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

    // initially, the variable values during taping are stored in f
    ok &= f.size_order() == 1;

    // Compute three forward orders at once
    size_t q = 2, q1 = q+1;
    CPPAD_TESTVECTOR(double) xq(n*q1), yq;
    xq[q1*0 + 0] = 3.;    xq[q1*1 + 0] = 4.; // x^0 (order zero)
    xq[q1*0 + 1] = 1.;    xq[q1*1 + 1] = 0.; // x^1 (order one)
    xq[q1*0 + 2] = 0.;    xq[q1*1 + 2] = 0.; // x^2 (order two)
    // X(t) =   x^0 + x^1 * t + x^2 * t^2
    //      = [ 3 + t, 4 ]
    yq  = f.Forward(q, xq);
    ok &= size_t( yq.size() ) == m*q1;
    // Y(t) = F[X(t)]
    //      = (3 + t) * (3 + t) * 4
    //      = y^0 + y^1 * t + y^2 * t^2 + o(t^3)
    //
    // check y^0 (order zero)
    CPPAD_TESTVECTOR(double) x0(n);
    x0[0] = xq[q1*0 + 0];
    x0[1] = xq[q1*1 + 0];
    ok  &= NearEqual(yq[q1*0 + 0] , x0[0]*x0[0]*x0[1], eps, eps);
    //
    // check y^1 (order one)
    ok  &= NearEqual(yq[q1*0 + 1] , 2.*x0[0]*x0[1], eps, eps);
    //
    // check y^2 (order two)
    double F_00 = 2. * yq[q1*0 + 2]; // second partial F w.r.t. x_0, x_0
    ok   &= NearEqual(F_00, 2.*x0[1], eps, eps);
    //
    // check number of orders per variable
    ok   &= f.size_order() == 3;
    //
    return ok;
}
Input File: example/general/forward_order.cpp