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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
# include <limits>
# include <cppad/cppad.hpp>
namespace { // --------------------------------------------------------
// define the template function ForwardCases<Vector> in empty namespace
template <class Vector>
bool ForwardCases(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;

    // zero order forward mode using notation in forward_zero
    // use the template parameter Vector for the vector type
    Vector x0(n), y0(m);
    x0[0] = 3.;
    x0[1] = 4.;
    y0    = f.Forward(0, x0);
    ok  &= NearEqual(y0[0] , x0[0]*x0[0]*x0[1], eps, eps);
    ok  &= f.size_order() == 1;

    // first order forward mode using notation in forward_one
    // X(t)           = x0 + x1 * t
    // Y(t) = F[X(t)] = y0 + y1 * t + o(t)
    Vector x1(n), y1(m);
    x1[0] = 1.;
    x1[1] = 0.;
    y1    = f.Forward(1, x1); // partial F w.r.t. x_0
    ok   &= NearEqual(y1[0] , 2.*x0[0]*x0[1], eps, eps);
    ok   &= f.size_order() == 2;

    // second order forward mode using notation in forward_order
    // X(t) =           x0 + x1 * t + x2 * t^2
    // Y(t) = F[X(t)] = y0 + y1 * t + y2 * t^2 + o(t^3)
    Vector x2(n), y2(m);
    x2[0]      = 0.;
    x2[1]      = 0.;
    y2         = f.Forward(2, x2);
    double F_00 = 2. * y2[0]; // second partial F w.r.t. x_0, x_0
    ok         &= NearEqual(F_00, 2.*x0[1], eps, eps);
    ok         &= f.size_order() == 3;

    return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool Forward(void)
{   bool ok = true;
    // Run with Vector equal to three different cases
    // all of which are Simple Vectors with elements of type double.
    ok &= ForwardCases< CppAD::vector  <double> >();
    ok &= ForwardCases< std::vector    <double> >();
    ok &= ForwardCases< std::valarray  <double> >();
    return ok;
}
Input File: example/general/forward.cpp