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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 .
Optimize Cumulative Sum Operations: Example and Test
# include <cppad/cppad.hpp>

namespace {
    struct tape_size { size_t n_var; size_t n_op; };

    template <class Vector> void fun(
        const Vector& x, Vector& y, tape_size& before, tape_size& after
    )
    {   typedef typename Vector::value_type scalar;

        // phantom variable with index 0 and independent variables
        // begin operator, independent variable operators and end operator
        before.n_var = 1 + x.size(); before.n_op  = 2 + x.size();
        after.n_var  = 1 + x.size(); after.n_op   = 2 + x.size();

        // operators that are identical, and that will be made part of the
        // cumulative summation. Make sure do not replace second variable
        // using the first and then remove the first as part of the
        // cumulative summation.
        scalar first  = x[0] + x[1];
        scalar second = x[0] + x[1];
        before.n_var += 2; before.n_op  += 2;
        after.n_var  += 0; after.n_op   += 0;

        // test that subtractions are also included in cumulative summations
        scalar third = x[1] - 2.0;
        before.n_var += 1; before.n_op  += 1;
        after.n_var  += 0; after.n_op   += 0;

        // the finial summation is converted to a cumulative summation
        // the other is removed.
        scalar csum = first + second + third;
        before.n_var += 2; before.n_op  += 2;
        after.n_var  += 1; after.n_op   += 1;

        // results for this operation sequence
        y[0] = csum;
        before.n_var += 0; before.n_op  += 0;
        after.n_var  += 0; after.n_op   += 0;
    }
}
bool cumulative_sum(void)
{   bool ok = true;
    using CppAD::AD;
    using CppAD::NearEqual;
    double eps10 = 10.0 * std::numeric_limits<double>::epsilon();

    // domain space vector
    size_t n  = 2;
    CPPAD_TESTVECTOR(AD<double>) ax(n);
    ax[0] = 0.5;
    ax[1] = 1.5;

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

    // range space vector
    size_t m = 1;
    CPPAD_TESTVECTOR(AD<double>) ay(m);
    tape_size before, after;
    fun(ax, ay, before, after);

    // create f: x -> y and stop tape recording
    CppAD::ADFun<double> f(ax, ay);
    ok &= f.size_order() == 1; // this constructor does 0 order forward
    ok &= f.size_var() == before.n_var;
    ok &= f.size_op()  == before.n_op;

    // Optimize the operation sequence
    f.optimize();
    ok &= f.size_order() == 0; // 0 order forward not present
    ok &= f.size_var() == after.n_var;
    ok &= f.size_op()  == after.n_op;

    // Check result for a zero order calculation for a different x,
    CPPAD_TESTVECTOR(double) x(n), y(m), check(m);
    x[0] = 0.75;
    x[1] = 2.25;
    y    = f.Forward(0, x);
    fun(x, check, before, after);
    ok  &= CppAD::NearEqual(y[0], check[0], eps10, eps10);

    return ok;
}
Input File: example/optimize/cumulative_sum.cpp