$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
# 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 // cummulative 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_var() == before.n_var; ok &= f.size_op() == before.n_op; // Optimize the operation sequence f.optimize(); 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; }