This example is intended to demonstrate how a_float and a2float can be used together to compute derivatives of functions that are defined in terms of derivatives of other functions.
For this example, the function $F : \B{R}^2 \rightarrow \B{R}$ is defined by $$F(u) = u_0^2 + u_1^2$$ It follows that $$\begin{array}{rcl} \partial_{u(0)} F(u) = 2 * u_0 \\ \partial_{u(1)} F(u) = 2 * u_1 \end{array}$$
For this example, the function $G : \B{R}^2 \rightarrow \B{R}$ is defined by $$G(x) = x_1 * \partial_{u(0)} F(x_0 , 1) + x_0 * \partial_{u(1)} F(x_0, 1)$$ where $\partial{u(j)} F(a, b)$ denotes the partial of $F$ with respect to $u_j$ and evaluated at $u = (a, b)$. It follows that $$\begin{array}{rcl} G (x) & = & 2 * x_1 * x_0 + 2 * x_0 \\ \partial_{x(0)} G (x) & = & 2 * x_1 + 2 \\ \partial_{x(1)} G (x) & = & 2 * x_0 \end{array}$$  from pycppad import * def pycppad_test_two_levels(): # start recording a_float operations x = numpy.array( [ 2. , 3. ] ) a_x = independent(x) # start recording a2float operations a_u = numpy.array( [a_x[0] , ad(1) ] ) a2u = independent(a_u) # stop a2float recording and store operations if f a2v = numpy.array( [ a2u[0] * a2u[0] + a2u[1] * a2u[1] ] ) a_f = adfun(a2u, a2v) # F(u0, u1) = u0 * u0 + u1 * u1 # evaluate the gradient of F a_J = a_f.jacobian(a_u) # stop a_float recording and store operations in g a_y = numpy.array( [ a_x[1] * a_J[0,0] + a_x[0] * a_J[0,1] ] ) g = adfun(a_x, a_y) # G(x0, x1) = x1 * F_u0(x0, 1) + x0 * F_u1(x0, 1) # evaluate the gradient of G J = g.jacobian(x) assert J[0,0] == 2. * x[1] + 2 assert J[0,1] == 2. * x[0]