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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 .
exp_2: CppAD Forward and Reverse Sweeps
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Purpose
Use CppAD forward and reverse modes to compute the partial derivative with respect to @(@ x @)@, at the point @(@ x = .5 @)@, of the function
    exp_2(x)
 as defined by the exp_2.hpp include file.

Exercises
  1. Create and test a modified version of the routine below that computes the same order derivatives with respect to @(@ x @)@, at the point @(@ x = .1 @)@ of the function
        exp_2(    x)
  2. Create a routine called
        exp_3(    x)
     that evaluates the function @[@ f(x) = 1 + x^2 / 2 + x^3 / 6 @]@ Test a modified version of the routine below that computes the derivative of @(@ f(x) @)@ at the point @(@ x = .5 @)@. 

# include <cppad/cppad.hpp>  // http://www.coin-or.org/CppAD/
# include "exp_2.hpp"        // second order exponential approximation
bool exp_2_cppad(void)
{   bool ok = true;
    using CppAD::AD;
    using CppAD::vector;    // can use any simple vector template class
    using CppAD::NearEqual; // checks if values are nearly equal

    // domain space vector
    size_t n = 1; // dimension of the domain space
    vector< AD<double> > X(n);
    X[0] = .5;    // value of x for this operation sequence

    // declare independent variables and start recording operation sequence
    CppAD::Independent(X);

    // evaluate our exponential approximation
    AD<double> x   = X[0];
    AD<double> apx = exp_2(x);

    // range space vector
    size_t m = 1;  // dimension of the range space
    vector< AD<double> > Y(m);
    Y[0] = apx;    // variable that represents only range space component

    // Create f: X -> Y corresponding to this operation sequence
    // and stop recording. This also executes a zero order forward
    // sweep using values in X for x.
    CppAD::ADFun<double> f(X, Y);

    // first order forward sweep that computes
    // partial of exp_2(x) with respect to x
    vector<double> dx(n);  // differential in domain space
    vector<double> dy(m);  // differential in range space
    dx[0] = 1.;            // direction for partial derivative
    dy    = f.Forward(1, dx);
    double check = 1.5;
    ok   &= NearEqual(dy[0], check, 1e-10, 1e-10);

    // first order reverse sweep that computes the derivative
    vector<double>  w(m);   // weights for components of the range
    vector<double> dw(n);   // derivative of the weighted function
    w[0] = 1.;              // there is only one weight
    dw   = f.Reverse(1, w); // derivative of w[0] * exp_2(x)
    check = 1.5;            // partial of exp_2(x) with respect to x
    ok   &= NearEqual(dw[0], check, 1e-10, 1e-10);

    // second order forward sweep that computes
    // second partial of exp_2(x) with respect to x
    vector<double> x2(n);     // second order Taylor coefficients
    vector<double> y2(m);
    x2[0] = 0.;               // evaluate second partial .w.r.t. x
    y2    = f.Forward(2, x2);
    check = 0.5 * 1.;         // Taylor coef is 1/2 second derivative
    ok   &= NearEqual(y2[0], check, 1e-10, 1e-10);

    // second order reverse sweep that computes
    // derivative of partial of exp_2(x) w.r.t. x
    dw.resize(2 * n);         // space for first and second derivatives
    dw    = f.Reverse(2, w);
    check = 1.;               // result should be second derivative
    ok   &= NearEqual(dw[0*2+1], check, 1e-10, 1e-10);

    return ok;
}
Input File: introduction/exp_2_cppad.cpp