exp_eps: 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 $x = .5$ and $\varepsilon = .2$, of the function       exp_eps(x, epsilon)  as defined by the exp_eps.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 $x = .1$ and $\varepsilon = .2$, of the function       exp_eps(x, epsilon) 
2. Create and test a modified version of the routine below that computes partial derivative with respect to $x$, at the point $x = .1$ and $\varepsilon = .2$, of the function corresponding to the operation sequence for $x = .5$ and $\varepsilon = .2$. Hint: you could define a vector u with two components and use       f.Forward(0, u)  to run zero order forward mode at a point different form the point where the operation sequence corresponding to f was recorded.
# include <cppad/cppad.hpp>  // http://www.coin-or.org/CppAD/
# include "exp_eps.hpp"      // our example exponential function approximation
{     bool ok = true;
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 = 2; // dimension of the domain space
U[0] = .5;    // value of x for this operation sequence
U[1] = .2;    // value of e for this operation sequence

// declare independent variables and start recording operation sequence

// evaluate our exponential approximation

// range space vector
size_t m = 1;  // dimension of the range space
Y[0] = apx;    // variable that represents only range space component

// Create f: U -> Y corresponding to this operation sequence
// and stop recording. This also executes a zero order forward
// mode sweep using values in U for x and e.

// first order forward mode sweep that computes partial w.r.t x
vector<double> du(n);      // differential in domain space
vector<double> dy(m);      // differential in range space
du[0] = 1.;                // x direction in domain space
du[1] = 0.;
dy    = f.Forward(1, du);  // partial w.r.t. x
double check = 1.5;
ok   &= NearEqual(dy[0], check, 1e-10, 1e-10);

// first order reverse mode 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_eps(x, epsilon)
check = 1.5;              // partial w.r.t. x
ok   &= NearEqual(dw[0], check, 1e-10, 1e-10);
check = 0.;               // partial w.r.t. epsilon
ok   &= NearEqual(dw[1], check, 1e-10, 1e-10);

// second order forward sweep that computes
// second partial of exp_eps(x, epsilon) w.r.t. x
vector<double> x2(n);     // second order Taylor coefficients
vector<double> y2(m);
x2[0] = 0.;               // evaluate partial w.r.t x
x2[1] = 0.;
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_eps(x, epsilon) w.r.t. x
dw.resize(2 * n);         // space for first and second derivative
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;
}