\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

erfc.cpp

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The AD erfc Function: Example and Test

# include <cppad/cppad.hpp>
# include <cmath>
# include <limits>

bool erfc(void)
{   bool ok = true;

    using CppAD::AD;
    using CppAD::NearEqual;
    double eps = 10. * CppAD::numeric_limits<double>::epsilon();

    // domain space vector
    size_t n  = 1;
    double x0 = 0.5;
    CPPAD_TESTVECTOR(AD<double>) ax(n);
    ax[0]     = x0;

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

    // range space vector
    size_t m = 1;
    CPPAD_TESTVECTOR(AD<double>) ay(m);
    ay[0] = CppAD::erfc(ax[0]);

    // create f: x -> y and stop tape recording
    CppAD::ADFun<double> f(ax, ay);

    // check relative error
    double erfc_x0 = 0.4795001221869534;
    ok &= NearEqual(ay[0] , erfc_x0,  0.,    4e-4);
    double tmp = std::max(1e-15, eps);
    ok &= NearEqual(ay[0] , erfc_x0,  0.,    tmp);

    // value of derivative of erfc at x0
    double pi     = 4. * std::atan(1.);
    double factor = 2. / sqrt(pi);
    double check  = - factor * std::exp(-x0 * x0);

    // forward computation of first partial w.r.t. x[0]
    CPPAD_TESTVECTOR(double) dx(n);
    CPPAD_TESTVECTOR(double) dy(m);
    dx[0] = 1.;
    dy    = f.Forward(1, dx);
    ok   &= NearEqual(dy[0], check,  0.,  1e-3);
    ok   &= NearEqual(dy[0], check,  0.,  eps);

    // reverse computation of derivative of y[0]
    CPPAD_TESTVECTOR(double)  w(m);
    CPPAD_TESTVECTOR(double) dw(n);
    w[0]  = 1.;
    dw    = f.Reverse(1, w);
    ok   &= NearEqual(dw[0], check,  0., 1e-1);
    ok   &= NearEqual(dw[0], check,  0., eps);

    // use a VecAD<Base>::reference object with erfc
    CppAD::VecAD<double> v(1);
    AD<double> zero(0);
    v[zero]           = x0;
    AD<double> result = CppAD::erfc(v[zero]);
    ok   &= NearEqual(result, ay[0], eps, eps);

    // use a double with erfc
    ok   &= NearEqual(CppAD::erfc(x0), ay[0], eps, eps);

    return ok;
}