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$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
The AD erf Function: Example and Test
 # include <cppad/cppad.hpp> # include <cmath> # include <limits> bool Erf(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::erf(ax[0]); // create f: x -> y and stop tape recording CppAD::ADFun<double> f(ax, ay); // check relative erorr double erf_x0 = 0.5204998778130465; ok &= NearEqual(ay[0] , erf_x0, 0., 4e-4); # if CPPAD_USE_CPLUSPLUS_2011 double tmp = std::max(1e-15, eps); ok &= NearEqual(ay[0] , erf_x0, 0., tmp); # endif // value of derivative of erf 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); # if CPPAD_USE_CPLUSPLUS_2011 ok &= NearEqual(dy[0], check, 0., eps); # endif // 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); # if CPPAD_USE_CPLUSPLUS_2011 ok &= NearEqual(dw[0], check, 0., eps); # endif // use a VecAD<Base>::reference object with erf CppAD::VecAD<double> v(1); AD<double> zero(0); v[zero] = x0; AD<double> result = CppAD::erf(v[zero]); ok &= NearEqual(result, ay[0], eps, eps); // use a double with erf ok &= NearEqual(CppAD::erf(x0), ay[0], eps, eps); return ok; } 
Input File: example/general/erf.cpp