// Example program
#include <iostream>
#include <string>

struct Dual {
  float real;
  float img;

  Dual(float x, float y=0)
    :real{x}, img{y}
  {}
};

Dual operator+(Dual l, Dual r) {
    return {
        l.real+r.real,
        l.img +r.img
    };
}

Dual operator*(Dual l, Dual r) {
    return {
        l.real*r.real,
        l.img * r.real + l.real * r.img
    };
}

std::ostream& operator<< (std::ostream& o, Dual x) {
    return o << "("<<x.real << "|" <<x.img<<")";
}


auto f(auto x) {
    return x*x + 2*x + 3;
}

auto df(auto x) {
    return 2*x + 2;
}

static Dual d = {0,1};

int main() {
  std::cout << "manual (f(3)|df(3)) = (" << f(3) << "|" << df(3) <<")"<< std::endl;
  std::cout << "using duals : (f(3)|df(3)) = " << f( 3 + 1*d ) << std::endl;
}