double nlopt_eval_enode(const double* x, void * extra) {
    auto extra_info = static_cast<tuple<Enode *, box const &, bool> *>(extra);
    Enode * e = get<0>(*extra_info);
    box const & b = get<1>(*extra_info);
    bool const polarity = get<2>(*extra_info);
    unordered_map<Enode *, double> var_map;
    unsigned i = 0;
    for (Enode * e : b.get_vars()) {
        if (e->isForallVar()) {
            var_map.emplace(e, x[i]);
            i++;
        } else {
            var_map.emplace(e, b[e].mid());
        }
    }
    try {
        double const ret1 = eval_enode(e->get1st(), var_map);
        double const ret2 = eval_enode(e->get2nd(), var_map);
        double ret = 0;
        if (e->isLt() || e->isLeq() || e->isEq()) {
            ret = ret1 - ret2;
        } else if (e->isGt() || e->isGeq()) {
            ret = ret2 - ret1;
        } else if (e->isEq()) {
            throw runtime_error("nlopt_obj: something is wrong.");
        }
        if (!polarity) {
            ret = - ret;
        }
        return ret;
    } catch (exception & e) {
        DREAL_LOG_FATAL << "Exception in nlopt_eval_enode: " << e.what() << endl;
        throw e;
    }
}
box refine_CE_with_nlopt_core(box counterexample, vector<Enode*> const & opt_ctrs, vector<Enode*> const & side_ctrs) {
    // Plug-in `a` into the constraint and optimize `b` in the counterexample `M` by solving:
    //
    //    ∃ y_opt ∈ I_y. ∀ y ∈ I_y. f(a, y_opt) >= f(a, y) — (2)
    //
    // using local optimizer (i.e. nlopt).
    // Let `M’ = (a, b_opt)` be a model for (2).

    DREAL_LOG_DEBUG << "================================" << endl;
    DREAL_LOG_DEBUG << "  Before Refinement              " << endl;
    DREAL_LOG_DEBUG << "================================" << endl;
    DREAL_LOG_DEBUG << counterexample << endl;
    DREAL_LOG_DEBUG << "================================" << endl;
    static bool initialized = false;
    static vector<double> lb, ub, init;
    init.clear();
    for (Enode * e : counterexample.get_vars()) {
        if (e->isForallVar()) {
            if (!initialized) {
                lb.push_back(e->getDomainLowerBound());
                ub.push_back(e->getDomainUpperBound());
            }
            init.push_back(counterexample[e].mid());
            DREAL_LOG_DEBUG << lb.back() << " <= " << init.back() << " <= " << ub.back() << endl;
        }
    }
    auto const n = init.size();
    static nlopt::opt opt(nlopt::LD_SLSQP, n);
    if (!initialized) {
        opt.set_lower_bounds(lb);
        opt.set_upper_bounds(ub);
        // set tollerance
        // TODO(soonhok): set precision
        // opt.set_xtol_rel(0.0001);
        opt.set_xtol_abs(0.001);
        opt.set_maxtime(0.01);
        initialized = true;
    }

    opt.remove_equality_constraints();
    opt.remove_inequality_constraints();

    // set objective function
    vector<tuple<Enode *, box const &, bool> *> extra_vec;
    Enode * e = opt_ctrs[0];
    bool polarity = false;
    while (e->isNot()) {
        e = e->get1st();
        polarity = !polarity;
    }
    auto extra = new tuple<Enode *, box const &, bool>(e, counterexample, polarity);
    extra_vec.push_back(extra);
    opt.set_min_objective(nlopt_obj, extra);
    opt.add_inequality_constraint(nlopt_side_condition, extra);
    DREAL_LOG_DEBUG << "objective function is added: " << e << endl;

    // set side conditions
    for (Enode * e : side_ctrs) {
        bool polarity = false;
        while (e->isNot()) {
            e = e->get1st();
            polarity = !polarity;
        }
        auto extra = new tuple<Enode *, box const &, bool>(e, counterexample, polarity);
        extra_vec.push_back(extra);
        DREAL_LOG_DEBUG << "refine_counterexample_with_nlopt: Side condition is added: " << e << endl;
        if (e->isEq()) {
            opt.add_equality_constraint(nlopt_side_condition, extra);
        } else if (e->isLt() || e->isLeq() || e->isGt() || e->isGeq()) {
            opt.add_inequality_constraint(nlopt_side_condition, extra);
        }
    }
    try {
        vector<double> output = opt.optimize(init);
        unsigned i = 0;
        for (Enode * e : counterexample.get_vars()) {
            if (e->isForallVar()) {
                counterexample[e] = output[i];
                i++;
            }
        }
    } catch (nlopt::roundoff_limited & e) {
    } catch (std::runtime_error & e) {
        DREAL_LOG_DEBUG << e.what() << endl;
    }

    for (auto extra : extra_vec) {
        delete extra;
    }
    DREAL_LOG_DEBUG << "================================" << endl;
    DREAL_LOG_DEBUG << "  After Refinement              " << endl;
    DREAL_LOG_DEBUG << "================================" << endl;
    DREAL_LOG_DEBUG << counterexample << endl;
    DREAL_LOG_DEBUG << "================================" << endl;
    return counterexample;
}