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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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
One Dimensional Romberg Integration: Example and Test

# include <cppad/utility/romberg_one.hpp>
# include <cppad/utility/vector.hpp>
# include <cppad/utility/near_equal.hpp>

namespace {
    class Fun {
    private:
        const size_t degree;
    public:
        // constructor
        Fun(size_t degree_) : degree(degree_)
        { }

        // function F(x) = x^degree
        template <class Type>
        Type operator () (const Type &x)
        {   size_t i;
            Type   f = 1;
            for(i = 0; i < degree; i++)
                f *= x;
            return f;
        }
    };
}

bool RombergOne(void)
{   bool ok = true;
    size_t i;

    size_t degree = 4;
    Fun F(degree);

    // arguments to RombergOne
    double a = 0.;
    double b = 1.;
    size_t n = 4;
    size_t p;
    double r, e;

    // int_a^b F(x) dx = [ b^(degree+1) - a^(degree+1) ] / (degree+1)
    double bpow = 1.;
    double apow = 1.;
    for(i = 0; i <= degree; i++)
    {   bpow *= b;
        apow *= a;
    }
    double check = (bpow - apow) / double(degree+1);

    // step size corresponding to r
    double step = (b - a) / exp(log(2.)*double(n-1));
    // step size corresponding to error estimate
    step *= 2.;
    // step size raised to a power
    double spow = 1;

    for(p = 0; p < n; p++)
    {   spow = spow * step * step;

        r = CppAD::RombergOne(F, a, b, n, p, e);

        ok  &= e < double(degree+1) * spow;
        ok  &= CppAD::NearEqual(check, r, 0., e);
    }

    return ok;
}
Input File: example/utility/romberg_one.cpp