from scipy.optimize import root_scalar


def get_f_df_df2(value=0):
    """Get target function and its first two derivatives."""
    if value < 0:
        raise ValueError("Neuman: epsilon for root finding needs to be >0.")

    def _f_df_df2(x):
        return (
            np.subtract(np.multiply(x, np.tan(x)), value),
            np.tan(x) + np.divide(x, np.cos(x) ** 2),
            2 * (np.multiply(x, np.tan(x)) + 1.0) * np.cos(x) ** -2,
        )

    return _f_df_df2


def nth_root(n, v):
    """Get n-th root of x*tan(x) = v."""
    x0 = np.sqrt(v) if (v < 1 and n < 1) else np.arctan(v) + n * np.pi
    f = get_f_df_df2(v)
    sol = root_scalar(f, x0=x0, fprime=True, fprime2=True)
    if not sol.converged:
        raise ValueError(f"Neuman: couldn't find {n}-th root for eps={v}")
    return sol.root