)<sup>3</sup>). Differentiating this function
      by <code class="computeroutput"><span class="identifier">finite_difference_derivative</span></code>
      in double precision at <span class="emphasis"><em>x=5.5</em></span> gives zero correct digits
      at order 4, 6, and 8, but recovers 5 correct digits at order 2. These are dangerous
      waters; use the error estimates to tread carefully.
    </p>
<p>
      For a finite-difference method of order <span class="emphasis"><em>k</em></span>, the error i