Balancing quadrature truncation and round-off error ($C{n}^{-p}\approx \delta ⟹n\approx (C/\delta {)}^{1/p}$)

If a quadrature error bound is $C{n}^{-p}$ and the relevant numerical resolution is $\delta$, equating these scales gives an estimate of the strip count beyond which that truncation guarantee reaches the rounding scale. For the composite Simpson's rule, $p=4$. This estimate is useful for choosing a sensible order of magnitude, but a worst-case bound need not equal the actual quadrature error.

For a normal floating-point number in base two, the spacing depends on the exponent of the result; half the spacing is a natural final-storage rounding scale. Machine epsilon instead measures the gap above one. Function-evaluation errors, accumulation of summation error, cancellation, and the arithmetic used for the nodes can impose an earlier practical accuracy limit. Thus final storage precision alone cannot give a universal optimal strip count.