Suppose s_j is the value of some integral obtained by use of the trapezoid rule for n_j intervals, with n_j = 2^j.

Then the quadratic terms in the Taylor expansion of the integrand will cause errors which will behave as the square of the size of the intervals; these will decrease by a factor 1 / 4 when j is replaced by j + 1. Further terms in the Taylor expansion decline even more rapidly with increasing j. If we apply the result above with c = 1 / 4, we obtain the new sequence, t_j :

t_j = (4 / 3)s_j - (1 / 3)s_{j -1} .

The sequence {t_j } is a better numerical integrator: in fact t_j is the estimate of the integral given by Simpson's Rule,and t₁ is exactly correct for any cubic integrand.

We can obtain a still better integration rule by repeating this process. Since the leading error in Simpson's rule occurs from the quartic terms in the Taylor expansion and these behave as the inverse fourth power of n, they will decrease by factor 1 / 16 when j increases by 1. The better sequence is b_j = (16 / 15)t_j - t_{j -1} / 15.

You can verify that b₂ gives the exactly correct integral with any quintic polynomial as integrand.

When the t_j represent Simpson's rule estimates, the b_j provide a Super Simpson's rule.