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Adaptive Simpson's method compares one Simpson's rule panel $S$ on $[a,b]$ with two panels totaling $S_2$ after bisection. In the smooth asymptotic regime, the leading error drops by a factor of sixteen, so $(S_2-S)/15$ estimates $I-S_2$, and $S_2+(S_2-S)/15$ is the corrected estimate.

If the discrepancy exceeds the local tolerance, subdivide again and split the error budget between the children. Endpoint and midpoint values can be reused. The factor fifteen follows by eliminating the leading fourth-order error; it is not a rigorous error bound without quantitative assumptions. A narrow peak missed by all sampled nodes can make both estimates agree while both are wrong, as finite samples do not certify quadrature accuracy explains.