Composite Simpson rule on an even uniform grid

Input real endpoints $a,b$, an integer $n$, and an evaluation function $f$. If $a\ge b$ or $n<2$, return invalid-grid without evaluating $f$. For the remaining inputs require $f:[a,b]\to \mathbb{R}$ to be evaluable at every point of $[a,b]$. If $n$ is odd, the procedure returns invalid-Simpson-panel-count. If $n$ is even, put $h=(b-a)\mathrm{/}n$ and $x_j=a+jh$ for $0\le j\le n$; return

$Q_S(f;a,b,n):=\frac{h}{3}{\sum }_{j<n\mathrm{/}2}(f(x_{2j})+4f(x_{2j+1})+f(x_{2j+2}))\mathrm{.}$

The sum is a finite sum of Finite sums and finite products, by recursion. Each two-panel block $[x_{2j},x_{2j+2}]$ uses its endpoints and midpoint. Expanding the block sum gives weights $1,4,2,\dots ,4,1$. The even condition $n\ge 2$ makes the number of blocks a positive integer. The formula is an exact-real procedure; floating point evaluation and summation error are separate from its quadrature error.