Pulse and elementary-function enclosures

Let $D(x)=(x-1/4)(3/4-x)$. Its extrema on a support cell are found at endpoints and, if present, $1/2$. Since the bump is increasing in $D>0$, these extrema give a pulse rectangle. Boundary-crossing cells include zero.

Summing lengths times pulse intervals encloses $B_0$. Analytically,

The lower bound uses $[3/8,5/8]$, and the upper uses the support length and maximum.

The checker encloses

by alternating series. For $e^{-x}$, scale to $y=x/2^s\le 1$, enclose its Taylor polynomial using

and square the interval $s$ times. For $x\ge b$, use $e^{-x}\le 2^{-b}$. Sine and cosine use alternating bounds and monotonicity on $[0,1]$.

Established ball arithmetic and rigorous integration provide alternative implementations JohanssonArb,JohanssonIntegration; they do not certify the reported run merely by existing.