  The contribution of the $\alpha^3$ single electron-loop vacuum-polarization
diagrams to the Lamb shift of the muonic hydrogen has been evaluated recently
by two independent methods. One uses the exact parametric representation of the
vacuum-polarization function while the other relies on the Pad\'{e}
approximation method. High precision of these values offers an opportunity to
examine the reliability of the Monte-Carlo integration as well as that of the
Pad\'{e} method. Our examination covers both muonic hydrogen atom and muon
$g-2$. We tested them further for the cases involving two-loop vacuum
polarization, where an exact analytic result is known. Our analysis justifies
the result for the Lamb shift of the muonic hydrogen and also resolves the
long-standing discrepancy between two previous evaluations of the muon $g-2$
value.
