Let $\varepsilon\in(0,1)$ and $c>0$. Note that $$\varphi(x):=e^{-(cx)^2}<\varepsilon\Leftrightarrow|x|>\frac{\sqrt{-\ln\varepsilon}}c\tag1$$ for all $x\in\mathbb R$. Now let $h\in(-1,1)$.
I need to numerically verify in a computer program whether $$\varphi(h-k)<\varepsilon$$ for all $k\in\mathbb Z$? It is crucial to perform this test as efficient as possible.
As explained in this answer, the desired condition is equivalent to $$\exists n\in\Bbb Z\quad n+a<h<n+1-a.\tag1$$ Unfortunately, I have no idea how we can check this efficiently.
The existence of such an $n$ is equivalent to $$h+a<\lceil h-a\rceil$$ and also to $$\lfloor h+a\rfloor<h-a.$$