A mathematical way for defining the Floor and Ceiling functions

Use $\bmod$ (not necessarily $\diamond$ mods) $$ \lfloor x \rfloor = x - \bmod(x,1) $$ To get $\bmod(x,1)$, use $\frac12\Biggr(\frac{\log\left(\exp\left(2\pi i(x- \frac12)\right)\right)}{\pi i}+1\Biggr)$, since $\exp\left(2\pi i(x- \frac12)\right)$ has a period of $1$.

Floor and ceiling functions are usually defined as

$$ \lfloor x \rfloor=\max\, \{m\in\mathbb{Z}\mid m\le x\} $$ and $$ \lceil x \rceil=\min\,\{n\in\mathbb{Z}\mid n\ge x\} $$ for $x\in\mathbb R$ (see Floor and ceiling functions for more details).