Explicit formula for floor(x)?

Note that $\lfloor x \rfloor = x - \{x\}$, where $\{x \}$ (the fractional part of $x$) is periodic with period $1$, with a jump discontinuity at integer points. Let's try to write $\{x\}=f\left(\cot \pi x\right)$, since $\cot \pi x$ has the same property. We need $f(y)\rightarrow 0$ as $y\rightarrow \infty$, $f(y)\rightarrow 1$ as $y\rightarrow -\infty$, and the correct arc-tangent-y interpolation in between. What works is $f(y)=\frac{1}{2}-\frac{1}{\pi}\tan^{-1}y$. Putting things together, $$ \lfloor x \rfloor=x-\frac{1}{2} + \frac{1}{\pi}\tan^{-1}(\cot \pi x). $$