I've been working through a derivation and have arrived at the following expression:
$$E = 1 - \frac{x}y \left( \bigg\lceil \dfrac{x}{y} \bigg\rceil \right)^{-1}$$
where $x,y \in \mathbb{R^+}$.
I would like to know whether this can be reduced further, by converting the floor function to a polynomial to simplify expression?
furthermore, lets say there was a constant k, where $k \in \mathbb{N^+}$ hence,
$$E = 1 - \frac{x}{k.y} \left( \bigg\lceil \dfrac{x}{k.y} \bigg\rceil \right)^{-1}$$
could this be reduced to,
$$E = 1 - \frac{x}y \left( \bigg\lceil \dfrac{x}{y} \bigg\rceil \right)^{-1}$$
Kind regards!
If I understand your question correctly then the answer is No. The function $$ f(x)=\frac{x}{\lceil x \rceil} $$ has infinitely many (jump) discontinuities.
If we can somehow convert $f(x)$ into a polynomial or a rational function then it would be continuous on the whole $\Bbb R$ except for only finitely many points. This is impossible since it'd contradict the fact that $f(x)$ has infinitely many discontinuity.