"The floor function furnishes an example of a function that couldn't be defined by a simple formula and for which (because it proved to be so useful) a new notation has been invented."

While searching for an expression for the Floor function online, I came across this expression:

floor(x)=x+(arctan(cot(pi*x))/pi) - 1/2

There is a short discussion about the function and analogs to it at this link:

https://nrich.maths.org/askedNRICH/edited/3225.html

I don't know if this meets the criteria for a "simple formula" or not, but it might be of interest to those trying to solve equations that use the floor function. Thanks!

-Andrew

Alex