Does there exist a smooth approximation of $x \bmod y$? I'm looking for a function $m(x,y)$ that smoothly approximates $x \bmod y$, and I'm assuming there would be some $n$ or $\varepsilon$ in the body of $m(x,y)$ that defines the degree of approximation such that as $n$ goes to infinity or $\varepsilon$ goes to zero, 
$$
m(x,y)= x \bmod y.
$$
I've looked at, for example, $\sin(x/y)$ and have been trying to figure out ways to make $\sin(x)$ more "steep" so that I can apply such techniques to $\sin(x/y)$ in order to perhaps more closely approximate $x \bmod y$.
The analogy I'm looking at is the approximation of $\mid x\mid$ as $\sqrt{x^2 + \varepsilon)}$ where epsilon is some small quantity.
Does a similarly smooth function exist for $x \bmod y$?

One approach I found was to find a smooth floor function, and then simply apply it to the equation:
$$
x \bmod y=x-{\lfloor}\frac{x}{y}{\rfloor}y
$$
And a definition for the smooth floor function I derived is:
$$
{\lfloor}\frac{x}{y}{\rfloor}=-\frac{1}{2} + x + a ln(1/a) -\frac{i}{2 \pi} \left[ln\left(1-e^\left(-2 i \pi \left(x+a 
 ln(1/a)\right)\right)+a\right) - ln\left(1-e^\left(2 i \pi \left(x+a 
 ln(1/a)\right)\right)+a\right) \right]
$$
As you make a smaller and smaller, this increasingly approximates the floor function.
Update
I realized after posting that @user856 had already posted a much simpler approximation quite a while ago.
