The commonly used smooth alternative for the function $\max(x,y)$ is the LogSumExp function, which can be (loosely) defined as: $$\frac{\ln(\exp(kx)+\exp(ky))}{k}$$ For higher values of $k$, this function tends towards the actual $\max$ function. Now I was thinking of an alternative, namely: $$\frac{y-x}{1+\exp(-k(y-x))}+x=(y-x)\sigma(-k(y-x))+x$$ where $\sigma$ is the standard logistic function. As shown here, this is also a good approximation of the $\max$ function.
Now my question is this:
What are the possible pros and cons of using my function instead of the LogSumExp function?
Context: I was thinking of this problem as alternatives to the common activation functions, but answers explaining the difference between the two functions based on other perspectives are also welcome.