Curious formula for minimum?

A few years ago I derived the following formula which I just came across in my notes:

$$\min(x,y)=\log\left(\frac{e^x+e^y}{1+e^{|x-y|}}\right)=y+\log\left(\frac{1+e^{x-y}}{1+e^{|x-y|}}\right).$$

Has anyone seen this before, and if so is there a reference? There is also a version for $\max$,

$$\max(x,y)=\log \left(\frac{e^x+e^y}{1+e^{-\left| x-y\right| }}\right).$$

At the time (rightly or wrongly!) I thought maybe it might shed light on "$\min$" and "$\max$" analogies for complex numbers, but apart from producing a new complex number (different to $x$ and $y$) it didn't yield much. Still I'd not seen this before so thought it might be of interest... Works perfectly for the reals.

UPDATE

Apparently this is related to the Softmax function.

Whether it's related to softmax or not, we have $$\log\left(\frac{1+e^{2t}}{1+e^{|2t|}}\right)= \log\left(\frac{e^t\cosh t}{e^{|t|}\cosh|t|}\right)= t-|t|$$ For $2t=x-y$, we get that your second expression for the minimum is $$y+\frac{x-y}{2}-\frac{|x-y|}{2}=\frac{1}{2}(x+y-|x-y|)$$ which is a well known expression for $\min\{x,y\}$ and much less heavy from a computational point of view.