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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.

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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.

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