Expressing maximum and minimum as $\frac12(x+y)\pm\frac12|x-y|$

Question

I'm looking at ways to get the max/min value of two numbers without using conditional statements, I found these two functions:

int Max(int x, int y)
{
    return (float)(x + y) / 2.0 + abs((float)(x - y) / 2);
}

int Min(int x, int y)
{
    return (float)(x + y) / 2.0 - abs((float)(x - y) / 2);
}

While they do work, what is the mathematical reason/proof behind them? why do they work?

In mathematical notation:

$$\max(x,y)=\frac12(x+y)+\frac12|x-y|$$ $$\min(x,y)=\frac12(x+y)-\frac12|x-y|$$

Answer 1

Without loss of generality, assume that $x>y$. Then, $$\max(x,y)=\frac12(x+y)+\frac12|x-y|=\frac12(x+y)+\frac12(x-y)=x.$$

Similarly, assuming $x>y$, then, $$\min(x,y)=\frac12(x+y)-\frac12|x-y|=\frac12(x+y)-\frac12(x-y)=y.$$

This is what you should expect, seeing as though $|x-y|=x-y$ for $x>y$.