Arithmetical representation $F(x,y,z)\Longleftrightarrow (x+z)=y \lor (y+z)=x$


I am kind of confused which function $ f:\mathbb{N}^2\longrightarrow\mathbb{N}$ is presented by $ F(x,y,z)\Longleftrightarrow (x+z)=y \lor (y+z)=x$

I know that $ f(x,y)=y-x$ is represented by $ F(x,y,z)\Longleftrightarrow (x+z=y)\lor ((z=0)\land \exists w(x+w)=y)$

But the or symbol confuses me … My guess was $ f(x,y)=max\{x-y,y-x\}$ but i think this is wrong due to a missing check in F(x,y,z)