Checking whether $X=\mathbb R$ with $d(x,y)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ is a metric space

Question

Examine whether $d$ is a metric on $X=\mathbb{R}$ where $d\left(x,y\right)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ for all $x,y\in \mathbb{R}$

I think it is not. Even though it satisfies all other properties but it doesn't satisfy triangle inequality.

Take $x=2,y=1.5,z=1$ $d\left(x,z\right)=1$

$d\left(x,y\right)=0.25$

$d\left(y,z\right)=0.25$

Can someone confirm and verify. It is correct or not.