Can you think of a $C^1$ bijection $f:\mathbb{R}\to\mathbb{R}$ with $f^{-1}$ also of class $C^1$ but $f$ is not of class $C^\infty$?

I have been trying to construct one from some usual examples of $C^1$ functions that are not $C^\infty$ (like $x\mapsto \mid x\mid^3$), but they are not even bijective or defined on all $\mathbb{R}$. And also some usual examples of smooth homeomorphisms that are not diffeomorphisms (like $x\mapsto x^3$) do not do the job here because their inverse is not even differentiable.