On the regularity of a semialgebraic function

Let $f$ be a semialgebraic function of class $C^1$, defined on an open set ${U \subset \mathbb R ^n}$. Let $P(x,y) \in \mathbb R [x_1,\dots , x_n, y]$ be a polynomial of degree $d$ such that ${P(x,f(x)) = 0}$, $x\in U$. We prove that if $f$ is of class $C^K$ with $K \gt \frac {1}{2}d^7$, then $f$ is analytic. If $n=1$, then it suffices that $K \gt \frac {1}{2}d^2$.