Minimal regular normal crossings models of superelliptic curves

Let $K$ be a complete discretely valued field with perfect residue field $k$. If $X\to {\mathbb{P}}_K^1$ is a $\mathbb{Z}\mathrm{/}d$-cover with $\text{char }k\nmid d$, we compute the minimal regular normal crossings model $\mathcal{X}$ of $X$ as the normalization of an explicit normal model $\mathcal{Y}$ of ${\mathbb{P}}_K^1$ in $K(X)$. The model $\mathcal{Y}$ is given using Mac Lane’s description of discrete valuations on the rational function field $K({\mathbb{P}}^1)$.