  Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$, the
edit distance function ${\rm ed}_{\mathcal{H}}(p)$ is asymptotically the
maximum proportion of edge-additions plus edge-deletions applied to a graph of
edge density $p$ sufficient to ensure that the resulting graph satisfies
$\mathcal{H}$. The edit distance function is directly related to other
well-studied quantities such as the speed function for $\mathcal{H}$ and the
$\mathcal{H}$-chromatic number of a random graph. Let $\mathcal{H}$ be the
property of forbidding an Erd\H{o}s-R\'{e}nyi random graph $F\sim
\mathbb{G}(n_0,p_0)$, and let $\varphi$ represent the golden ratio. In this
paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$, then a.a.s. as
$n_0\to\infty$,
  \begin{align*}
  {\rm ed}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0}
  \cdot\min\left\{
  \frac{p}{-\log(1-p_0)},
  \frac{1-p}{-\log p_0}
  \right\}.
  \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in
(0,1)$.
