The variable $r$, defined for $r \in [0,\infty)$ needs to be related to to random numbers $x'\in [0,1]$. To achieve that, we introduce a new variable $$\begin{equation*} u=\frac{1}{2}r^2, \end{equation*}$$ and define a PDF $$\begin{equation*} \exp{(-u)}du, \end{equation*}$$ with $u\in [0,\infty)$. Using the results from example 2 for the exponential distribution, we have $$\begin{equation*} u=-\ln{(1-x')}, \end{equation*}$$ where $x'$ is a random number generated for $x'\in [0,1]$.