Avoiding catastrophic cancellation

To avoid catastrophic cancellation occurring when $x \to 0$, then as Raymond Manzoni's question comment suggests, rationalize $1 - \sqrt{1 - x^2}$ to get

$$\begin{equation}\begin{aligned} 1 - \sqrt{1 - x^2} & = (1 - \sqrt{1 - x^2})\left(\frac{1 + \sqrt{1 - x^2}}{1 + \sqrt{1 - x^2}}\right) \\ & = \frac{1 - (1 - x^2)}{1 + \sqrt{1 - x^2}} \\ & = \frac{x^2}{1 + \sqrt{1 - x^2}} \end{aligned}\end{equation}\tag{1}\label{eq1A}$$