I found somebody using the following approximation for $\sqrt{x^2 + y^2}$:
$\sqrt{x^2 + y^2} \approx (x+y) \sqrt{2/\pi},$
where $x$ and $y$ are positive numbers smaller than 1. It appears in this context:
http://www.cepii.fr/CEPII/en/publications/wp/abstract.asp?NoDoc=2726; p.18
So $x^2$ and $y^2$ are variances of a random value which describes the reporting error in trade data (assumed to be log-normally distributed).
What's the rationale for this approximation?
I tried several values for $x$ and $y$ and the approximation isn't too bad. E.g. $x=0.5$ and $y=0.3$ yields 0.64 instead of 0.58, and even for $x=0.9$ and $y=0.1$, the error is still acceptable (0.80 instead of 0.91). In most practical cases, we expect $x$ and $y$ to be of somewhat similar size (let's say a maximum ratio $x/y$ of 2).
Thanks!