In the pre-MO era, I once realized that on the integers, the function

$$ d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ , $$

is a metric (all properties are easily verified; in fact this is a Hilbertian metric).

Now that I got reminded of it, I wanted to ask

Does anyone know about applications where such a metric can be useful?

PS: I remember that this metric had some relations to Addition chains.