Nifty! I've been doing an approximate `sqrt(n)` recently with this iterative approach:
a = n
for 1..m:
a = (a + n / a) / 2

So I tried this using your approach for an initial guess:
a = max(abs(x),abs(y)) + min(abs(x),abs(y))/2;
s = x*x + y*y;
if (a != 0.0) {
a = (a + s / a) / 2;
a = (a + s / a) / 2;
}

Within a range of -100 to 100 this takes your approximation from a max error of about 34% to a max error of 0.4%. The average error goes from 9% to 0.4%.

It's still just 7% max error and 0.4% average error if you only do one iteration.

Sorry, no fancy visualisations, but here’s a picture of them both overlayed (I considered a “they're the same picture" meme instead):