On the Limiting Density of a gcd Map

The function $$f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}$$ is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $\rho=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/\zeta(2)=6/\pi^2$.