Importance sampling of a Cauchy tail ($Y=2/(1-U),W=Y^2/[2\pi (1+Y^2)]$)

For $U$ uniform on $(0,1)$, $Y$ has density $q(y)=2y^{-2}$ on $y\ge 2$. The importance sampling weight for the standard Cauchy distribution tail is $W=f(Y)/q(Y)$, and its expectation is the target tail probability. Writing $A=\arctan (1/2)$ gives $\mathbb{E}W=A/\pi$ and $\mathbb{E}{W}^2=(A+2/5)/(4{\pi }^2)$ by integrating $y^2/(1+y^2{)}^2$. The weight varies only between $2/(5\pi )$ and $1/(2\pi )$, making its variance much smaller than that of a rare-event indicator. This is a comparison of sampling variances, not a deterministic ordering of every pair of empirical variance estimates.