 Antithetic Cauchy-tail integration on a finite interval

For standard Cauchy distribution probability density function $f$ and $k>0$, this estimates its upper-tail probability without sampling a heavy tail. The function $f(ku)$ decreases on $[0,1]$, while $f(k(1-u))$ increases, so their covariance is nonpositive. The paired estimator is unbiased and has no larger variance than an estimator using the same number $2r$ of independent probability density function evaluations.