  We perform large-scale Monte Carlo simulations of the classical XY model on a
three-dimensional $L\times L \times L$ cubic lattice using the graphics
processing unit (GPU). By the combination of Metropolis single-spin flip,
over-relaxation and parallel-tempering methods, we simulate systems up to
L=160. Performing the finite-size scaling analysis, we obtain estimates of the
critical exponents for the three-dimensional XY universality class:
$\alpha=-0.01293(48)$ and $\nu=0.67098(16)$. Our estimate for the
correlation-length exponent $\nu$, in contrast to previous theoretical
estimates, agrees with the most recent experimental estimate $\nu_{\rm
exp}=0.6709(1)$ at the superfluid transition of $^4$He in a microgravity
environment.
