  We present a new algorithm for the numerical evaluation of five-point
conformal blocks in $d$-dimensions, greatly improving the efficiency of their
computation. To do this we use an appropriate ansatz for the blocks as a series
expansion in radial coordinates, derive a set of recursion relations for the
unknown coefficients in the ansatz, and evaluate the series using a Pad\'e
approximant to accelerate its convergence. We then study the
$\langle\sigma\sigma\epsilon\sigma\sigma\rangle$ correlator in the 3d critical
Ising model by truncating the operator product expansion (OPE) and only
including operators with conformal dimension below a cutoff $\Delta\leqslant
\Delta_{\rm cutoff}$. We approximate the contributions of the operators above
the cutoff by the corresponding contributions in a suitable disconnected
five-point correlator. Using this approach, we compute a number of OPE
coefficients with greater accuracy than previous methods.
