  We present an algorithm to efficiently sample first-passage times for
fractional Brownian motion. To increase the resolution, an initial coarse
lattice is successively refined close to the target, by adding exactly sampled
midpoints, where the probability that they reach the target is non-negligible.
Compared to a path of $N$ equally spaced points, the algorithm achieves the
same numerical accuracy $N_{\rm eff}$, while sampling only a small fraction of
all points. Though this induces a statistical error, the latter is bounded for
each bridge, allowing us to bound the total error rate by a number of our
choice, say $P_{\rm error}^{\rm tot}=10^{-6}$. This leads to significant
improvements in both memory and speed. For $H=0.33$ and $N_{\rm eff}=2^{32}$,
we need $5\,000$ times less CPU time and $10\, 000$ times less memory than the
classical Davies Harte algorithm. The gain grows for $H=0.25$ and $N_{\rm eff}
= 2^{42}$ to $3\cdot 10^{5}$ for CPU and $10^6$ for memory. We estimate our
algorithmic complexity as ${\cal C}^{\rm ABSec}(N_{\rm eff}) = {\cal
O}\left(\left( \ln N_{\rm eff}\right)^{3}\right)$, to be compared to Davies
Harte which has complexity ${\cal C}^{\rm DH}(N) = {\cal O}\left(N \ln N
\right)$. Decreasing $P_{\rm error}^{\rm tot}$ results in a small increase in
complexity, proportional to $\ln (1/P_{\rm error}^{\rm tot})$. Our current
implementation is limited to the values of $N_{\rm eff}$ given above, due to a
loss of floating-point precision. The algorithm can be adapted to other extreme
events and arbitrary Gaussian processes. It enables one to numerically validate
theoretical predictions that were hitherto inaccessible.
