  Brent and McMillan introduced in 1980 a new algorithm for the computation of
Euler's constant $\gamma$, based on the use of the Bessel functions I\_0(x) and
K\_0(x). It is the fastest known algorithm for the computation of $\gamma$. The
time complexity can still be improved by evaluating a certain divergent
asymptotic expansion up to its minimal term. Brent-McMillan conjectured in 1980
that the error is of the same magnitude as the last computed term, and
Brent-Johansson partially proved it in 2015. They also gave some numerical
evidence for a more precise estimate of the error term. We find here an
explicit expression of that optimal estimate, along with a complete
self-contained formal proof and an even more precise error bound.
