  We give an apparently new proof of Stirling's original asymptotic formula for
the behavior of $\ln z!$ for large $z$. Stirling's original formula is not the
formula widely known as "Stirling's formula", which was actually due to De
Moivre. We also show by experiment that this old formula is quite effective for
numerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with the
sequence acceleration method known as Levin's $u$-transform. As an homage to
Stirling, who apparently used inverse symbolic computation to identify the
constant term in his formula, we do the same in our proof.
