  In this paper, we develop Windschitl's approximation formula for the gamma
function to two asymptotic expansions by using a little known power series. In
particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*}
\Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left(
x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left(
2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}}
+R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left|
R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left(
2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all $x>0$, where $B_{2n}$
is the Bernoulli number. Moreover, we present some approximation formulas for
gamma function related to Windschitl's approximation one, which have higher
accuracy.
