Deep within an MCMC algorithm, I need to evaluate this expression:
$$\log(\Gamma(x + c)) - \log(\Gamma(x + 1/2))$$
where $x$ is a variable, and $c$ is a constant such that $c\geq1/2$.
This expression will be evaluated millions of times, so I need to simplify/approximate it as much as possible with as little cost on its accurateness as possible.
Can this be done or is this as good as it gets?
You are computing
$$ \int_{x+1/2}^{x+c}\psi(x)\,dx $$ where by Binet's second integral $$ \psi(x) = \log(x)-\frac{1}{2x}-2\int_{0}^{+\infty}\frac{t\,dt}{(t^2+x^2)(e^{2\pi t}-1)}. $$ This leads to Stirling's formula for $\log\Gamma$, which can be used with a suitable number of terms, depending on the tolerated error.