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We present an algorithm for generating approximations for the logarithm ofBarnes $G$-function in the half-plane $Re(z)\ge 3/2$. These approximationsinvolve only elementary functions and are easy to implement. The algorithm isbased on a two-point Pad\'e approximation and we use it to provide twoapproximations to $\ln(G(z))$, accurate to $3 \times 10^{-16}$ and $3 \times10^{-31}$ in the half-plane $Re(z)\ge 3/2$; a reflection formula is then usedto compute Barnes $G$-function in the entire complex plane. A by-product of ouralgorithm is that it also produces accurate approximations to the gammafunction.

Computing the Barnes $G$-function and the gamma function in the entire complex plane