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We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities 
$\lambda$ for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval 
$[0,\lambda ]$. As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time deterministic algorithm for all 
$\lambda \lt 1/(e^{B + 1} \hat C_\phi )$ where 
$\hat C_\phi$ is a temperedness parameter and 
$B$ is the stability constant of 
$\phi$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least 
$e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all 
$\lambda \lt e/\Delta _\phi$, where 
$\Delta _\phi$ is the potential-weighted connective constant of the potential 
$\phi$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.
