  Four-center two-electron Coulomb integrals routinely appear in electronic
structure algorithms. The resolution-of-the-identity (RI) is a popular
technique to reduce the computational cost for the numerical evaluation of
these integrals in localized basis-sets codes. Recently, Duchemin and Blase
proposed a separable RI scheme [J. Chem. Phys. 150, 174120 (2019)], which
preserves the accuracy of the standard global RI method with the Coulomb metric
(RI-V) and permits the formulation of cubic-scaling random phase approximation
(RPA) and $GW$ approaches. Here, we present the implementation of a separable
RI scheme within an all-electron numeric atom-centered orbital framework. We
present comprehensive benchmark results using the Thiel and the GW100 test set.
Our benchmarks include atomization energies from Hartree-Fock, second-order
M{\o}ller-Plesset (MP2), coupled-cluster singles and doubles, RPA and
renormalized second-order perturbation theory as well as quasiparticle energies
from $GW$. We found that the separable RI approach reproduces RI-free HF
calculations within 9 meV and MP2 calculations within 1 meV. We have confirmed
that the separable RI error is independent of the system size by including
disordered carbon clusters up to 116 atoms in our benchmarks
