  We present randomized algorithms for sampling the standard Gaussian
distribution restricted to a convex set and for estimating the Gaussian measure
of a convex set, in the general membership oracle model. The complexity of
integration is $O^*(n^3)$ while the complexity of sampling is $O^*(n^3)$ for
the first sample and $O^*(n^2)$ for every subsequent sample. These bounds
improve on the corresponding state-of-the-art by a factor of $n$. Our
improvement comes from several aspects: better isoperimetry, smoother
annealing, avoiding transformation to isotropic position and the use of the
"speedy walk" in the analysis.
