  We construct an approximate expression for the cross section for Higgs
production in gluon fusion at next-to-next-to-next-to-leading order (N$^3$LO)
in $\alpha_s$ with finite top mass. We argue that an accurate approximation can
be constructed by exploiting the analiticity of the Mellin space cross section,
and the information on its singularity structure coming from large N (soft
gluon, Sudakov) and small N (high energy, BFKL) all order resummations. We
support our argument with an explicit comparison of the approximate and the
exact expressions up to the highest (NNLO) order at which the latter are
available. We find that the approximate N$^3$LO result amounts to a correction
of 16% to the NNLO QCD cross section for production of a 125 GeV Higgs at the
LHC (8 TeV), larger than previously estimated, and it significantly reduces the
scale dependence of the NNLO result.
