  We revisit the method of small subgraph conditioning, used to establish that
random regular graphs are Hamiltonian a.a.s. We refine this method using new
technical machinery for random $d$-regular graphs on $n$ vertices that hold not
just asymptotically, but for any values of $d$ and $n$. This lets us estimate
how quickly the probability of containing a Hamiltonian cycle converges to 1,
and it produces quantitative contiguity results between different models of
random regular graphs. These results hold with $d$ held fixed or growing to
infinity with $n$. As additional applications, we establish the distributional
convergence of the number of Hamiltonian cycles when $d$ grows slowly to
infinity, and we prove that the number of Hamiltonian cycles can be
approximately computed from the graph's eigenvalues for almost all regular
graphs.
