  Universal scaling laws only apply asymptotically near critical phase
transitions. We propose a general scheme, based on normal form theory of
renormalization group flows, for incorporating corrections to scaling that
quantitatively describe the entire neighboring phases. Expanding Onsager's
exact solution of the 2D Ising model about the critical point, we identify a
special coordinate with radius of convergence covering the entire physical
temperature range, $0<T<\infty$. Without an exact solution, we demonstrate that
using solely the critical singularity with low- and high-temperature expansions
leads to exponentially converging approximations across all temperatures for
both the 2D and 3D Ising free energies and the 3D magnetization. We discuss
challenges and opportunities for future work.
