  The accuracy of equation-of-state formulations is compared for theoretical
total energies or experimental pressure-volume measurements for H$_{2}$, Ne,
Pt, and Ta. This spans the entire range of compression found for minerals and
vo latiles in the Earth. The Vinet equation is found to be most accurate. The
origin of the behavior of different equation-of-state formulations is
discussed. It is shown that subtle phase transitions can be detected by
examining the residuals from an equation-of-state fit. A change in the
electronic structure of Ta is found at high pressures using this procedure, and
a possible new transition in H$_{2}$.
