I tested the successive residual polynomial fit method against NIST saturation vapor pressure data to see how it works with more accurate data. It's easiest to fit the log of the pressure, logp, as a function of the inverse of the temperature, U, with a polynomial fit. See Karapetoff, Engineering Applications of Higher Mathematics (1916) for more details on fit procedures. The best fit that I was able to get was for a 35 degree polynomial which added roughly 3 digits of precision to the values of the pressure in the NIST tables.
δ is the error in log(p) for the fit and appears to be just a round off error distorted by the log function. The temperature, T, ranges from the triple point of water to its critical temperature.