Combined equation of state for liquids and gases, which includes the classical and scaling parts

A new equation of state is suggested, which describes the $P$—$\rho$—$T$ data for $^4$He and SF$_6$ in the ranges of reduced densities $-1 < (\rho;-\rho_c)/\rho_c < 1$ and of reduced temperatures $-0.3 < (T-T_c)/T_c < 0.3$ ($\rho_c$ and $T_c$ are the critical values). This equation includes the regular equation of state approximating the $P$—$\rho$—$T$ data outside of the critical region and the nonparametric scaling equation of state adequately describing the $P$—$\rho$—$T$ data in the vicinity of the critical points, which are combined by the crossover function. The classical function of damping of fluctuations of density and temperature when moving away from the critical point is suggested as the crossover function. Two equations of state are used for the regular part of combined equation, namely, the new cubic equation of state suggested by us and the equation of state of Kaplun and Meshalkin. The nonparametric scaling equation of state with three system-dependent constants is used as the scaling part of the combined equation. The conditions $(\partial P/\partial v)_T= 0$ and $(\partial^2 P/\partial v^2)_T = 0$ are valid for the combined equation at the critical point; binodal and spinodal are present, as is the case in classical equations of state. The approximation of the most exact data on $^4$He and SF$_6$ using the new equation reveals that the latter equation correctly describes the $P$—$\rho$—$T$ data with mean-square error with respect to pressure of $\pm 0.5\ \%$.