The critical density as a function of the inverse of the critical temperature as obtained from our simulations and the aforementioned theories is shown in Fig. 1. The critical simu- lation data obtained by Singh using GC-TMMC, also displayed in the figure, are in excellent agreement with ours, which shows that both techniques give similar results within the error bar. At small κ , all theories give similar results, and are in good agreement with the simu- lations. In fact, in the long-range regime even a simple mean-field approach along the lines of the van der Waals theory provides an accurate description of the system, which has been shown to become exact in the limit where the range of the tail potential goes to infinity, and its strength goes to zero [47]. As the interaction range gets shorter, discrepancies become more and more pronounced, especially as far as ρ ∗ c is concerned. In particular, the data for ρ ∗ c reported by Duh and Mier-Y-Teran [22] using their equation of state (EOS) for HCAY fluid strongly overestimate the simulations. The TPT by Zhou [14] and the S-SCOZA show a better agreement with the simulation data. The latter are bracketed by the S-SCOZA and M-SCOZA results: S-SCOZA overestimates both ρ ∗ c and T ∗ c , while the converse is true for M-SCOZA. For both approaches, the relative error with respect to the simulations is in general considerably smaller for T ∗ c than for ρ ∗ $_{c}$. The most remarkable feature of the simulation results is the linear trend for ρ ∗ c vs. 1 /T ∗ c up to inverse-range parameters as large as κ = 15, corresponding to 1 /T ∗ c = 3 . 425. A similar behavior is displayed by S-SCOZA, as well as by TPT. As κ is further increased, the slopes 10 TABLE I: Critical parameters of HCAY Fluid. $^{a}$Gran canonical Monte Carlo with finite-size scaling. $^{b}$The calculation for this value of κ had already been per- formed in Ref. [1], but the result had not been tabulated. $^{c}$ρ$_{c}$ was obtained using Eq. (4) from Ref. [35]. 15.0 0.293 0.551 S-SCOZA this work 15.0 0.291 0.455 M-SCOZA this work 25.0 0.235 0.555 MC-TI [7] c 25.0 0.236 0.600 S-SCOZA this work 25.0 0.241 0.459 M-SCOZA this work 25.0 0.239 0.632 TPT [14]