The article explores the relationship between the masses of elementary particles and the fine-structure constant α. In addition to the Koide formula, a new formula is proposed, which is as accurate as the Koide formula. The new formula demonstrates the connection between the masses of four elementary particles (electron, proton, tau lepton, and muon) with the fine-structure constant α: Considering the high precision of the fine-structure constant and the accurate experimental values of the electron, proton, and muon masses, the new formula provides a significantly more accurate value for the mass of the tau lepton compared to the current experimental value:mτ/me=3477.0298u2026 (1776.7586u2026 MeV)The obtained value of the tau lepton mass (1776.7586... MeV/c2) is within the range of the experimental value but does not coincide with the result obtained by the Koide formula (1776.9688... MeV/c2). The Koide formula and the new formula yield different values for the mass of the tau lepton with high precision in both results. Both mass values are significantly more accurate than the current experimental value. The discrepancy between the results of the two formulas requires explanation.