  Experimental $\alpha$-decay half-life, spin, and parity of 398 nuclei in the
range 50$\leq$Z$\leq$118 are utilized to propose a new formula (QF) with only 4
coefficients as well as to modify the Tagepera-Nurmia formula with just 3
coefficients (MTNF) by employing nonlinear regressions. These formulas, based
on reduced mass ($\mu$) and angular momentum taken away by the
$\alpha$-particle, are ascertained very effective for both favoured and
unfavoured $\alpha$-decay in addition to their excellent match with all (Z, N)
combinations of experimental $\alpha$-decay half-lives. After comparing with
similar other empirical formulas of $\alpha$-decay half-life, QF and MTNF
formulas are purported with accuracy, minimum uncertainty and deviation,
dependency on least number of fitted coefficients together with less
sensitivity to the uncertainties of $Q$-values. The QF formula is applied to
predict $\alpha$-decay half-lives for 724 favoured and 635 unfavoured
transitions having experimentally known $Q$-values. Moreover, these available
$Q$-values are also employed to test various theoretical approaches viz. RMF,
FRDM, WS4, RCHB, etc. along with machine learning method XGBoost for
determining theoretical $Q$-values, incisively. Thereafter, using $Q$-values
from the most precise theoretical treatment mentioned above along with the
proposed formulas, probable $\alpha$-decay chains for Z$=$120 isotopes are
identified.
