Recently, several statistically significant tensions between different cosmological datasets have raised doubts about the standard Lambda cold dark matter ($Lambda$CDM) model. A recent letter~citet{Huang:2020mub} suggests to use Parameterization based on cosmic Age (PAge) to approximate a broad class of beyond-$Lambda$CDM models, with a typical accuracy $sim 1%$ in angular diameter distances at $zlesssim 10$. In this work, we extend PAge to a More Accurate Parameterization based on cosmic Age (MAPAge) by adding a new degree of freedom $eta_2$. The parameter $eta_2$ describes the difference between physically motivated models and their phenomenological PAge approximations. The accuracy of MAPAge, typically of order $10^{-3}$ in angular diameter distances at $zlesssim 10$, is significantly better than PAge. We compare PAge and MAPAge with current observational data and forecast data. The conjecture in~citet{Huang:2020mub}, that PAge approximation is sufficiently good for current observations, is quantitatively confirmed in this work. We also show that the extension from PAge to MAPAge is important for future observations, which typically requires sub-percent accuracy in theoretical predictions.
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