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Cosmography becomes non-predictive when cosmic data span beyond the red shiftlimit $z\simeq1 $. This leads to a \emph{strong convergence issue} thatjeopardizes its viability. In this work, we critically compare the two mainsolutions of the convergence problem, i.e. the $y$-parametrizations of theredshift and the alternatives to Taylor expansions based on Pad\'e series. Inparticular, among several possibilities, we consider two widely adoptedparametrizations, namely $y_1=1-a$ and $y_2=\arctan(a^{-1}-1)$, being $a$ thescale factor of the Universe. We find that the $y_2$-parametrization performsrelatively better than the $y_1$-parametrization over the whole redshiftdomain. Even though $y_2$ overcomes the issues of $y_1$, we get that the mostviable approximations of the luminosity distance $d_L(z)$ are given in terms ofPad\'e approximations. In order to check this result by means of cosmic data,we analyze the Pad\'e approximations up to the fifth order, and compare theseseries with the corresponding $y$-variables of the same orders. We investigatetwo distinct domains involving Monte Carlo analysis on the Pantheon SuperovaeIa data, $H(z)$ and shift parameter measurements. We conclude that the (2,1)Pad\'e approximation is statistically the optimal approach to explain low andhigh-redshift data, together with the fifth-order $y_2$-parametrization. Athigh redshifts, the (3,2) Pad\'e approximation cannot be fully excluded, whilethe (2,2) Pad\'e one is essentially ruled out.

High-redshift cosmography: auxiliary variables versus Pad\'e polynomials