  We perform the combined analysis of the double Higgs production via gluon
fusion in the $b\bar{b} \gamma\gamma$ and $b\bar{b}\tau^+\tau^-$ decay channels
at the High-Luminosity LHC (HL-LHC). To validate our analysis, we reproduce the
ATLAS result of the $b\bar{b} \gamma\gamma$ process including all contributions
from fakes. For the $b\bar{b}\tau^+\tau^-$ decay channel, we perform the
similar analysis to the CMS one. As an improvement, we also perform the
multivariate analysis employing the boosted decision tree algorithm. Then, we
derive 68% probability contours on anomalous Higgs couplings in the effective
field theory (EFT) approach for various analyses. We find that the
$b\bar{b}\tau^+\tau^-$ process outperforms the $b\bar{b}\gamma\gamma$ for the
measurement of energy-growing operators, while adding the
$b\bar{b}\tau^+\tau^-$ process is least beneficial for improving the precision
of the Higgs self-coupling (mainly set by the $b\bar{b}\gamma\gamma$ process).
We illustrate that the double Higgs production alone can be comparable to the
single Higgs process in constraining the modification of the top Yukawa
coupling in the positive direction. Focusing on the Higgs self-coupling as a
special interest, we derive the precision as a function of various improvable
parameters such as tag and mistag rates of tau leptons, heavy flavor jets,
photon identification, diphoton mass resolution, and jet energy resolution to
take into account future phenomenological studies. As an optimistic benchmark
scenario, we illustrate that the 68% and 95% probability intervals of the Higgs
self-coupling, $\lambda_3/\lambda_{3}^{SM}$, at the HL-LHC can reach $[0.2,\,
2.3]$ and $[-0.1,\, 3.5] \cup [4.0,\, 6.5]$, respectively, where the
correlation among the EFT coefficients is taken into account.
