We perform the combined analysis of the double Higgs production via gluon fusion in the b (b) over bar gamma gamma and b (b) over bar tau(+)tau(-) btthornt-decay channels at the High-Luminosity LHC (HL-LHC). To validate our analysis, we reproduce the ATLAS result of the b (b) over bar gamma gamma process including all contributions from fakes. For the b (b) over bar tau(+)tau(-) btthornt-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 (b) over bar tau(+)tau(-) btthornt-process outperforms the b (b) over bar gamma gamma for the measurement of energy-growing operators, while adding the b (b) over bar tau(+)tau(-) btthornt-process is least beneficial for improving the precision of the Higgs self-coupling (mainly set by the b (b) over bar 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, and diphoton mass 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(SM)(3), at the HL-LHC can reach [0.2, 2.3] and [-0.1; 3.5] boolean OR [4.0; 6.5], respectively, where the correlation among the EFT coefficients is taken into account.