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The band gap is an important parameter of semiconductor materials that influences several functional properties, in particular optical properties. However, a fast and reliable first-principles prediction of band gaps remains a challenging problem. Standard DFT approximations tend to strongly underestimate band gaps, while the more accurate $GW$ and hybrid functionals are much more computationally demanding and unsuitable for high-throughput screening. In this work, we have performed an extensive benchmark of several approximations with different computational complexity ($G_{0}W_{0}$@PBEsol, HSE06, PBEsol, mBJ, PBEsol$-1/2$, and ACBN0) to evaluate and compare their performance in predicting the band gap of semiconductors. The benchmark is based on 114 binary semiconductors of different compositions and crystal structures, where about half of them have experimental band gaps. We find that, as expected, $G_{0}W_{0}$@PBEsol performs well relative to the experiment, with a noticeable underestimation of the band gaps by about 14% on average. Surprisingly, $G_{0}W_{0}$@PBEsol is followed closely by the much computationally cheaper pseudo-hybrid ACBN0 functional, showing an excellent performance with respect to experimental data. The meta-GGA mBJ functional also performs well relative to the experiment, even slightly better than $G_{0}W_{0}$@PBEsol in terms of mean absolute (percentage) error. The HSE06 and PBEsol$-1/2$ schemes perform overall worse than ACBN0 and mBJ schemes but much better than PBEsol. Comparing the calculated band gaps on the whole data set (including the samples with no experimental band gap), we find that HSE06 and mBJ have excellent agreement with respect to the reference $G_{0}W_{0}$@PBEsol band gaps. Thus, we propose the mBJ band gaps as economic descriptors when developing artificial intelligence models for high-throughput screening of semiconductor band gaps.