Recent advances in high dynamic range (HDR) capture and display technologies have attracted a lot of interest from scientific, professional, and artistic communities. As in any technology, the evaluation of HDR systems in terms of quality of experience is essential. Subjective evaluations are time consuming and expensive, and thus objective quality assessment tools are needed as well. In this paper, we report and analyze the results of an extensive benchmarking of objective quality metrics for HDR image quality assessment. In total, 35 objective metrics were benchmarked on a database of 20 HDR contents encoded with 3 compression algorithms at 4 bit rates, leading to a total of 240 compressed HDR images, using subjective quality scores as ground truth. Performance indexes were computed to assess the accuracy, monotonicity, and consistency of the metric estimation of subjective scores. Statistical analysis was performed on the performance indexes to discriminate small differences between metrics. Results demonstrated that metrics designed for HDR content, i.e., HDR-VDP-2 and HDR-VQM, are the most reliable predictors of perceived quality. Finally, our findings suggested that the performance of most full-reference metrics can be improved by considering non-linearities of the human visual system, while further efforts are necessary to improve performance of no-reference quality metrics for HDR content.