Innovations in rendering of high dynamic range (“HDR”) video on a display device having enhanced dynamic range (“EDR”). The peak brightness for an EDR display device is lower than the peak brightness for a typical HDR display device but higher than the peak brightness for a typical display device having standard dynamic range. The increased range of brightness values in an EDR display device can be utilized effectively to show bright highlights of the HDR video. For example, decision logic is configured to evaluate a peak brightness of a target display device and select an HDR-to-EDR tone mapping mode. A tone mapper is configured to apply tone mapping to input values for the sample values of HDR video, according to a tone mapping function, thereby producing output values for sample values of EDR video.