Human cognitive performance is critical to productivity, learning, and accident avoidance. Cognitive performance varies throughout each day and is in part driven by intrinsic, near 24-hour circadian rhythms. Prior research on the impact of sleep and circadian rhythms on cognitive performance has typically been restricted to small-scale laboratory-based studies that do not capture the variability of real-world conditions, such as environmental factors, motivation, and sleep patterns in real-world settings. Given these limitations, leading sleep researchers have called for larger in situ monitoring of sleep and performance. We present the largest study to date on the impact of objectively measured real-world sleep on performance enabled through a reframing of everyday interactions with a web search engine as a series of performance tasks. Our analysis includes 3 million nights of sleep and 75 million interaction tasks. We measure cognitive performance through the speed of keystroke and click interactions on a web search engine and correlate them to wearable device-defined sleep measures over time. We demonstrate that real-world performance varies throughout the day and is influenced by both circadian rhythms, chronotype (morning/evening preference), and prior sleep duration and timing. We develop a statistical model that operationalizes a large body of work on sleep and performance and demonstrates that our estimates of circadian rhythms, homeostatic sleep drive, and sleep inertia align with expectations from laboratory-based sleep studies. Further, we quantify the impact of insufficient sleep on real-world performance and show that two consecutive nights with less than six hours of sleep are associated with decreases in performance which last for a period of six days. This work demonstrates the feasibility of using online interactions for large-scale physiological sensing.