Working overtime doesn't increase your output. It makes you stupid.
The reason sleep is so important is because fatigue isn't simple. When we are tired, our performance doesn't degrade equally. Instead, when you lose a night's sleep, the parietal and occipital lobes in your brain become less active. The parietal lobe integrates information from the senses and is involved in our knowledge of numbers and manipulation of objects. The occipital lobe is involved in visual processing. So the parts of our mind responsible for understanding the world and the data around us start to slow down. This is because the brain is prioritizing the thalamus—the part of your brain responsible for keeping you awake. In evolutionary terms, this makes sense. If you're driven to find food, you need to stay awake and search, not compare recipes.