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Sleep cycles
Wake at the end of a cycle, not in the middle of one.
Up at
In bed by
Assumptions
How this is worked out
Sleep runs in cycles of light, deep and dreaming sleep, and an alarm that lands at the end of one is easier to get up to than one that lands in deep sleep. The times here are whole cycles back from your alarm, or forward from bedtime, plus the minutes it takes you to fall asleep.
It is a heuristic, not physiology. Cycles run anywhere from 70 to 120 minutes, differ between people and get shorter through the night, so the 90 above is an average you can move. Five cycles is about seven and a half hours, the range most adults do best on; six is nine.
The short nap stops before deep sleep, which is why twenty minutes leaves you clear and forty leaves you foggy; the long one is a full cycle.