Current Bitcoin mining hardware is about a quintillion times faster than manual hasing and about 67 quadrillion times more energy efficient (assuming you feed yourself with $0.23 a pop donuts).
Bitcoins are mined using a cryptographic algorithm called SHA-256. This algorithm is simple enough to be done with pencil and paper, as I show in this video. Not surprisingly, this is a thoroughly impractical way to mine. One round of the algorithm takes 16 minutes, 45 seconds which works out to a hash rate of 0.67 hashes per day.