REM helps you sleep better by showing you the best time to wake up.
Studies show that waking up in the middle of REM (rapid eye movement) sleep makes you feel drowsy. REM tells you the best time to set your alarm by dividing your sleep in 1½ hour cycles and multiplying it X times (you should sleep through at least 5 o 6 cycles per night).
Rest (ah!) assured, REM works offline too, just load the page once and you'll be all set. If you find yourself using the app a lot you can also add the app to your iPhone homescreen.
I've been using sleepyti.me for a while and loved it, but I quickly grew tired of its table based mobile–sucky layout, so I created my own. I also needed offline support, so here we are. If something's not working be sure to hit me up on Twitter!