My iPhone 5 on iOS 7.0.4 keeps turning off early at around 7-20%.

I tried restoring it twice and resetting the battery cycle by following these directions: "What you need to do given your current problem: When the phone shuts off and it isn't down around 1%, connect to a charger for just a minute or 2, or wait 15 minutes, then reboot the phone (hold HOME and SLEEP until an Apple logo appears). Continue running the battery down until it shuts off around 1%. If it goes off above that value repeat the above process until you really do get to 1% (this will rarely take more than 2 tries). Once it has turned off and won't restart or reboot charge it with the wall charger for 4 hours without using the phone"

I also restored in DFU mode and recovery mode before, not from backup any of the times.