Lately I've been trying to learn reapers by muse on guitar and the intro has fast tapping with fairly big stretches for the left hand and I have been struggling to play it at the right speed and sometimes it can sound a bit sloppy.
What kind of exercises could I do to both improve the strength of my fingers and help me get up to speed?
Man honestly... just do it real slow for a while. Accuracy is the most important thing so do it slow, get it right (building your finger strength all the while) and then once you have it down you can work on speed. (try a metronome and work way up to the speed you want). Plenty of free metronomes online and in the app store for your phone.
You can keep trying tapping for hours on end.
You can also get one of these:
or one of these (which are particularly good due to the fact the little finger gets the most leverage and is therefore most likely to be trained while using this):
http://www.dolphinfitness.co.uk/en/fitness-mad-power-grip-hand-exerciser/56317/?ladid=uk
There are also some sports that will also increase your finger strength massively, such as rock climbing.
It would help to do some stretches. I know these are for piano, but they will still help significantly. Practice makes perfect, so I'd suggest getting a metronome and starting off very slow. Make sure to hit every note accurately. Gradually speed up. If you get to where you can't do it anymore, slow back down. Don't push yourself as you'll only become frustrated. One of my teachers told me if you "sleep on it" after you've practiced, your brain will let it settle in and muscle memory will start to take effect. It really works, too. Anywho, main points: Stretch, practice, don't push yourself.
In terms of pure strength training I do trills for each finger permutation on a random string. So if we assume 1 = index and 4 = pinky. I would trill with fingers 1-2, then 1-3, 1-4, 2-3, 2-4, 3-4 for a minimum of 30 seconds each (adjust according to strength and gradually buildup to 3 minutes or so). Furthermore, these trills would be nothing but hammer ons and pull offs (so no picking), and you must keep a certain dynamic. An example can found here.
In terms of accuracy as mentioned by @lilsam138 the metronome is the way to go and I think their answer explains it well enough.
In terms of increase reach this is something you acquire with time as it require you stretch it just like any other muscle. In terms of an exercise I use this one quite often. Here is a video from such said site explaining the exercise.
