Sometimes I am surprised that some of the pitches Wade Davis throws are not considered illegal in at least the continental 48 states. I can't really speak for Alaska, Hawaii, and Canada. I've never been to the first two and barely been to the third one. Considering Davis' 2013 season, it's a strange opinion that Davis' pitches might be considered illegal. Davis was awful in 2013. Davis was historically amazing in 2014. Quite the turnaround.
You know the ERA thing. You need only 1 finger to count his ERA. You probably know the WAR thing, for which you need three point one fingers. Instead, I'll use another finger to point you to Davis' cutter. Oh, his cutter. I don't think it's been called Public Enemy No. 1, but it's nastier than things which cannot be uttered on this platform. From Brooks Baseball, check out what hitters did with his cutter in 2013, when he was a miserable starter:
.333/.496 BA/SLG with a .424 BABIP. Woof.
Now, look at what hitters did with his cutter in 2014, when Davis was a golden reliever:
.115/.115 BA/SLG with a .212 BABIP.
From 2013 to 2014, the cutter's velocity increased, its horizontal movement increased, and its vertical movement increased. All that resulted in the numbers above as well as an increase in whiff rate from about 12% to about 19%, an increase of over 50%.
I'll point you to Davis' four-seam fastball. His rising bread and butter. Like an increasingly growing group of others, Davis throws a "rising" fastball in that it doesn't quite sink as much as other fastballs. Davis kept the pitch up and away against lefties and threw it generally anywhere not inside against righties. The velocity of the pitch also increased by about 3.6mph. Here is what hitters did with the pitch in 2013:
.279/.443 BA/SLG with a .314 BABIP. This isn't great, but there are worse fastballs.
Now, look at the same stuff for 2014:
.141/.170 BA/SLG with a .253 BABIP.
It's fairly hard to hit a 96mph pitch that's higher than you expect or a ~92mph pitch that looks like a slider but moves much faster. Speaking of being fairly hard to hit, I was looking over Davis' PitchF/X plate discipline numbers, because why not. I saw that he had a huge drop in contact rate both inside the zone and outside the zone. I wanted to know how this compared to other pitchers, so I made a little visual. To make this visual, I used FanGraphs to look at all pitchers who threw at least 50 innings in both 2013 and 2014 (the 426 nulls are the guys who didn't have at least 50 IP in both seasons). I was then able to take the difference between the 2014 and 2013 O-contact% and Z-contact% and put them on a scatterplot. Davis' position in this scatterplot will be fairly obvious.
As you can see, there was plenty of room between Davis and the other guys for me to scribble his name like a 1st grader in Paint. Aroldis Chapman and Randall Delgado are the only ones close enough even to sniff Davis' level of changes, and those two guys need a really strong wind and a dog's nose to catch Davis' filthy reliever stench. Davis had the largest decrease in Z-contact% and the second largest decrease in O-contact%. Joaquin Benoit had the largest decrease in O-contact%.
Now, Davis had quite a bit of room to make the change, since he was starting from a place of misery, but that shouldn't detract from what he accomplished. Plenty of guys improve when making the change from a starter to a reliever, but Davis made the change in 2014 with something not unlike aplomb.