What does Special attack do???

I made a 3 element team with a useless Eidolon that did 2.5-2.7k dmg on average

Then I used diabolos with 100% special attack bonus and the dmg numbers didn't change??? Shouldn't it double??

Compare to changing Eidolon to Python and my dmg numbers went up to ~3.4k for the dark kamihimes

Also what are the specifics of "commensurate"?? I know it's proportional but does anyone have the exact math on it???