Go to this forum and open up "The ultimate bottlenecking guide", you will find all info there. (I gave a link in a previous msg)
According to that chart, i5-4440 is compatible and will not bottleneck a RX 480. You might feel some bottleneck because i5-4440 is still in "will not bottleneck" colour, but any other weaker CPU like i5-4440S are in "will bottleneck in the worst case scenario / very cpu demanding games" colour.