Multi entry Schengen visa

If you don't want to depend on the calculator, you can also note that by the time you enter on May 6, you will have been outside of Schengen for more than 90 consecutive days (namely, say, the full months of February, March, and April, plus May 1st).

This, in itself, is enough to make everything that came before those 90 days irrelevant for the purpose of complying with the 90/180 day rule.

Thus, as far as 90/180 is concerned, when you enter on May 6 you're in the same situation as if you have no prior trips at all.

Today's date is populated automatically in "Date of entry/Control" (and needs to be a specific format):

The above (as reference by @Dorothy) is the official calculator and does appear to be working. What it is saying is that on July 14, 2017 you will not have overstayed, given the data as shown, and indeed could extend your proposed trip by a further 20 days without exceeding the limit - provided (as @phoog has pointed out!) you have a visa that is valid for the further 20 days (at least).

On the proposed date of entry (May 6) of previous trips only the 25-day January trip is relevant since the one in 2016 ended more than 180 days earlier. So on May 6 you would have used up 26 (25 +1) of the allowed 90 days. This keeps increasing up to 84 days utilised on July 3 (25+59) and then levels off, as further days for your proposed trip are offset by days from your January trip that cease to count, for being longer ago than 180 days. This would continue to be the case until July 28, 14 days after the end of your proposed trip at which point your utilisation would go up again (if you extended that far) so you would only be allowed a further 6 days (84 + 6 = 90) hence the 20 'spare' (14 + 6).

All the above though just concerns counting days. You are only allowed in the Schengen area while your visa is valid and since your current visa expires July 15 you must leave before July 16, unless you obtain a valid new visa.