This in D5 and copied down will deal with the holiday days:
Formula: copy to clipboard
=C5-SUMIFS('Employee Leave Tracker'!F:F,'Employee Leave Tracker'!B:B,B5,'Employee Leave Tracker'!E:E,"Holiday Full")-(SUMIFS('Employee Leave Tracker'!F:F,'Employee Leave Tracker'!B:B,B5,'Employee Leave Tracker'!E:E,"Holiday Half")/2)

If you need to to account for sick / medical etc. leave then we'll be needing more info on your need.

BSB