[ujjain]

I've never seen this either.

I do sometimes charge half days or 3/4th days if I have to leave work a few hours early.

[runako]

Do you ever do the reverse and charge e.g. 1.5X if you have to stay late?

Is there any advantage to this system versus hourly billing?

[ujjain]

I never had a contracting job that requires me to stay late, just 8 hours a day. I would have billed 1/4th more day if I had to stay 2 more hours.