You need to get out of the mindset of charging for your time. You are charging for the value you create - and time is just a crude approximation of that at best.

What if you gave a client two choices: fixed price for $1,000 or hourly at $100/hour with an estimate of 8-12 hours. They're on the fence, but opt for the fixed price deal. You luck out and knock the work out in 3 hours. Do you feel bad? No. You gave them $1000 worth of value.

Now what if they had chosen hourly? Now you get $300 in exchange for the $1000 of value you delivered. Don't do that.

No, what's going to piss them off is when you deliver $1000 of value, but take 20 hours to do it and charge them $2000. Your goal is to match the value you create with the money they pay. That's how you build lasting profitable relationships.

You're not charging for time. You're charging for software.

That is their problem and not yours. Establish a sane pricing system, communicate it to your client, and move on. I would (again) argue that hourly billing is almost never a sane system.

I've never had a client ask or complain about that, and the ones who would probably aren't clients you want, either.

There is a difference between cost and value. The client valued your ability to solve their problem as equal to your daily rate. It shouldn't matter if it took you 1 hour or 8 hours, they are paying for a result, not for you to punch a clock.

The client is not paying you for your hours. It really doesn't matter to them whether you take an hour or a day to do the job, as long as it gets done.