The short answer is if they are offering you $X, you need to factor in a lot of different taxes
Withholding - depends upon how much after expenses you can pay yourself. I generally figure 25%.
You are also responsible for both ends of Social Security (12.4%) and Medicare (2.9%) which equates to 15.3% (2018) at the very least. So you should factor in about 40% for taxes (which makes me sick to think about). Because you can expense a lot as your own business, that figure "may" come down quite a bit depending upon your circumstances.
Another way to look at $X is from the perspective of what a salaried employee would get.
In addition to taxes you will also want to factor in your overhead in running your own business, i.e. doing your own paperwork or hiring it out, making your own sales calls, handling your own contracts, etc. It takes a lot of personal time that nobody pays you for. As a developer myself, I look at what a regular salaried employee might get paid for the same job, add 25% for benefits, then add 50% on top of that. For a bid like you mentioned, something like this:
Benefits for salaried employee (25%) - makes it .80X$/hr (X / 1.25)
Extra taxes (~10%) - makes it .73X$/hr
Other overhead (~30%) - makes it somewhere in the neighborhood of .56X$/hr
So by that calculus your "take" will be a little better than half what a salaried employee would get being paid the same amount.
You should be able, with sites like glassdoor, to get a decent read on what the market will pay for your services. Pricing is tricky, though, but I have found government agencies tend to be pretty realistic and pay pretty well. All the companies that have lowballed me to the point of walking away have been private companies sadly. Good luck.