It depends on a lot of factors. The formula of just doubling hourly employee pay rate works well for short-term contracts. The intent for such a high pay rate is to cover not only the contractor basics (self-employment tax, health care, etc), but also to smooth the transition between working and non-working periods. An employee would be paid regardless of the amount of work available, whereas a contractor is only paid when working, so they're paid extra to account for the time between contracts.
For longer-term contracts, employers often expect a discount. That seems fair, as the contractor can expect to be working for multiple years without contact lapses, and it can be expected that the contractor will have some downtime (ie, not dealing with a constant barrage of looming deadlines). I usually charge 75% of my hourly rate for long-term contracts.
There's also a number of other reasons for discounts as well. I give non-profits discounts, or projects that I think are really cool/fun to work on. The last non-profit I contacted for, they got 55% of my hourly rate because I liked what they were doing AND it was a long-term contract.
Note that this is assuming your hourly rate is doubled. For example, if you make $120k/yr full time, that's roughly $60/hr. If it's a short-term contract, I'd charge $120/hr. For a longer-term contract, I'd charge $90/hr. For the long-term non-profit (or fun) contract, I'd charge $65/hr.
The $120k salary is what you see, but the employer is paying additional fees on top that are hidden from you. Employer tax, health care fees, 401k matching, not to mention the cost of office space, computer equipment, etc. As a contractor, the extra pay takes that into account.
