In The Almost People, and The Eleventh Hour, the Doctor's screwdriver is destroyed and later regenerated. However, in The Day of the Doctor, the Doctor(s), run a calculation running over 400 years.
How is this possible, if the screwdriver was destroyed between the start and finish?
This is technically trivial. Anyone running a very long computation (and some real computations can last for weeks) will regularly save the state of the computation on some reliable medium that does not need power, so that the computation can be restarted after any kind of failure. Actually there can even be several levels of back-up, balancing reliability and cost.
The computation does not need to be restarted with exactly the same software, as long as the intermediate results that have been saved can be interpreted and reused by the new software. Actually this can even be a way to improve the software (or the hardware) while the computation is on-going. This is standard technology, even for people who are not time-lords.
The screwdriver could simply save its state on the Tardis whenever there are close enough to communicate by whatever means. It could also be saved in various places known to be stable in time by someone who travels through time.
If the screwdriver is destroyed, only that part of the computation done since the last back-up is lost and must be redone.
By the way, since the Doctor travels through time, he could well organize the computation and back-up to get a lot more than 400 years worth of computation, depending on the structure of the algorithm and the computational capacity of the screwdriver.
It is to be expected that the computation does not take all the computational power of the screwdriver. If a computational thread actually leaves a lot of untapped computational power, it can be parallelized with a future fragment of the same thread by getting from the future the starting state for the later thread. Time travel extends the possibilities for parallelizing computations.
Like Rose said in The Day of The Doctor
" Same Software different case"
Therefore the Doctor's screwdriver was not completely destroyed, only the case containing it was; the software was safe making it possible for the TARDIS to repair the case and give the Doctor a seemingly "new" screwdriver with the computation still intact.
In "Let's Kill Hitler", it's explained that instead of having settings, this version operates through a telepathic interface, linked to the Doctor. Hence he can use it to do such a varied number of things without needing to continually change settings on it.
WARNING GUESSWORK*
Therefore the internal state of the screwdriver is probably stored in the Tardis (on the cloud) as this is telepathically linked to the Dr. too. When it is regenerates it just syncs and carries on without minor disruption.
