special functions should use mpfr when available

[AlexGhitza]

MPFR has fast implementations for restricted types of arguments in some special functions, e.g. Bessel J and Y with integer order and positive real argument. We should be using these instead of Pari or Maxima or Scipy whenever that is feasible.

Example:

sage: a = RR(2)
sage: timeit("bessel_J(1, a)")
625 loops, best of 3: 370 µs per loop
sage: timeit("a.j1()")
625 loops, best of 3: 13.9 µs per loop

That's 26 times faster than Pari.

[kcrisman]

Or maybe we should use mpmath - there are a number of tickets about that. Such as Alex's own comment in #3426 :)

[burcin]

This ticket is too broad, I suggest we close it as invalid.

At the time it was created, there was no general framework to handle these functions. The (not so) new pynac-based symbolics provide this framework. It is true that a lot of work is still needed to sort these numerical evaluation issues out, but we need a separate specific ticket for each issue.

See symbolics/functions for an overview on the progress of symbolic functions.

[kcrisman]

Agreed. This wiki page solves the problem.