In article <31BD7983.5C85@cliff.backbone.uoknor.edu>, <firstname.lastname@example.org> writes:
> > email@example.com (Rene Bos) writes: > > >> >Please define "closed form formula". > > >> A function that is finite in length and allowing only > > >> predetermined constants,+,-,*,/,exponents, and roots. > > >To me, this definition seems arbitrary. Why do you include roots, but exclude > > >logarithms, hyperbolic functions, etc.? > > Aren't all definitions arbitrary? I suppose roots did not even need to be > > included, since xth roots are just 1/xth powers. > > > .... > > It seems to me (though I have no proof) that the virtue of a closed-form > formula lay in its being able to be evaluated with pencil and paper (or > stylus and clay, or whatever) to any desired precision in a reasonably
Derivable by WHOM?
The problem with your "definition" is that it depends on the level of knowledge of the person undertaking the calculation.
If one knows or can derive the Taylor expansion for f(x) or (say) a Pade approximate or Chebyshev polynomial approximation one can quickly do what you say. Also helpful is knowing accelerated convergence techniques etc.
What *I* can compute quickly with pencil/paper is very probably a large superset of what you can do quickly.
Can you compute (say) log(37.00000) to 10 digits quickly with pencil/paper? It is not hard to someone who has the right knowledge.