The way to do this is probably to find a hypergeometric formula
that can be run to high precision, perhaps with an error bound,
and map all special functions to this. It is not beautiful or
fast, but almost all prior art has to do with
(a) fix some precision, e.g. 8 decimals, or 18 decimals or 53 bits
(b) figure out a particular algorithm for various ranges.
The paradigm for bigfloat is
(a) figure out an extensible (maybe slow) algorithm that always
converges and has arbitrary accuracy. E.g. taylor series, or
a reduction to taylor series.
(b) run it until you are tired of it.
Note that no good library would do Taylor series for
sine / cos/ exponential for known-in-advance precision.
There would be entirely different routines for single, double, extended
precision.
RJF
Stavros Macrakis wrote:
>http://math.nist.gov/mcsd/Reports/2001/nesf/paper.pdf seems to have a
>good bibliography of numerical evaluation algorithms, including
>everything from the small-value integer-order case to the
>arbitrary-order, arbitrary-argument case.
>
>Of course, that leaves you with the usual problem of smoothness where
>you shift from one algorithm to another -- is besselj(1,1) close to
>besselj(1+eps,1+delta), where eps and delta are very small complex
>numbers....
>
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