[Python-Dev] Deprecating float.is_integer()

[Tim Peters]

[Tim]
>> from trig functions doing argument reduction as if pi were represented
>> with infinite precision,

[Greg Ewing <greg.ewing at canterbury.ac.nz>]
> That sounds like an interesting trick! Can you provide
> pointers to any literature describing how it's done?
>
> Not doubting it's possible, just curious.

As I recall, when it was first done a "lazy" routine produced as many
bits of pi as a given argument required, doing gonzo arbitrary
precision arithmetic.

Later, computer-aided analysis based on continued fraction expansions
identified the worst possible case across all IEEE doubles (&
singles).  For example, it's possible in reasonable time to find the
IEEE double that comes closest to being an exact integer multiple of
pi/4 (or whatever other range you want to reduce to).  Then it's only
necessary to precompute pi to as many bits as needed to handle the
worst case.

In practice, falling back to that is necessary only for "large"
arguments, and the usual double-precision numeric tricks suffice for
smaller arguments.

Search the web for "trig argument reduction" for whatever the state of
the art may be today ;-)

For actual code, FDLIBM does "as if infinite precision" trig argument
reduction, using a precomputed number of pi bits sufficient to handle
the worst possible IEEE double case, and is available for free from
NETLIB:

    http://www.netlib.org/fdlibm/

The code is likely to be baffling, though, as there's scant
explanation.  Reading a paper or two first would be a huge help.