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Date:         Tue, 5 Nov 2013 21:39:06 +1300
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From: Andrew Parsloe <aparsloe@CLEAR.NET.NZ>
Subject: Re: Bug in l3fp for sines near 2pi
To: LATEX-L@LISTSERV.UNI-HEIDELBERG.DE
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Status: R

On 5/11/2013 2:44 p.m., Bruno Le Floch wrote:
> Hello Andrew,
>
> On 10/4/13, Joseph Wright <joseph.wright@morningstar2.co.uk> wrote:
>> On 04/10/2013 08:50, Andrew Parsloe wrote:
>>> \fp_eval:n { sin( 2pi - .0001 ) } ==> 9999.999000000002
>>
>> Thanks for the bug report: I've logged it as
>> https://github.com/latex3/svn-mirror/issues/155, and assigned it to
>> Bruno :-) (He wrote the FPU!). As I've said in the issue database, this
>> looks like a range-reduction failure to me. Range reduction is known to
>> be 'hard', so that's not that surprising.
>
> Thank you for reporting the bug (now fixed), which led me to
> reconsider the implementation of trigonometric functions.  I ended up
> adding trigonometric functions expecting their argument in degrees
> (sind, cosd, tand, cotd, secd, cscd),for which range reduction is
> almost trivial, as we want to reduce modulo rational numbers
> (integers, even!) rather than an irrational number.  The result of
> sind, cosd, etc. can thus be guaranteed to lie within 0.6 ulp of the
> correct sine, cosine, etc.  For the functions in radians, I would need
> to either store EMAX+16 decimals of pi, where EMAX is the maximal
> exponent and 16 is the precision of floating point numbers, or be able
> to compute them.  Right now, I am cheating, using a fixed-precision
> value of 2pi (equal to twice \c_pi_fp).
>

"fixed precision" makes me think of the LaTeX2e package fp, 18 places 
before and after the decimal point. In using it I became alarmed at how 
rapidly errors propagated inwards when exponents were involved.

> Comments welcome on the addition of sind and friends (and similarly
> asind for arc-trigonometric functions, newly added), and on range
> reduction in radians.
>
> Regards,
> Bruno
>

sind etc, and the inverse functions, are very welcome. (But as a student 
of hyperbolic geometry, I especially look forward to the hyperbolic 
functions and inverses!)

Andrew