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Re: math function accuracy -- was Re: text2interval again /

On Mar 12 2013, Vincent Lefevre wrote:

On 2013-03-12 02:41:18 +0100, Siegfried M. Rump wrote:


The trick described in
M. Payne and R. Hanek: Radian Reduction for Trigonometric Function, SIGNUM Neweleter, 18:19-24, 1983 

can be used (apparently, I rediscovered and used it in
S.M. Rump. Rigorous and portable standard functions. BIT Numerical Mathematics, 41(3):540-562, 2001. 

which you can download from my homepage). It uses the limited exponent
range, and only a computation with few extra digits is necessary, even
for huge arguments like sin(2^1000). The method is implemented in
INTLAB with rigorous error bounds.


That's OK when the exponent range is limited enough, but not for
the exponent ranges used in multiple precision with MPFR (31-bit
exponent by default, possible 63-bit exponent on 64-bit machines),
for instance.


In any case, range reduction is relevant only to essentially periodic
functions (i.e. where f(x+P) is simple to calculate given f(x)), and
few real programs use only those.

This whole debate strikes me as being an analogue to rearranging the
deck chairs on the Titanic.


Regards,
Nick Maclaren.