[PATCH] Fix for k_tan.c specific inputs

[Corinna Vinschen]

On Mar 17 15:48, Fabian Schriever wrote:
> This fix for k_tan.c is a copy from fdlibm version 5.3 (see also
> http://www.netlib.org/fdlibm/readme), adjusted to use the macros
> available in newlib (SET_LOW_WORD).
> 
> This fix reduces the ULP error of the value shown in the fdlibm readme
> (tan(1.7765241907548024E+269)) to 0.45 (thereby reducing the error by
> 1).
> 
> This issue only happens for large numbers that get reduced by the range
> reduction to a value smaller in magnitude than 2^-28, that is also
> reduced an uneven number of times. This seems rather unlikely given that
> one ULP is (much) larger than 2^-28 for the values that may cause an
> issue.  Although given the sheer number of values a double can
> represent, it is still possible that there are more affected values,
> finding them however will be quite hard, if not impossible.
> 
> We also took a look at how another library (libm in FreeBSD) handles the
> issue: In FreeBSD the complete if branch which checks for values smaller
> than 2^-28 (or rather 2^-27, another change done by FreeBSD) is moved
> out of the kernel function and into the external function. This means
> that the value that gets checked for this condition is the unreduced
> value. Therefore the input value which caused a problem in the
> fdlibm/newlib kernel tan will run through the full polynomial, including
> the careful calculation of -1/(x+r). So the difference is really whether
> r or y is used. r = y + p with p being the result of the polynomial with
> 1/3*x^3 being the largest (and magnitude defining) value. With x being
> <2^-27 we therefore know that p is smaller than y (y has to be at least
> the size of the value of x last mantissa bit divided by 2, which is at
> least x*2^-51 for doubles) by enough to warrant saying that r ~ y.  So
> we can conclude that the general implementation of this special case is
> the same, FreeBSD simply has a different philosophy on when to handle
> especially small numbers.
> ---
>  newlib/libm/math/k_tan.c | 29 +++++++++++++++++++++--------
>  1 file changed, 21 insertions(+), 8 deletions(-)

Pushed.


Thanks,
Corinna

-- 
Corinna Vinschen
Cygwin Maintainer
Red Hat
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