The state of glibc libm

[Joseph S. Myers]

On Wed, 14 Mar 2012, Vincent Lefevre wrote:

> For double-double (IBM long double), I don't think the notion of
> correct rounding makes much sense anyway. Actually the double-double
> arithmetic is mainly useful for the basic operations in order to be
> able to implement elementary functions accurately (first step in
> Ziv's strategy, possibly a second step as well). IMHO, on such a
> platform, if expl() (for instance) just calls exp(), this is OK.

expl just calling exp - losing 53 bits of precision - seems rather 
extreme.  But I'd think it would be fine to say: when asked to compute 
f(x), take x' within 10ulp of x, and return a number within 10ulp of 
f(x'), where ulp is interpreted as if the mantissa were a fixed 106 bits 
(fewer bits for subnormals, of course).  (And as a consequence, accurate 
range reduction for large arguments would be considered not to matter for 
IBM long double; sin and cos could return any value in the range [-1, 1] 
for sufficiently large arguments.)

> > (b) Where functions do make attempts at being correctly rounded 
> > (especially the IBM Accurate Mathematical Library functions), they tend to 
> > be sufficiently slow that the slowness attracts bug reports.  Again, this 
> > would likely be addressed by new implementations that use careful error 
> > bounds and information about worst cases to reduce the cost of being 
> > correctly rounding.
> 
> I'm not sure that the complaints are about worst cases. More probably
> software implementation vs hardware implementation in the average
> case. But a new software implementation (better in average) could
> help.

Various bugs do complain about particular cases being slow (as well as 
about such things as sinf being slower than sin - there, if you 
automatically generate functions based not just on the type for the 
function being generated but also on what wider types are available and 
efficient in hardware, you could generate a version of sinf that uses 
double or long double computations internally to speed things up).

-- 
Joseph S. Myers
joseph@codesourcery.com