Re: Accuracy of Mathematical Functions

[Paul Zimmermann <Paul.Zimmermann@inria.fr>]

       Dear Joseph,

> Thanks.  Some further comments on the document (some of which I may have 
> made before):
> 
> * The bug referenced as a reason for not testing other rounding modes was 
> fixed long ago, results in other rounding modes would be interesting as 
> well (they would of course make the document longer), and worst-case 
> inputs for other rounding modes would be worth adding to the glibc tests.

in the next update we'll point out that these disastrous results with
other rounding modes were in the past. Surely results with other rounding modes
would be interesting, volunteers to implement that are welcome!

I did a quick test with exp and GNU libc 2.40:

    -0x1.49f33ad2c1c58p+9,   /* GNU libc 0.510359 */
    -0x1.9df9aecc00001p-29,  /* GNU libc RNDZ 1.24399 */
    0x1.50f5a68c91c18p-1,    /* GNU libc RNDU 1.01597 */
    -0x1.9df9aecc00001p-29,  /* GNU libc RNDD 1.24399 */

> * I don't think "dummy implementations" is a useful description for 
> implementations that do actually implement the function (within a few 
> ulp).  There are many possible ways to implement a function, including 
> building on other functions, and those ones are optimized for getting the 
> APIs into glibc for all supported formats rather than for speed or 
> accuracy (and replacing by faster, more accurate implementations for 
> particular formats is welcome).

agreed, maybe "trivial" would be more accurate. By the way, if one wants
to replace say exp10m1f by a more accurate version, what are the steps to
follow?

> * As previously mentioned, I still think it would be of interest to see 
> such data for complex functions (a function with one complex argument and 
> a complex result is effectively two bivariate real functions.

we also got similar feedback for complex functions. Indeed I believe one might
have surprises, even for basic operations since there is no standard. But
again, time is limited, and volunteers are welcome! (This might be a different
document, testing less libraries, and updated less often.)

Paul