bin/170206: complex arcsinh, log, etc.

[Stephen Montgomery-Smith]

On 07/28/2012 11:15 AM, Stephen Montgomery-Smith wrote:
> On 07/28/2012 10:46 AM, Stephen Montgomery-Smith wrote:
>> OK.  This clog really seems to work.
>>
>> x*x + y*y - 1 is computed with a ULP less than 0.8.  The rest of the
>> errors seem to be due to the implementation of log1p.  The ULP of the
>> final answer seems to be never bigger than a little over 2.
>>
>>
>
>
> Also, I don't think the problem is due to the implementation of log1p.
> If you do an error analysis of log(1+x) where x is about exp(-1)-1, and
> x is correct to within 0.8 ULP, I suspect that about 2.5 ULP is the best
> you can do for the final answer:
>
> relative_error(log(1+x)) = fabs(1/((1+x) log(1+x))) * relative_error(x)
>                    = 1.58 * relative_error(x)
>
> Given that log1p has itself a ULP of about 1, and relative error in x is
> 0.8, and considering x=exp(-1)-1, this gives a ULP at around 1.58*0.8+1
> = 2.3.  And that is what I observed.
>
> (Here "=" means approximately equal to.)

And I should add that I just realized that ULP isn't quite the same as 
relative error, so an extra factor of up to 2 could make its way into 
the calculations.