Subject: Increasing the accuracy of Gamma for double float
From: Raymond Toy
Date: Wed, 22 Oct 2008 13:59:14 -0400
Dieter Kaiser wrote:
> -----Urspr?ngliche Nachricht-----
> Von: raymond.toy at ericsson.com [mailto:raymond.toy at ericsson.com]
>
>> FWIW, I played around with the dlngam function in slatec. For z from 10
>> to 150, dlngam differs from log_gamma by about 2.75d-16 (relative
>> error). Not too bad compared to double-float-epsilon of about 1.1d-16.
>
>> If we compare gamma(z) to exp(dlngam(z)), the max rel error is about
>> 1.3d-13. Assuming gamma(z) with fpprec = 32 is correct, the error comes
>>from computing the exp. Not much we can do about that, I think.
>
Our conclusion about the accuracy of gamma is wrong. According to this
<http://www.netlib.org/cephes/doubldoc.html#gamma>;, the peak relative
error of 2.3d-15 can be achieved:
* Relative error:
* arithmetic domain # trials peak rms
* DEC -34, 34 10000 1.3e-16 2.5e-17
* IEEE -170,-33 20000 2.3e-15 3.3e-16
* IEEE -33, 33 20000 9.4e-16 2.2e-16
* IEEE 33, 171.6 20000 2.3e-15 3.2e-16
We should examine the algorithm used there.
Ray