Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-09-2017 11:16 PM)Gerson W. Barbosa Wrote:  I can only praise your striving for perfection!

It's not perfect until it's perfect. ;-)
Here is an even more optimized coefficient set:

c0 =  0,577215664857
c1 =  0,072815841271
c2 = -0,004845236463
c3 = -0,000342577145
c4 =  0,000096241083
c5 = -0,000007415866
c6 = -0,000000821217

With this set it seems less likely that the error rounds to 6 ULP at two crucial points (e.g. at x=0,34537... where Zeta changes from –0,9999... to –1).
The results you posted do not change much, exept that Zeta(0,2) now is dead on. ;-)

In order to avoid unneccessary roundoff errors I would suggest the following way of evaluating the approximation: Calculate the polynomial in u first. If u>0 add 1/u. Else add (u+1)/u and subtract 1 afterwards. This also is the method in the proposed program.

Dieter