Re: [Maxima-discuss] significant difference?

[Raymond T. <toy...@gm...>]

>>>>> "Schorsch" == Schorsch  <Dr.> writes:

    >> Maxima uses Carlson's Rf function to compute the elliptic_k value.
    Schorsch> thanks Ray, for clarifying that

    Schorsch> IIRC, I determined the 2ULPs error bound on a system with intermediate
    Schorsch> extended double precision on the x87 stack...
 
Ah.  So you got some extra 11 bits of precision for intermediate
calculations before saving the final result as a double?

    Schorsch> because only K is needed, one could give the old Gaussian algo 
    Schorsch> involving the https://en.wikipedia.org/wiki/Arithmetic?geometric_mean 
    Schorsch> a try

Good point.  I knew about that, but perhaps I was lazy since I already
had Rf implemented.

    Schorsch> long time ago, I found that (1-x)*(1+x) is more accurate than 1-x^2 in
    Schorsch> case abs(x)>1/sqrt(2), but is outperformed by 2*(1?x)?(1?x)^2 in case 
    Schorsch> x> 1/2, because then, 1-x does not produce any roundoff error (see 
    Schorsch> Lemma of Sterbenz, which according to Sylvie Boldo in fact had been 
    Schorsch> coined by Kahan).

Ooh, those are neat little tricks when you want to squeeze out the
most accuracy with a fixed precision.

--
Ray