Re: [Maxima-discuss] e, pi, gamma, log(2)

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

>>>>> "Volker" == Volker van Nek <vol...@gm...> writes:

    Volker> Am 20.09.2014 22:53, schrieb Raymond Toy:
    >> 
    >> ​Ok. I haven't looked at your code yet, but if they're significantly faster
    >> and not (much) less accurate, we should consider adding these.  I'm sure we
    >> can come up with a nice set of tests to cover the code to show that it's
    >> good.​
    >> 

    Volker> I made some timing measurements and to summarize the results I have to
    Volker> say that these algorithms are only of interest if you want to use
    Volker> precisions of $fpprec > 1000 and (much) higher. In "normal" ranges, i.e.
    Volker> 16 <= $fpprec <= let's say 100, they are generally much slower.

    Volker> The accuracy depends on the binary representation of the argument. A
    Volker> simple argument like 0.5b0 results in slightly better accuracy but e.g.
    Volker> bs_sin(bs_asin(0.5b0)) leads to significantly worse results compared to
    Volker> sin(asin(0.5b0)) .

    Volker> fpprec:30$
    Volker> x:0.5b0$
    Volker> asin(x);            --> 5.23598775598298873077107230547b−1
    Volker> bs_asin(x);         --> 5.23598775598298873077107230547b−1
    Volker> sin(asin(x));       --> 5.00000000000000000000000000001b−1
    Volker> bs_sin(bs_asin(x)); --> 5.00000000000000000023968760547b−1

Interesting. I'm sure this can be fixed, but it seems like an odd
artifact.

Thanks for the detailed performance results. I wasn't actually asking
for them, but it's really nice to see them.

    Volker> To merge them into src one has to create precision transition points
    Volker> from the current versions to the binary splitting versions and these
    Volker> transitions will vary from Lisp to Lisp, from 32 to 64 bit and from
    Volker> platform to platform. I am not inclined to open this can of worms.

    Volker> I will put these functions to /share/contrib, so that someone who needs
    Volker> $fpprec > 1000 can use them. Or for someone who wants to look at the
    Volker> implementation details. But nothing more at the moment.


    Volker> My functions have a bug where I need help. In case the return value is a
    Volker> power of 2 and fpround has rounded from below I have the problem to fix
    Volker> the exponent. It seems that I do not understand the mechanism of fpround
    Volker> and *m in float.lisp. Can someone help?

Can you give an example of the problem? I have not looked at fpround
in a very long time and don't really remember how it works anymore.

Ray