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

[Volker v. N. <vol...@gm...>]

Hendrik, some Details on e and pi.

The computation of e and pi is done by the functions fpe1 resp. fppi1
which are - as Richard Fateman already posted - in src/float.lisp .

Before 2007 e was computed by it's Taylor series and pi by the formula
pi = 6*asin(1/2) and the asin-series.

In 2007 I replaced the asin-formula by the Chudnovsky-formula and for
the computation of e I implemented a mechanism that combines a lot of
Taylor summands to one. This mechanism uses the fact that division is
expensive and multiplication of large integers is much cheaper.
At that time I didn't know binary splitting. Binary splitting is also a
mechanism to combine Taylor summands and also uses large integer
multiplication but in contrast to my approach it is a recursive divide
and conquer algorithm and provides better timings for high precisions.

Very soon I will replace the current versions by the binary splitting
variants (which are attached to my first mail in this thread).

Volker van Nek

Am 21.09.2014 09:59, schrieb Jan Hendrik Mueller:
> Maybe I overlooked that: What was the "old" way maxima computed e and pi? 
> Is there a possibility to look at these codes for non developers?
> bw
> Jan
> 
>> Am 20.09.2014 um 22:53 schrieb Raymond Toy <toy...@gm...>:
>>
>>
>>
>>> On Sat, Sep 20, 2014 at 2:58 AM, Volker van Nek <vol...@gm...> wrote:
>>> Am 19.09.2014 20:35, schrieb Raymond Toy:
>>>>>>>>> "Volker" == Volker van Nek <vol...@gm...> writes:
>>>>
>>>>     Volker> [ I also implemented exp(x), log(x), sin(x) and friends for bigfloat x.
>>>>     Volker> In contrast to fpe1, .. these implementions are still some kind of raw
>>>>     Volker> material and need more error analysis. And of course it's a lot more
>>>>     Volker> difficult to merge them into Maxima. My idea is to put them into
>>>>     Volker> /share/contrib for now. ]
>>>>
>>>> Is there something wrong with the current implementations of these
>>>> functions?
>>>>
>>>> Ray
>>>>
>>>
>>> Ray, as I wrote in my last sentence I do not feel any pressure to
>>> replace the current functions in src.
>>>
>>> After coding the binary splitting versions for the constants I was
>>> curious to check out the other algorithms described in this paper.
>>>
>>> I thought it is worth to share the implementation of these algorithms.
>>> Perhaps someone can use them or is simply interested to study them.
>>>
>>> The binary splitting variants are very fast and in some far future these
>>> implementations might be worth to consider. At the moment I don't want
>>> to risk breaking anything in src for the sake of speed advantages.
>>
>> ​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
>>>
>>>>
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>>
>>
>> -- 
>> Ray
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