Re: Power (^) function speed depending on argument types

[Denis Simon <denis.simon@unicaen.fr>]

Hi Karim and Bill,

An increase of 64 bits ??? 
I find it a little optimistic.
I would understand 1 or 2, or up to 14 when I compute 20000 + s
(because log_2(i) ~= 14 and log_2(s) ~= 0), but not 64.

In the example of Bill
precision(Pi)
precision(100+Pi)
I suspect that the huge increase of precision comes from the hidden digits of Pi ?
Otherwise, where would the last digits of 100+Pi come from ?

Denis SIMON.



----- Mail original -----
> De: "Karim Belabas" <Karim.Belabas@u-bordeaux.fr>
> À: "Denis Simon" <denis.simon@unicaen.fr>
> Cc: "pari-users" <pari-users@pari.math.u-bordeaux.fr>
> Envoyé: Lundi 11 Septembre 2023 13:56:10
> Objet: Re: Power (^) function speed depending on argument types

> Hi Denis,
> 
>  actually, the only command that increases the precision there is
> the addition s + i; we add to the t_REAL s the t_INT i which has
> - infinite accuracy (obviously)
> - larger exponent than s after the first few loops
> 
> So the result s + i has a larger precision than s (in fact, the
> bitprecision increases by 64 which is the minimal amount), raising that
> to the power 1/5 keeps that precision. But the next loop with the new s
> increases it again.
> 
> In practice, the precision increases by 64 bits every loop.
> 
> Cheers,
> 
>  K.B.
> 
> * Denis Simon [2023-09-11 13:05]:
>> Hi,
>> 
>> what we learn with this example, is that the function
>> (x) -> x^(1/5)
>> can increase the precision of x when x is a t_REAL.
>> 
>> Is it true for all x t_REAL ?
>> For which values is this still true, instead of 1/5 ?
>> 
>> Denis SIMON.
>> 
>> ----- Mail original -----
>> > De: "Bill Allombert" <Bill.Allombert@math.u-bordeaux.fr>
>> > À: "pari-users" <pari-users@pari.math.u-bordeaux.fr>
>> > Envoyé: Dimanche 10 Septembre 2023 23:02:15
>> > Objet: Re: Power (^) function speed depending on argument types
>> 
>> > On Sun, Sep 10, 2023 at 11:40:22PM +0300, Дмитрий Рыбас wrote:
>> >> Hi All,
>> >> 
>> >> I observe the following
>> >> 
>> >> ? s=0.0;n=2000;for(i=1,n,s=(s+i)^(1/5));
>> >> *cpu time = 4,148 ms,* real time = 4,148 ms.
>> >> ? s=0.0;n=2000;for(i=1,n,s=(s+i+0.0)^(1/5));
>> >> *cpu time = 51 ms,* real time = 51 ms.
>> >> ? s=0.0;n=2000;for(i=1,n,s=(s+i)^(0.2));
>> >> *cpu time = 70 ms,* real time = 73 ms.
>> >> ? s=0.0;n=2000;for(i=1.0,n,s=(s+i)^(1/5));
>> >> *cpu time = 50 ms,* real time = 49 ms.
>> >> ?
>> >> 
>> >> Please advise why the difference in computation time is so drastic?
>> > 
>> > You are not computing s with the same accuracy!
>> > 
>> > ? s=0.0;n=2000;for(i=1,n,s=(s+i)^(1/5));precision(s)
>> > %10 = 38281
>> > ? s=0.0;n=2000;for(i=1,n,s=(s+i+0.0)^(1/5));precision(s)
>> > %11 = 57
>> > 
>> > Cheers,
>> > Bill.
> --
> Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
> Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
> http://www.math.u-bordeaux.fr/~kbelabas/