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

[Karim Belabas <Karim.Belabas@u-bordeaux.fr>]

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/