Re: guarantees on digits of floating point values

[American Citizen <website.reader3@gmail.com>]

Bill:

Which one is faster?

On 11/6/25 14:38, Bill Allombert wrote:
> On Sat, Nov 01, 2025 at 08:11:06PM +0100, Ruud H.G. van Tol wrote:
> > As a test, I checked 1000021 digits of exp(1),
> >
> > my(N=10^6+21); localprec(N); exp(1) * 10^(N-1)\1
> >
> > against https://apod.nasa.gov/htmltest/gifcity/e.1mil
> > and all digits matched. :)
> You can avoid floating points entirely and do the computation with
> rational numbers. The formula PARI/GP uses is
>
> /* binary spliiting */
> exprat_rec(a,b)=
> {
>    if(a==b,[0,1],
>       a==b-1,[1,b],
>       my(c=(a+b)\2,[u,v]=exprat_rec(a,c),[u1,v1]=exprat_rec(c,b));[u*v1+u1,v*v1])
> }
> /* compute sum(i=0,n,1/i!) as a rational number */
> exprat(n)=my([p,q]=exprat_rec(0,n));1+p/q
>
> We have
> sum(i=0,n,1/i!) < exp(1) < sum(i=0,n,1/i!) + 1/(n*n!)
>
> so for n = 205022, B = 10^(10^6+1)
>
> floor(B*(exprat(n))) < B * exp(1) < ceil(B*(exprat(n) + 1/(n*n!)))
>
> ? r=exprat(n);a=floor(B*r);b=ceil(B*(r+1/(n*n!)));
> ? b-a
> %59 = 1
> ? a%10^20
> %60 = 37981764476942281888
> ? b%10^20
> %61 = 37981764476942281889
>
> so to 10^6 digits, it should end with 3798176447694228188
>
> Cheers,
> Bill