Calculating e

[sifukurt]

I need some help. I've written a script that will calculate e using Euler's formula. I'm using Math::NumberCruncher to facilitate the factorial process, and Math::BigFloat which should, in theory, allow me to take e out to an arbitrary number of places. However, for some reason it seems to stop after about 320 places, or so. Here is the code:

use Math::BigFloat;
use Math::NumberCruncher;

$| = 1;
$N = shift || 100;
$e = Math::BigFloat->new("1");
for ( $i = 1; $i <= $N; $i++ ) {
    $fac = Math::NumberCruncher::Factorial( $i );
    $e += 1 / $fac;
}

print $e, "\n";

Any thoughts on how I can get the above script to go beyond the 320-odd places?

Thanks for your time, Monks.

[mdillon]

i just used this with N=400 and got a 1638 decimal place e, then with N=500 and got a 2145 decimal place e:

use Math::BigFloat;
$N = shift || 100;
$e = Math::BigFloat->new(1);
$fac = Math::BigFloat->new(1);
for $i (1 .. $N)
{
        $fac *= $i;
        $e += 1 / $fac;
}
print $e, $/;

my computer is too slow to wait for it to go any higher. i don't know how far you can take it out.

the number of digits seems to increase with the size of N but i don't know why, nor at what rate this happens.

p.s. the result of this code looks to me like e, but i have no idea how accurate it is (beyond being basically accurate, i.e. about 2.71828). i compared it to a 10000 digit calculation of e i found on the Net, and it was only the same up to about 40 decimal places. if you're doing this to get an accurate value of e for later use, i would recommend just finding a calculated and verified version of it on the web (or use the version posted by IO in another response to this parent thread, which, as far as i can tell, reliably generates the actual digits of e). if you're just doing this for fun, then have a blast.

[I0]

Unfortunately, 1/3! was only accurate to about 40 decimal places

[Albannach]

Not only is mdillon's solution correct, it's also much more efficient as each factorial is simply one multiply on the previous one, whereas calling the function each time means a lot more work is being done.

Now the reason for the failure is that the Math::NumberCruncher function overflows for 171!. Unfortunately it returns '1.#INF' which Perl happily treats as 1.0 in your division, so you simply divide by one from that point on, not making much more progress.

Update: While Math::NumberCruncher does not use BigFloats for it's factorial function, it does already use the Math::BigFloat module for storing its very large version of $PI (though strangely BigFloats are not used in calculations with that $PI so it's probably pointless...), so it is a trivial matter to patch Math::NumberCruncher to use a BigFloat for the factorial. This is of course for occasional uses as for an iterative use of factorial mdillon's answer is still much faster.

--
I'd like to be able to assign to an luser

[I0]

$|=1;
use integer;
my @e=(1)x450;  #represents sum{$e[$i]/$i!}
for(1..1000){
    for(reverse 1..$#e){    #Normalize
        $e[$_-1] += $e[$_]/$_;
        $e[$_] %= $_;
    }
    print $e[0];
    $e[0] = 0;
    for( @e ){ $_ *= 10; }  #shift decimal point
}

[Boots111]

This is just a guess based on my knowledge of C and how it treats mathematical terms, but...

I think that the line $e += 1 / $fac; is the culprit. While $e is a BigFloat, my guess would be that the result of the division of 1 and $fac would be a number of the same type as $fac. Thus while $e supports arbitrary precision, 1 / $fac does not support such a high precision

Matt