Calculating e

sifukurt has asked for the wisdom of the Perl Monks concerning the following question:

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";
[download]

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

Thanks for your time, Monks.

Comment on Calculating e Download Code

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Re: Calculating e by mdillon (Priest) on Jul 18, 2001 at 06:01 UTC
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, $/;` [download] 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.	[reply] [d/l]
Re: Re: Calculating e by I0 (Priest) on Jul 18, 2001 at 11:19 UTC
Unfortunately, 1/3! was only accurate to about 40 decimal places	[reply]
Re: Calculating e by Albannach (Monsignor) on Jul 18, 2001 at 06:21 UTC
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 $PI so it's probably pointless...), so it is a trivial matter to patch Math::NumberCruncher to use a BigFloat for the factorial. This is of course for occasional uses as for an iterative use of factorial mdillon's answer is still much faster. -- I'd like to be able to assign to an luser	[reply]
Re: Calculating e by I0 (Priest) on Jul 18, 2001 at 09:58 UTC
`$\|=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 ){ $_ *= 10; } #shift decimal point }` [download]	[reply] [d/l]
Re: Calculating e by Boots111 (Hermit) on Jul 18, 2001 at 18:40 UTC
This is just a guess based on my knowledge of C and how it treats mathematical terms, but... I think that the line `$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	[reply] [d/l]