Re: Stirling Approx to N! for large Number?

[tall_man]

There was a more accurate formula than Stirling's in tilly's post. Wolfram's Mathematica attributes it to Gosper:

sub factorial_gosper {
  # sqrt(2*n*PI + 1/3)*n**n/exp(n)
  my $n = shift;
  return exp($n*log($n) - $n + 0.5*log(2*$n*PI + 1.0/3.0));
}

This gives better accuracy for small numbers. For 8!, I get 40034.48, compared to the correct 40320.

There are several problems with your implemenation of Stirling's formula. For one thing, log(e^(-n)), which you give as -e*log(n). Since the log is just the exponent base e, the correct answer is -n.

Tilly's formula is closer to being right; there was only one parenthesis mistake. I corrected that and tried them all in a modified version of your code:

#!/usr/bin/perl -w
use strict;
                                                                      
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my $h =8;
my $pure     = factorial_pure($h);
my $stirling = factorial_stirling($h);
my $real_stirling = real_stirling($h);
my $gosper = factorial_gosper($h);
                                                                      
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print "PURE     : $pure\n";
print "STIRLING : $stirling\n";
print "REAL     : $real_stirling\n";
print "GOSPER   : $gosper\n";
                                                                      
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#------Subroutines------------
                                                                      
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sub factorial_stirling{
    my $n = shift;
    use constant PI => 4*atan2 (1,1);
    use constant e => exp(1);
                                                                      
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    my $log_nfact =
       $n * log ($n) - e* log($n) + 0.5 * (log(2*PI)+log($n));
                                                                      
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    # Below is Tilly's suggestion, with less accuracy
    # $n * log ($n) - ($n) + (0.5 * (log(2+ log(PI))+log($n)));
                                                                      
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    return exp($log_nfact);
}
                                                                      
+                     
                                                                      
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sub factorial_pure {
                                                                      
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     my ($n,$res) =  (shift,1);
     return undef unless $n>=0 and $n == int($n);
     $res *= $n-- while $n>1;
     return $res;
                                                                      
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    }
                                                                      
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sub real_stirling {
  my $n = shift;
  my $log_nfact = $n*log($n) - $n + 0.5*(log(2) + log(PI)+ log($n));
  return exp($log_nfact);
}                  
                                                                      
+   
sub factorial_gosper {
  # sqrt(2*n*PI + 1/3)*n**n/exp(n)
  my $n = shift;
  return exp($n*log($n) - $n + 0.5*log(2*$n*PI + 1.0/3.0));
}

I get the answers below:

PURE     : 40320
STIRLING : 417351.253768542
REAL     : 39902.3954526567
GOSPER   : 40034.4823215513