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));
}

##</code><code>##

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

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