#!perl
use bignum;

# This is Stirling's approximation for log(n!), but
# without the "-n" piece that cancels out below in any
# case.

sub xe{
 $_[0]*log($_[0])+log(atan2(1,1)*8*$_[0])/2
  + 1/(12*$_[0]) - 1/(360*($_[0]**3)) + 1/(1260*($_[0]**5));
}

if (!@ARGV) {
  @ARGV = qw{ 2**96 3*(10**6) 1*(10**6) };
}
my($m,$n,$r)=map {eval "use bignum;$_"}@ARGV;

my $e=exp(xe($m-$n)+xe($m-$r)-xe($m)-xe($m-$n-$r));

printf "%3.5g   %3.5g   %3.5g\n",$e,1-$e,(1-"$e");
[download]

The bug is that the numbers printed in the last line don't make sense. Somehow, both $e and 1-$e print as "1", though 1-"$e" properly prints as "0". (The true value of $e should in fact be somewhere between 1-2**(-54) and 1)

When I use Data::Dumper, then $e is shown as '1' (note the quotes). However, that doesn't really help isolate the bug, as this also creates something that dumps as '1', but doesn't show the "both $e and 1-$e are 1" anomaly:

use bignum;
$e = exp(2 ** (-128));
use Data::Dumper; 
print Dumper($e);
print $e,"\n",1-$e,"\n";
[download]

My perl environment is perl 5.8.7 on cygwin.

@/=map{[/./g]}qw/.h_nJ Xapou cets krht ele_ r_ra/;
map{y/X_/\n /;print}map{pop@$_}@/for@/
[download]