in reply to Re (tilly) 1: In theory, theory and practice are the same... in thread Binomial Expansion
I stand by my (quite conservative) statement. It does introduce errors for fairly small values of n. It is also at least twice as slow (it can be orders of magnitude slower).
But the greatest weakness to my mind is the following output:
200 choose 1: 200 vs. 1.#IND (1.#QNAN).
Rate easy nice
easy 1464/s  96%
nice 34687/s 2270% 
Now that is quite a large error term, don't you think?
Yes, there is a useful space over which its accuracy is quite good (though not always as good), and thanks for exploring that.
Here is the code I used:
#!/usr/bin/perl w
use strict;
use Benchmark qw( cmpthese );
sub fact {
my $fact = 1;
$fact *= $_ for 1..$_[0];
return $fact;
}
sub nice {
my ($n, $r) = @_;
my $res = 1;
for my $i (1..$r) {
$res *= $n;
$res /= $i;
}
return $res;
}
while( <> ) {
my( $n, $m )= split ' ';
my $nice= nice($n,$m);
my $easy= fact($n)/fact($m)/fact($n$m);
print "$n choose $m: $nice vs. $easy (",abs($nice$easy),").\n";
cmpthese( 3, {
nice=>sub{nice($n,$m)},
easy=>sub{fact($n)/fact($m)/fact($n$m)},
} );
}

tye
(but my friends call me "Tye")
Re (tilly) 3: In theory, theory and practice are the same... by tilly (Archbishop) on Apr 01, 2001 at 18:56 UTC 
It is a valid performance note. True But it isn't
producing (visibly) wrong answers to take the naive
approach.
And the improvement is subtle. For instance consider
the following slightly different version of your code:
sub not_so_nice {
my ($n, $r) = @_;
my $res = 1;
for my $i (1..$r) {
$res *= ($n / $i);
}
return $res;
}
Someone was trying to simplify. But oops. In Perl you
have just slowed down instead. In other languages you
are going to get wrong answers. It takes a fair amount of
knowledge to understand what changed. In fact it takes
a fair amount of knowledge to understand
why the original formula can be sped up by using your
alternate approach.
Now I appreciate that you have both pieces of knowledge.
In fact the necessary analysis is probably almost a
reflex for you, just like it is for me. But not
everyone has the math background that we do...  [reply] [d/l] 
