Note: In the code below:

$cnt_rex          x      The sample number
$VecSizeBits      N      Number of bits in the vector
$num_vecs         h      Number of vectors/hash functions
@vectors                 Content of vectors
@cnt_bits                Number of bits set to 1 in vector
@cnt_colbits             Number of collisions:  I've broken it out to
                         [0] = Total document collisions
                         [1] = Count of documents colliding with 1 vec
+tor
                         [2] = Count of docs colliding with 2 vectors
                         ...
$first_coll              Index of first collision...useless, should re
+move
$first_coll_f            Index of first false positive (all vectors co
+llide)
[download]

$ cat analyze_bloom.pl
#!/usr/bin/perl
#
# analyze_bloom.pl <NumBits> <NumVecs> <ln2(Samples)>
#
# Experiment with Bloom filter variant by BUk
#
use strict;
use warnings;
use autodie;
use Bit::Vector;
use Math::BigInt;

my $VecSizeBits=shift or die "Missing # of bits!";
my $num_vecs   =shift or die "missing # of vectors!";
my $iterations =shift or die "missing iteration count!";
if ($iterations<33) {
    $iterations = 1<<$iterations;
}

my $VecSize = 1 << $VecSizeBits;
my $VecMask = $VecSize-1;

print "NumVecs=$num_vecs\n";

my @col2numbits = (0, 1);
while (@col2numbits < 1<<$num_vecs) {
    @col2numbits = (@col2numbits, map { $_+1 } @col2numbits);
}
my @cnt_colbits = (0) x (1+$num_vecs);

my %report_cnts = map { (1<<$_)=>0 } (8 .. 24);

my @vectors;
my @cnt_bits = (0) x $num_vecs;
$vectors[$_] = Bit::Vector::new(0,$VecSize) for 0 .. $num_vecs-1;

{
    my @HDR = (
        q{                avg      avg   calc  },
        q{BI  Samples   one bits pct 1s pct 1s },
        q{-- ---------- -------- ------ ------ },
    );
    my $t = int(($num_vecs+1) * 8 - length(" # collisions "))/2;
    $HDR[0] .= "-"x$t . " # collisions " . "-"x$t;
    $HDR[1] .= "  all   ";
    $HDR[1] .= sprintf("   % 2u   ", $_) for 1 .. $num_vecs;
    $HDR[1] .= "   p(coll)";
    $HDR[2] .= ("-"x7 . " ") x ($num_vecs+1) . "------------";
    print join("\n", @HDR, "");
}

my $first_coll=0;
my $cnt_coll=0;
my $first_coll_f=0;
my $cnt_coll_f=0;
my $cnt_rex=0;
while (++$cnt_rex < $iterations) {
    my @h = map { int(rand($VecSize)) } 0 .. $VecMask;
    my $bits=32;

    my $collision=0;
    for my $i (0 .. $num_vecs-1) {
        if ($vectors[$i]->bit_test($h[$i])) {
            $collision |= 1<<$i
        }
        else {
            ++$cnt_bits[$i]
        }
        $vectors[$i]->Bit_On($h[$i]);
    }
    if ($collision) {
        ++$cnt_colbits[$col2numbits[$collision]];
        ++$cnt_colbits[0];          # 'any' collision
        ++$cnt_coll;
        $first_coll=$cnt_rex        if !$first_coll;
        if ($collision == $VecMask) {
            ++$cnt_coll_f;
            $first_coll_f=$cnt_rex  if !$first_coll_f;
        }
    }

    if (exists $report_cnts{$cnt_rex}) {
        my $ttl_bits=0;
        $ttl_bits += $_ for @cnt_bits;
        my $avg_bits = $ttl_bits / $num_vecs;
        my $avg_pct_full = 100.0 * $ttl_bits / ($num_vecs*$VecSize);
        my $calc_pct_full = 100.0 * (1-exp(-$cnt_rex/$VecSize));
        printf "% 2u %10u: % 7u %6.2f %6.2f",
               $VecSizeBits, $cnt_rex, $avg_bits, $avg_pct_full, $calc
+_pct_full;
        print " ",
              join(" ",
                   map {sprintf"% 7u",$_} @cnt_colbits
              );
        my $expected = (1-exp(-$cnt_rex/$VecSize))**$num_vecs;
        printf " %.10f", $expected;
        print "\n";
    }
}

$ ./analyze_bloom.pl 10  3  13
NumVecs=3
                avg      avg   calc  --------- # collisions ---------
BI  Samples   one bits pct 1s pct 1s   all       1       2       3    
+  p(coll)
-- ---------- -------- ------ ------ ------- ------- ------- ------- -
+-----------
10        256:     222  21.74  22.12      86      72      14       0 0
+.0108230772
10        512:     405  39.62  39.35     250     186      59       5 0
+.0609161842
10       1024:     647  63.22  63.21     699     351     265      83 0
+.2525804578
10       2048:     892  87.11  86.47    1711     494     677     540 0
+.6464623148
10       4096:    1012  98.83  98.17    3758     511    1000    2247 0
+.9460533270

$ ./analyze_bloom.pl 12  5  15
NumVecs=5
                avg      avg   calc  ----------------- # collisions --
+---------------
BI  Samples   one bits pct 1s pct 1s   all       1       2       3    
+   4       5      p(coll)
-- ---------- -------- ------ ------ ------- ------- ------- ------- -
+------ ------- ------------
12        256:     248   6.06   6.06      37      36       1       0  
+     0       0 0.0000008164
12        512:     479  11.70  11.75     138     114      23       1  
+     0       0 0.0000223999
12       1024:     908  22.17  22.12     434     311     103      17  
+     3       0 0.0005295634
12       2048:    1604  39.17  39.35    1296     634     447     170  
+    44       1 0.0094309292
12       4096:    2572  62.81  63.21    3282     915    1038     788  
+   443      98 0.1009251903
12       8192:    3542  86.47  86.47    7372     979    1392    1775  
+  1968    1258 0.4833243641
12      16384:    4020  98.15  98.17   15564     980    1420    2052  
+  3718    7394 0.9117155503
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