### comment on

 Need Help??

Hi Laurent_R and fellow monks,

Nice speedup! I applied 3 optimizations in preparation for a follow-up post combining caching with parallelization. This was done because File::Map adds overhead (i.e. it will not be as fast as an array). So, I asked myself can anything be done to further improve the serial implementation. And if yes, is it helpful. I provide the run time for each stage at the end of this post. It turns out that improvements were possible and will be converting collatz3_d.pl to use File::Map for the parallel demonstration.

Update: Further improvements, Step 4.

1. Replaced division by 2.

```\$n >> 1;

2. Removed one level of branching.

```    while (\$n != 1) {
\$result += \$cache[\$n], last
if defined \$cache[\$n];

my \$new_n = \$n % 2 ? 3 * \$n + 1 : \$n >> 1;
\$result++;
\$cache[\$n] = \$cache[\$new_n] + 1
if defined \$cache[\$new_n] and \$n < \$max;

\$n = \$new_n;
}

3. Then reduced the number of loop iterations. Credit for reducing the # of loop iterations was from watching Notation and compressed dynamics, one minute into it (i.e. the T(x) notation).

```    while (\$n != 1) {
\$result += \$cache[\$n], last
if defined \$cache[\$n];

\$n % 2 ? ( \$result += 2, \$new_n = (3 * \$n + 1) >> 1 )
: ( \$result += 1, \$new_n = \$n >> 1 );

\$cache[\$n] = \$cache[\$new_n] + (\$n % 2 ? 2 : 1)
if defined \$cache[\$new_n] and \$n < \$max;

\$n = \$new_n;
}

4. Finally, less caching.

```    while (\$n != 1) {
\$result += \$cache[\$n], last
if defined \$cache[\$n];

\$n % 2 ? ( \$result += 2, \$new_n = (3 * \$n + 1) >> 1 )
: ( \$result += 1, \$new_n = \$n >> 1 );

\$n = \$new_n;
}

The final code optionally takes an argument.

```#!/usr/bin/perl
use strict;
use warnings;
use feature qw/say/;

my \$max = shift || 1e6;
\$max = 1e6 if \$max < 1e6;

my @cache = (0, 1, 2);

sub collatz_seq {
my \$input = shift;
my \$n = \$input;
my \$result = 0;
my \$new_n;

while (\$n != 1) {
\$result += \$cache[\$n], last
if defined \$cache[\$n];

\$n % 2 ? ( \$result += 2, \$new_n = (3 * \$n + 1) >> 1 )
: ( \$result += 1, \$new_n = \$n >> 1 );

\$n = \$new_n;
}

\$cache[\$input] = \$result if \$input < \$max;
return \$result;
}

my @long_seqs;
for my \$num (1..\$max) {
my \$seq_length = collatz_seq \$num;
push @long_seqs, [ \$num, \$seq_length ] if \$seq_length > 400;
}

@long_seqs = sort { \$b->[1] <=> \$a->[1]} @long_seqs;
say "\$_->[0]: \$_->[1]" for @long_seqs[0..19];

Results:

```\$ time perl collatz3_a.pl 1e7

# Intel i7 laptop, Docker Container, Ubuntu + Perlbrew Perl 5.30.1

collatz3_a.pl 1e7   32.291s  (a) original code, accepts argument
collatz3_b.pl 1e7   30.134s  (b) a + replaced division with >> 1
collatz3_c.pl 1e7   28.503s  (c) b + removed 1 level of branching
collatz3_d.pl 1e7   21.464s  (d) c + reduced loop iterations
collatz3_e.pl 1e7   19.357s  (e) d + less caching

# AMD 3970x, Docker Container, Ubuntu + Perlbrew Perl 5.30.1

collatz3_a.pl 1e7   13.130s  (a) original code, accepts argument
collatz3_b.pl 1e7   12.394s  (b) a + replaced division with >> 1
collatz3_c.pl 1e7   12.261s  (c) b + removed 1 level of branching
collatz3_d.pl 1e7    9.170s  (d) c + reduced loop iterations
collatz3_e.pl 1e7    7.681s  (e) d + less caching

8400511: 686
8865705: 668
6649279: 665
9973919: 663
6674175: 621
7332399: 616
7532665: 616
5649499: 613
8474249: 611
6355687: 608
8847225: 606
9533531: 606
6635419: 603
9953129: 601
7464846: 598
7464847: 598
3732423: 597
5598635: 595
8397953: 593
6298465: 590

Regards, Mario

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