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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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