logarithmic_root.pl

#!/usr/bin/perl

# Author: Daniel "Trizen" Șuteu
# License: GPLv3
# Date: 17 July 2016
# Website: https://github.com/trizen

# Logarithmic root of n.

# Solves c = x^x, where "c" is known.
# (based on Newton's method for the nth-root)

# Example: 100 = x^x
#          x = lgrt(100)
#          x =~ 3.59728502354042

use 5.010;
use strict;
use warnings;

use Math::AnyNum qw(:overload);

sub lgrt {
    my ($c) = @_;

    my $p = 1 / 10**($Math::AnyNum::PREC >> 2);
    my $d = log($c);

    my $x = 1;
    my $y = 0;

    while (abs($x - $y) > $p) {
        $y = $x;
        $x = ($x + $d) / (1 + log($x));
    }

    $x;
}

say lgrt( 100);   # 3.59728502354041750549765225178228606913554305489
say lgrt(-100);   # 3.70202936660214594290193962952737102802777010583+1.34823128471151901327831464969872480416292147614i