$ cat choose_axes.pl

#!/usr/bin/env perl
use strict;
use warnings;

my @series = (
    [qw( 5 5 34 44 114 169 177 184 270 339 361 364 442
        511 530 554 555 587 709 709 735 778 791
        859 871 899 903 926 933 952 )],

    [  1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047,
      4095, 8191, 16383, 32767, 65535, 131071, 262143,
      524287, 1048575, 2097151, 4194303, 8388607,
      16777215, 33554431, 67108863, 134217727, 268435455,
      536870911, 1073741823 ],


    [ 1.713125e-005, 1.748086e-006, 2.101463e-006,
      1.977405e-006, 3.597675e-006, 3.725492e-006,
      3.924736e-006, 2.902199e-006, 3.988645e-006,
      8.210367e-006, 3.360837e-006, 5.202907e-006,
      7.082570e-006, 8.778026e-006, 7.079562e-005,
      9.100576e-005, 5.258545e-005, 9.292677e-005,
      1.789815e-004, 2.113948e-003, 7.229146e-004,
      1.428995e-003, 2.742045e-003, 5.552746e-003,
      1.822390e-002, 2.220999e-002, 4.316067e-002,
      8.876963e-002, 1.751072e-001, 3.494051e-001,
      7.155960e-001, 1.347822e+000 ],
);

for my $ar (@series) {
    my ($type, $min, $max) = choose_axis_params($ar);
    print "($min .. $max)  $type\n";
}

sub minmax {
    my $min = my $max = shift;
    while (@_) {
        my $t = shift;
        $min = $t<$min ? $t : $min;
        $max = $t>$max ? $t : $max;
    }
    return $min, $max;
}

sub check_axis {
    my $name = shift;
    my @points = @_;
    my ($min, $max) = minmax(@points);
    my $midpoint = ($min+$max)/2;

    my $cnt = 0;
    for my $t (@points) {
        ++$cnt if $t > $midpoint;
    }

    my $err = abs(@points/2 - $cnt);
    return $name, $min, $max, $midpoint, $cnt, $err;
}

sub choose_axis_params {
    my $r = shift;

    my ($min,$max) = minmax(@$r);
    $r = [ sort @$r ];

    my @axes;
    push @axes, [ check_axis('linear',@$r) ];
    push @axes, [ check_axis('log', map { log($_) } @$r) ];

    @axes = sort { $a->[-1] <=> $b->[-1] } @axes;

    #for my $r (@axes) {
    #    printf "%-8.8s (%s .. %s) %s, %s, %s\n", @$r;
    #}

    return @{$axes[0]};
}

$ perl choose_axes.pl
(5 .. 952)  linear
(0 .. 20.794415415867)  log
(-13.2569890828565 .. 0.298489956293335)  log