#! perl -slw
use strict;

sub log10{ log( $_[0] ) / log( 10 ) }

sub ticks {
    my( $start, $end, $n ) = @_;
    $n ||= 10;
    my $range = $end - $start;
    $range += $range / $n;
    my $interval = $range / $n;
    $interval  = 10**int( log10( $interval ) );
    $interval *= 10 while ( $range / $interval ) > ( $n * 2 );
    $interval /=  2 while ( $range / $interval ) < ( $n / 2 );
    $start = int( $start / $interval ) * $interval;

    my @ticks;
    while( $start < ( $end + $interval ) ) {
        push @ticks, $start;
        $start += $interval;
    }
    return @ticks;
}


for my $range (
    [0,9], [0.123,8.673], [0.022, 0.086],
    [1_123_345_456, 1_124_246_357.5], [1_123_345_456, 1_123_346_357.5],
    [ 0.00011, 0.00041],
) {
    for my $n ( 0, 4, 5, 8 , 10, 15, 16, 20, 21 ) {
        my @ticks = ticks( @$range, $n );

        printf "from:%12.12g to %12.12g by:%2u -> [%2u][ %s ]\n",
            @$range, $n, scalar( @ticks ), join ' ', map{
                my $s = sprintf( "%.8g", $_ ); $s =~ s[e([+-])0*([1-9]+)]['e' . ($1 eq '-' ? '-' : '') . $2]e;
                $s;
            } @ticks;

    }
}

##</code><code>##

C:\test>calcTicks.pl
from:           0 to            9 by: 0 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:           0 to            9 by: 4 -> [ 3][ 0 5 10 ]
from:           0 to            9 by: 5 -> [ 5][ 0 2.5 5 7.5 10 ]
from:           0 to            9 by: 8 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:           0 to            9 by:10 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:           0 to            9 by:15 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:           0 to            9 by:16 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:           0 to            9 by:20 -> [19][ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 ]
from:           0 to            9 by:21 -> [19][ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 ]
from:       0.123 to        8.673 by: 0 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:       0.123 to        8.673 by: 4 -> [ 3][ 0 5 10 ]
from:       0.123 to        8.673 by: 5 -> [ 5][ 0 2.5 5 7.5 10 ]
from:       0.123 to        8.673 by: 8 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:       0.123 to        8.673 by:10 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:       0.123 to        8.673 by:15 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:       0.123 to        8.673 by:16 -> [10][ 0 1 2 3 4 5 6 7 8 9 ]
from:       0.123 to        8.673 by:20 -> [19][ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 ]
from:       0.123 to        8.673 by:21 -> [19][ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 ]
from:       0.022 to        0.086 by: 0 -> [ 8][ 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ]
from:       0.022 to        0.086 by: 4 -> [ 5][ 0 0.025 0.05 0.075 0.1 ]
from:       0.022 to        0.086 by: 5 -> [ 5][ 0 0.025 0.05 0.075 0.1 ]
from:       0.022 to        0.086 by: 8 -> [ 8][ 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ]
from:       0.022 to        0.086 by:10 -> [ 8][ 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ]
from:       0.022 to        0.086 by:15 -> [15][ 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 ]
from:       0.022 to        0.086 by:16 -> [15][ 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 ]
from:       0.022 to        0.086 by:20 -> [15][ 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 ]
from:       0.022 to        0.086 by:21 -> [15][ 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 ]
from:  1123345456 to 1124246357.5 by: 0 -> [11][ 1.1233e9 1.1234e9 1.1235e9 1.1236e9 1.1237e9 1.1238e9 1.1239e9 1.124e9 1.1241e9 1.1242e9 1.1243e9 ]
from:  1123345456 to 1124246357.5 by: 4 -> [ 4][ 1.123e9 1.1235e9 1.124e9 1.1245e9 ]
from:  1123345456 to 1124246357.5 by: 5 -> [ 5][ 1.12325e9 1.1235e9 1.12375e9 1.124e9 1.12425e9 ]
from:  1123345456 to 1124246357.5 by: 8 -> [11][ 1.1233e9 1.1234e9 1.1235e9 1.1236e9 1.1237e9 1.1238e9 1.1239e9 1.124e9 1.1241e9 1.1242e9 1.1243e9 ]
from:  1123345456 to 1124246357.5 by:10 -> [11][ 1.1233e9 1.1234e9 1.1235e9 1.1236e9 1.1237e9 1.1238e9 1.1239e9 1.124e9 1.1241e9 1.1242e9 1.1243e9 ]
from:  1123345456 to 1124246357.5 by:15 -> [11][ 1.1233e9 1.1234e9 1.1235e9 1.1236e9 1.1237e9 1.1238e9 1.1239e9 1.124e9 1.1241e9 1.1242e9 1.1243e9 ]
from:  1123345456 to 1124246357.5 by:16 -> [11][ 1.1233e9 1.1234e9 1.1235e9 1.1236e9 1.1237e9 1.1238e9 1.1239e9 1.124e9 1.1241e9 1.1242e9 1.1243e9 ]
from:  1123345456 to 1124246357.5 by:20 -> [20][ 1.1233e9 1.12335e9 1.1234e9 1.12345e9 1.1235e9 1.12355e9 1.1236e9 1.12365e9 1.1237e9 1.12375e9 1.1238e9 1.12385e9 1.1239e9 1.12395e9 1.124e9 1.12405e9 1.1241e9 1.12415e9 1.1242e9 1.12425e9 ]
from:  1123345456 to 1124246357.5 by:21 -> [20][ 1.1233e9 1.12335e9 1.1234e9 1.12345e9 1.1235e9 1.12355e9 1.1236e9 1.12365e9 1.1237e9 1.12375e9 1.1238e9 1.12385e9 1.1239e9 1.12395e9 1.124e9 1.12405e9 1.1241e9 1.12415e9 1.1242e9 1.12425e9 ]
from:  1123345456 to 1123346357.5 by: 0 -> [11][ 1.1233454e9 1.1233455e9 1.1233456e9 1.1233457e9 1.1233458e9 1.1233459e9 1.123346e9 1.1233461e9 1.1233462e9 1.1233463e9 1.1233464e9 ]
from:  1123345456 to 1123346357.5 by: 4 -> [ 4][ 1.123345e9 1.1233455e9 1.123346e9 1.1233465e9 ]
from:  1123345456 to 1123346357.5 by: 5 -> [ 6][ 1.1233453e9 1.1233455e9 1.1233458e9 1.123346e9 1.1233463e9 1.1233465e9 ]
from:  1123345456 to 1123346357.5 by: 8 -> [11][ 1.1233454e9 1.1233455e9 1.1233456e9 1.1233457e9 1.1233458e9 1.1233459e9 1.123346e9 1.1233461e9 1.1233462e9 1.1233463e9 1.1233464e9 ]
from:  1123345456 to 1123346357.5 by:10 -> [11][ 1.1233454e9 1.1233455e9 1.1233456e9 1.1233457e9 1.1233458e9 1.1233459e9 1.123346e9 1.1233461e9 1.1233462e9 1.1233463e9 1.1233464e9 ]
from:  1123345456 to 1123346357.5 by:15 -> [11][ 1.1233454e9 1.1233455e9 1.1233456e9 1.1233457e9 1.1233458e9 1.1233459e9 1.123346e9 1.1233461e9 1.1233462e9 1.1233463e9 1.1233464e9 ]
from:  1123345456 to 1123346357.5 by:16 -> [11][ 1.1233454e9 1.1233455e9 1.1233456e9 1.1233457e9 1.1233458e9 1.1233459e9 1.123346e9 1.1233461e9 1.1233462e9 1.1233463e9 1.1233464e9 ]
from:  1123345456 to 1123346357.5 by:20 -> [20][ 1.1233455e9 1.1233455e9 1.1233456e9 1.1233456e9 1.1233457e9 1.1233457e9 1.1233458e9 1.1233458e9 1.1233459e9 1.1233459e9 1.123346e9 1.123346e9 1.1233461e9 1.1233461e9 1.1233462e9 1.1233462e9 1.1233463e9 1.1233463e9 1.1233464e9 1.1233464e9 ]
from:  1123345456 to 1123346357.5 by:21 -> [20][ 1.1233455e9 1.1233455e9 1.1233456e9 1.1233456e9 1.1233457e9 1.1233457e9 1.1233458e9 1.1233458e9 1.1233459e9 1.1233459e9 1.123346e9 1.123346e9 1.1233461e9 1.1233461e9 1.1233462e9 1.1233462e9 1.1233463e9 1.1233463e9 1.1233464e9 1.1233464e9 ]
from:     0.00011 to      0.00041 by: 0 -> [ 8][ 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 ]
from:     0.00011 to      0.00041 by: 4 -> [ 5][ 0.0001 0.0002 0.0003 0.0004 0.0005 ]
from:     0.00011 to      0.00041 by: 5 -> [ 5][ 0.0001 0.0002 0.0003 0.0004 0.0005 ]
from:     0.00011 to      0.00041 by: 8 -> [ 8][ 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 ]
from:     0.00011 to      0.00041 by:10 -> [ 8][ 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 ]
from:     0.00011 to      0.00041 by:15 -> [14][ 0.0001 0.000125 0.00015 0.000175 0.0002 0.000225 0.00025 0.000275 0.0003 0.000325 0.00035 0.000375 0.0004 0.000425 ]
from:     0.00011 to      0.00041 by:16 -> [14][ 0.0001 0.000125 0.00015 0.000175 0.0002 0.000225 0.00025 0.000275 0.0003 0.000325 0.00035 0.000375 0.0004 0.000425 ]
from:     0.00011 to      0.00041 by:20 -> [14][ 0.0001 0.000125 0.00015 0.000175 0.0002 0.000225 0.00025 0.000275 0.0003 0.000325 0.00035 0.000375 0.0004 0.000425 ]
from:     0.00011 to      0.00041 by:21 -> [14][ 0.0001 0.000125 0.00015 0.000175 0.0002 0.000225 0.00025 0.000275 0.0003 0.000325 0.00035 0.000375 0.0004 0.000425 ]