Maybe I'm misunderstanding, but why not simply return the max value when the distance falls within the radius, and apply the dropoff to what falls outside?  In code: 

sub boost {
    my ($dist, $imp, $rad, $drop) = @_;
    if ($dist <= $rad) {
        return $imp;
    } else {
        return $imp / ($dist - $rad + 1) ** $drop;
    }
}

my $imp  = 1000;
my $rad  = 125;
my $drop = 0.5;

for (map 2**$_, 4..13) {
    my $boost = boost($_, $imp, $rad, $drop);
    my $bar = "@" x ($boost/($imp*0.02));
    printf "%6d %s %.1f\n", $_, $bar, $boost;
}

__END__

    16 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ 1000.0
    32 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ 1000.0
    64 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ 1000.0
   128 @@@@@@@@@@@@@@@@@@@@@@@@@@@@ 577.4
   256 @@@@ 87.4
   512 @@ 50.8
  1024 @ 33.4
  2048 @ 22.8
  4096  15.9
  8192  11.1

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In reply to Re: [OT] Normalizing the return result of an exponential formula by Eliya
in thread [OT] Normalizing the return result of an exponential formula by clinton

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