The simplest way to speed this up would be to abandon the use of Math::BigInt. You use this in two places to raise numbers to a power:

$power=Math::BigInt->new($initial_index);
$distance_index=($power->bpow($distance))->numify();
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but you then immediately retrieve the result as a standard scalar without performing any check for overflow.

Assuming the value '4' is typical scale of magnitude for $distance aka $self{ distance } aka $distance_index,

then using Perl's built-in exponentiation operator,  $power ** $distance

cmpthese -3, {
    BigInt => q[
        my $x = Math::BigInt->new( $_ )->bpow( 4 )->numify() for 2 .. 
+1e6; 
    ],
    standard => q[ 
        my $x =  $_ ** 4 for 2 .. 1e6;
    ],
};;
            (warning: too few iterations for a reliable count)

         s/iter   BigInt standard
BigInt      172       --    -100%
standard  0.625   27472%       --
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will be at least 250 times quicker.

For sequences of length 10,000, standard math will safely accomodate distances of up to 78.

• 1e4 => 78
• 1e5 => 61
• 1e6 => 51
• 1e7 => 44

If you really need to use Math::BigInt for this calculation, then you should be doing the 1 / $distance_index part of your calculation before you numify the result of the previous step.