I stumbled across it while trying to add some speedups to Math-GMP

Nice post ++, btw.
I've always assumed that if you want to get the maximum performance out of Math::GMP, then you'll avoid the overloading altogether and explicitly make the method calls.
I think this is generally true and the following script seems to support that view, though it only compares the 2 ways of performing a right shift.

use warnings;
use Benchmark;
use Math::GMP;

no warnings 'once';

$str1 = '123456789' x 1000;

$gmp1 = Math::GMP->new($str1);

timethese(100000, {
'Math::GMP 1' => '$ret1 = $gmp1 >> 1000;',
'Math::GMP 2' => '$ret2 = $gmp1->div_2exp_gmp(1000)',
});

print "gmp calculation ok\n" if $ret1 == $ret2;

__END__


Outputs:
Benchmark: timing 100000 iterations of Math::GMP 1, Math::GMP 2...
Math::GMP 1:  0 wallclock secs ( 0.19 usr +  0.00 sys =  0.19 CPU) @ 5
+34759.36/s (n=100000)
            (warning: too few iterations for a reliable count)
Math::GMP 2:  0 wallclock secs ( 0.12 usr +  0.00 sys =  0.12 CPU) @ 8
+00000.00/s (n=100000)
            (warning: too few iterations for a reliable count)
gmp calculation ok
[download]

I might as well showcase my own Math::GMPz module while I'm at it, though I see it's main benefit as being that it enwraps much more of the gmp library than is covered by Math::GMP.

use warnings;
use Benchmark;
use Math::GMP;
use Math::GMPz qw(:mpz);

no warnings 'once';

$str1 = '123456789' x 1000;

$gmp1  = Math::GMP->new ($str1);
$gmpz1 = Math::GMPz->new($str1);
$ret4 = Math::GMPz->new();

timethese(100000, {
'Math::GMP 1' =>  '$ret1 = $gmp1 >> 1000;',
'Math::GMP 2' =>  '$ret2 = $gmp1->div_2exp_gmp(1000);',
'Math::GMPz 1' => '$ret3 = $gmpz1 >> 1000;',
'Math::GMPz 2' => 'Rmpz_div_2exp($ret4, $gmpz1, 1000);',
});

print "gmp calcualtion ok\n"      if  $ret1  ==  $ret2;
print "gmpz calculation ok\n"     if  $ret3  ==  $ret4;
print "both gmp and gmpz agree\n" if "$ret1" eq "$ret3";

__END__

Outputs:

Benchmark: timing 100000 iterations of Math::GMP 1, Math::GMP 2, Math:
+:GMPz 1, Math::GMPz 2...
Math::GMP 1:  0 wallclock secs ( 0.19 usr +  0.00 sys =  0.19 CPU) @ 5
+34759.36/s (n=100000)
            (warning: too few iterations for a reliable count)
Math::GMP 2:  0 wallclock secs ( 0.12 usr +  0.00 sys =  0.12 CPU) @ 8
+00000.00/s (n=100000)
            (warning: too few iterations for a reliable count)
Math::GMPz 1:  0 wallclock secs ( 0.09 usr +  0.00 sys =  0.09 CPU) @ 
+1063829.79/s (n=100000)
            (warning: too few iterations for a reliable count)
Math::GMPz 2:  0 wallclock secs ( 0.03 usr +  0.00 sys =  0.03 CPU) @ 
+3225806.45/s (n=100000)
            (warning: too few iterations for a reliable count)
gmp calcualtion ok
gmpz calculation ok
both gmp and gmpz agree
[download]

Both Math::GMP and Math::GMPz were built against the very same gmp-6.2.1 library.

Cheers,
Rob