use strict;
use warnings;
use Math::MPFR qw(:mpfr);

# Set precision to precisely 107 bits
Rmpfr_set_default_prec(107);

# Set $bin_str to the 107-bit representation of
# 3.1415926535897932384626433832795
my $bin_str = '11001001000011111101101010100010001000010110100011000010001101001100010011000110011000101000101110000000111';

my $check = Math::MPFR->new(0);

for(my $i = 0; $i < 107; $i++) {
  if(substr($bin_str, $i, 1)) {
     print "$i ", Math::MPFR->new(2 ** (-($i + 1))), "\n";
     $check += 2 ** (-($i + 1));
  }
}

print "\nSum =\n$check\nSum * 4 =\n", $check * 4, "\n";