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0x1.921fb54442d18p+1  => 3.1415926535897931
0x1.1a62633145c07p-53 => 1.2246467991473532e-16

0x1.921fb54442d18p+1  => 3.14159265358979311599796346854419
0x1.1a62633145c07p-53 => 1.2246467991473532071737640294584e-16

use Math::BigFloat;;

...
$n -= $bfd;;
print $n;;
printf "%a\n", $n;;

3.14159265358979310
0x1.921fb54442d18p+1
3.1415926535897931
0.0000000000000001384626433832795
0x1.3f45eb146ba31p-53