... not sure what a TPRNG is ... I meant TRNG ...

You can correct your OP. Please see How do I change/delete my post? for site etiquette and protocol regarding changing a post.

Update 1: Added example using Inline::C.

Update 2: Do more work, ... for my $e ( 1..8 ) { ... }

Well, the Stolz-Cesaro theorem is about convergence of series. There are a lot of series that converge to pi, did you have a specific one in mind? And I don't see where random numbers come into the picture at all.

Some time ago, a University Professor emailed me a serial implementation and asked if possible to run parallel. The number sequence generator in MCE makes it seamless. Also, enabling the bounds_only option further decreases latency by providing workers just the begin and end values per each chunk/segment.

Perl

use strict;
use warnings;

use MCE::Flow;

my $N;

# Workers receive [ begin, end ] values.

MCE::Flow::init(
  max_workers => MCE::Util::get_ncpu(),
  chunk_size  => 100000,
  bounds_only => 1,
  user_begin  => sub { $N = MCE->user_args()->[0] }
);

sub func {
  my ( $beg_seq, $end_seq ) = @{ $_ };
  my ( $pi, $t ) = ( 0.0 );

  for my $i ( $beg_seq .. $end_seq ) {
    $t = ( $i + 0.5 ) / $N;
    $pi += 4.0 / ( 1.0 + $t * $t );
  }

  MCE->gather($pi);
}

# The user_args option is how to pass arguments.
# Workers persist between each run.

for my $e ( 1..8 ) {
  my $n   = 10 ** $e;
  my @ret = mce_flow_s { user_args => [$n] }, \&func, 0, $n - 1;
  my $pi  = 0.0;   $pi += $_ for @ret;

  printf "%9d  %0.14f\n", $n, $pi / $n;
}
[download]

Output

       10  3.14242598500110
      100  3.14160098692313
     1000  3.14159273692312
    10000  3.14159265442313
   100000  3.14159265359816
  1000000  3.14159265358988
 10000000  3.14159265358979
100000000  3.14159265358979
[download]

Inline::C

use strict;
use warnings;

use Inline 'C' => <<'END_C';

  unsigned int N = 0;

  void c_init( unsigned int n )
  {
    N = n;
  }

  double c_func( unsigned int beg_seq, unsigned int end_seq )
  {
    double t, pi = 0.0;
    unsigned int i;

    for ( i = beg_seq ; i <= end_seq ; i++ ) {
      t = (double) i / (double) N;
      pi += 4.0 / ( 1.0 + t * t );
    }

    return pi;
  }

END_C

use MCE::Flow;

# Workers receive [ begin, end ] values.

MCE::Flow::init(
  max_workers => MCE::Util::get_ncpu(),
  chunk_size  => 100000,
  bounds_only => 1,
  user_begin  => sub { c_init( MCE->user_args()->[0] ) }
);

sub func {
  my ( $beg_seq, $end_seq ) = @{ $_ };
  my $pi = c_func($beg_seq, $end_seq);

  MCE->gather($pi);
}

# The user_args option is how to pass arguments.
# Workers persist between each run.

for my $e ( 1..8 ) {
  my $n   = 10 ** $e;
  my @ret = mce_flow_s { user_args => [$n] }, \&func, 0, $n - 1;
  my $pi  = 0.0;   $pi += $_ for @ret;

  printf "%9d  %0.14f\n", $n, $pi / $n;
}
[download]

Output

       10  3.23992598890716
      100  3.15157598692313
     1000  3.14259248692312
    10000  3.14169265192314
   100000  3.14160265357315
  1000000  3.14159365358964
 10000000  3.14159275358979
100000000  3.14159266358980
[download]

The C code is greater than 5 times faster on the Windows platform (32-bit), beyond 15 times faster on Unix platforms (64-bit).

Regards, Mario

That's great, but it runs in two seconds single threaded. A tenth of a second in C.

basically i have to write a code that generates a list of number pairs using the formula provided to estimate pie the theorm being what i posted a few min ago . (Cesaro's theorem states that given two random integers, x and y, the probability that gcd(x, y) = 1 is 6/(Pi^2).) and I to mark down the number pairs that are equate to one I believe from what my teacher said and the project states.

I dont know how to write it so it generates the numbers in pairs however.

In your loop, call the RNG twice:

for (1 .. 500)
{
    my $x = rng();
    my $y = rng();
    ...
}
[download]