My entry was,

perl -MMath::Pari -e'Math::Pari::setprecision(10000); my $foo = PARI(5
+); my $bar = $foo->sqrt; print $bar,$/,$bar**2,$/'
[download]

You can do essentially the same with Math::MPFR:


 
perl -MMath::MPFR=:mpfr -e "Rmpfr_set_default_prec(10000); $foo = Math
+::MPFR->new(5); $bar = $foo ** 0.5; print $bar, $/, $bar ** 2, $/"
[download]


With Math::MPFR, setting a default precision of 10000, means you have 10000 bits of precision, not 10000 decimal digits of precision. To be guaranteed 10000 decimal digits of precision I think you need to ask for 33220 bits - ie ceil(10000 / log(2)). I haven't checked whether Math::PARI sets its precision in bits or decimal digits.

Cheers,
Rob

With Math::MPFR, setting a default precision of 10000, means you have 10000 bits of precision, not 10000 decimal digits of precision.

Doh! You are right, I was wondering last night why I was getting a 1/3 factor of digits. 10000 gave me ~ 3000 decimal places, while 100000 precision gave me ~30000 decimal places. I thought it might be the print accuracy needed to do the inverse, but I was wrong. Thanks for clearly that up. :-)

---

I'm not really a human, but I play one on earth. Cogito ergo sum a bum

Computing 10k digits the hard slow way (at this time it is still computing, climbing through digit 2000 as I type.

use strict;
use warnings;

=pod Algorithm

http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Digit_b
+y_digit_calculation

Write the decimal expansion of the square. The numbers are laid out in
+ a fashion similar to the long division algorithm, and the root will 
+appear above its square. Divide the digits of the square into pairs, 
+starting from the decimal point and going both left and right. The de
+cimal point of the root should be placed above the decimal point of t
+he square. One digit of the root will appear above each pair of digit
+s of the square.

Begin at the left-most (non-zero) pair of digits of the square. Initia
+lize the remainder to zero. Initialize p to zero.
For each iteration (pair of digits) do:

   1. Bring down the most significant (leftmost) pair of digits of the
+ square not yet used (if all the digits have been used, use the digit
+s 00) and append them to the remainder from the previous step, i.e. m
+ultiply the remainder by one hundred and add the two digits. This wil
+l be the current value c.
   2. Let p be the part of the root found so far, ignoring any decimal
+ point (for the first step, p = 0). Determine the greatest digit x su
+ch that y = (20 \cdot p + x) \cdot x does not exceed c (notice 20p + 
+x is simply twice p, with the digit x appended to the right). You can
+ find x by guessing what c/(20·p) is and doing a trial calculation of
+ y, then adjusting x upward or downward as necessary. Place the digit
+ x as the next digit of the root, i.e above the two digits of the squ
+are which you just brought down. Thus the next p will be the old p ti
+mes 10 plus x.
   3. Subtract y from c to form a new remainder.
   4. If the remainder is zero and there are no more digits to bring d
+own, then the algorithm has terminated. Otherwise go back to step 1 f
+or another iteration.

=cut

$|++;
use Data::Dumper;
use bignum;

my $number = "5.0";
my ($front, $back) = split '\.', $number;
$front = '0' . $front if length $front % 2;
$back .= '0'          if length $back % 2;
print "Front $front, Back $back\n";
my @digits;
push @digits, grep {length} split(/(..)/, $front), ".", grep {length} 
+split(/(..)/, $back);
print "Digits ", join("-", @digits), "\n";

my $i = 0;
my $remainder = 0;
my $p = 0;    # root without decimal
my $root = 0; # root with decimal
my $x = 0;
my $y = 0;
my $c = 0;
while ($i < 10_000) {
    my $pair = shift @digits;
    $pair ||= '00';
    if ($pair eq '.') {
        $root .= '.';
        print "$i\t.\n";
        next;
    }
    $i++;
    $c = (100*$remainder) + $pair;
    $x = int($c / (20 - $p));
    $x = 0 if $x < 0;
    #print "Guessing X is $x when C = $c and P = $p\n";
    $y = ((20 * $p) + $x)  * $x;
    if ($y <= $c) {
        while (1) {
            $x++;
            $y = ((20 * $p) + $x)  * $x;
            #print "X = $x, Y = $y, C = $c\n" if $i == 308;
            if ($y > $c) {
                $x--;
                $y = ((20 * $p) + $x)  * $x;
                last;
            }
        } 
    } elsif ($y > $c) {
        while (1) {
            #print "X = $x, Y = $y, C = $c\n" if $i == 308;
            $x--;
            die if $x < 0;
            $y = ((20 * $p) + $x)  * $x;
            last if ($y < $c)
        } 
    }

    $p = $p . $x;
    $root = $root . $x;
    print "$i\t$x";
    #print "X = $x, Y = $y, C = $c";
    print "\n";
    $remainder = $c -$y;
    last if ($remainder == 0 and scalar @digits == 0);
}
print "\nAfter $i iterations:\nRoot=$root\n";
[download]

It completed about 5 hours later and it returns 4.999999999999999.....9999999......9999 ;)

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[download]

#!/usr/bin/perl
use warnings;
use strict;
use POSIX qw(ceil);
use Math::MPFR qw(:mpfr); 

sub log10{ my $n = shift;
       return log($n)/log(10);
  }

my $req_digits = 10000;
my $acc = ceil( $req_digits/log10(2) );
print "$acc\n";

$|++;
# 33220 is bits needed for 10000 digits
Rmpfr_set_default_prec($acc);

my $rop = Rmpfr_init2($acc); # return                                 
+                        
my $op = Rmpfr_init2($acc);  # operand                                
+                         
my $flip = Rmpfr_init2($acc);  # return test                          
+                               

Rmpfr_set_d ($op, 5.0 , GMP_RNDN);                                    
+            

#Set $rop to the square root of the 2nd arg rounded in the
#direction $rnd. Set $rop to NaN if 2nd arg is negative.
#Return 0 if the operation is exact, a non-zero value otherwise.
my $bool = Rmpfr_sqrt($rop, $op, GMP_RNDN);
print "$bool\n";
if($bool == 0){print "Exact\n";}

print "$rop\n\n"; 
print "length ",length $rop,"\n\n";


# see if return is accurate
#Set $flip to the square of $op, rounded in direction $rnd.
my $si = Rmpfr_sqr($flip, $rop, GMP_RNDN );

print "go back\n";
print "$flip\n";
# pretty close :-)
[download]