Ramanujan's constant

Iterative calculations

To generate a high-precision value for Ramanujan's constant, code is borrowed from three other Rosettacode tasks (with some modifications) for performing calculations of the value of π, Euler's number, and integer roots. Additional custom routines for exponentiation are used to ensure all computations are done with rationals, specifically FatRats (rational numbers stored with arbitrary size numerator and denominator). The module Rat::Precise makes it simple to display these to a configurable precision.

use Rat::Precise;

# set the degree of precision for calculations
constant D = 54;
constant d = 15;

# two versions of exponentiation where base and exponent are both FatRat
multi infix:<**> (FatRat $base, FatRat $exp where * >= 1 --> FatRat) {
    2 R** $base**($exp/2);
}

multi infix:<**> (FatRat $base, FatRat $exp where * <  1 --> FatRat) {
    constant ε = 10**-D;
    my $low  = 0.FatRat;
    my $high = 1.FatRat;
    my $mid  = $high / 2;
    my $acc  = my $sqr = sqrt($base);

    while (abs($mid - $exp) > ε) {
      $sqr = sqrt($sqr);
      if ($mid <= $exp) { $low  = $mid; $acc ×=   $sqr }
      else              { $high = $mid; $acc ××= 1/$sqr }
      $mid = ($low + $high) / 2;
    }
    $acc.substr(0, D).FatRat;
}

# calculation of π
sub π (--> FatRat) {
    my ($a, $n) = 1, 1;
    my $g = sqrt 1/2.FatRat;
    my $z = .25;
    my $pi;

    for ^d {
        given [ ($a + $g)/2, sqrt $a × $g ] {
            $z -= (.[0] - $a)**2 × $n;
            $n += $n;
            ($a, $g) = @$_;
            $pi = ($a ** 2 / $z).substr: 0, 2 + D;
        }
    }
    $pi.FatRat;
}

multi sqrt(FatRat $r --> FatRat) {
    FatRat.new: sqrt($r.nude[0] × 10**(D×2) div $r.nude[1]), 10**D;
}

# integer roots
multi sqrt(Int $n) {
    my $guess = 10**($n.chars div 2);
    my $iterator = { ( $^x + $n div ($^x) ) div 2 };
    my $endpoint = { $^x == $^y|$^z };
    min ($guess, $iterator … $endpoint)[*-1, *-2];
}

# 'cosmetic' cover to upgrade input to FatRat sqrt
sub prefix:<√> (Int $n) { sqrt($n.FatRat) }

# calculation of 𝑒
sub postfix:<!> (Int $n) { (constant f = 1, |[\×] 1..*)[$n] }
sub 𝑒 (--> FatRat) { sum map { FatRat.new(1,.!) }, ^D }

# inputs, and their difference, formatted decimal-aligned
sub format ($a,$b) {
    sub pad ($s) { ' ' x ((34 - d - 1) - ($s.split(/\./)[0]).chars) }
    my $c = $b.precise(d, :z);
    my $d = ($a-$b).precise(d, :z);
    join "\n",
        (sprintf "%11s {pad($a)}%s\n", 'Int', $a) ~
        (sprintf "%11s {pad($c)}%s\n", 'Heegner', $c) ~
        (sprintf "%11s {pad($d)}%s\n", 'Difference', $d)
}

# override built-in definitions
constant π = &π();
constant 𝑒 = &𝑒();

my $Ramanujan = 𝑒**(π × √163);
say "Ramanujan's constant to 32 decimal places:\nActual:     " ~
    "262537412640768743.99999999999925007259719818568888\n" ~
    "Calculated: ", $Ramanujan.precise(32, :z), "\n";

say "Heegner numbers yielding 'almost' integers";
for 19, 96, 43, 960, 67, 5280, 163, 640320 -> $heegner, $x {
    my $almost = 𝑒**(π × √$heegner);
    my $exact  = $x³ + 744;
    say format($exact, $almost);
}

Output:

Ramanujan's constant to 32 decimal places:
Actual:     262537412640768743.99999999999925007259719818568888
Calculated: 262537412640768743.99999999999925007259719818568888

Heegner numbers yielding 'almost' integers
        Int             885480
    Heegner             885479.777680154319498
 Difference                  0.222319845680502

        Int          884736744
    Heegner          884736743.999777466034907
 Difference                  0.000222533965093

        Int       147197952744
    Heegner       147197952743.999998662454225
 Difference                  0.000001337545775

        Int 262537412640768744
    Heegner 262537412640768743.999999999999250
 Difference                  0.000000000000750

Continued fractions

Ramanujan's constant can also be generated to an arbitrary precision using standard continued fraction formulas for each component of the 𝑒**(π*√163) expression. Substantially slower than the first method.

use Rat::Precise;

sub continued-fraction($n, :@a, :@b) {
    my $x = @a[0].FatRat;
    $x = @a[$_ - 1] + @b[$_] / $x for reverse 1 ..^ $n;
    $x;
}

#`{ √163 } my $r163 =           continued-fraction( 50, :a(12,|((2*12) xx *)),      :b(19 xx *));
#`{ π    } my $pi   =         4*continued-fraction(140, :a( 0,|(1, 3 ... *)),       :b(4, 1, |((1, 2, 3 ... *) X** 2)));
#`{ e**x } my $R    = 1 + ($_ / continued-fraction(170, :a( 1,|(2+$_, 3+$_ ... *)), :b(Nil,  |(-1*$_, -2*$_ ... *)  ))) given $r163*$pi;

say "Ramanujan's constant to 32 decimal places:\n", $R.precise(32);

Output:

Ramanujan's constant to 32 decimal places:
262537412640768743.99999999999925007259719818568888