Ramanujan's constant
Calculate Ramanujan's constant (as described on the OEIS site) with at least 32 digits of precision, by the method of your choice. Optionally, if using the 𝑒**(π*√x) approach, show that when evaulated with the last four Heegner numbers the result is almost an integer.
Perl 6
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. <lang perl6>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 = $x**3 + 744; say format($exact, $almost);
}</lang>
- 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