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 evaluated with the last four Heegner numbers the result is almost an integer.

Go

Library: bigfloat

The standard library's math/big.Float type lacks an exponentiation function and so I have had to use an external library to provide this function.

Also the math.Pi built in constant is not accurate enough to be used with big.Float and so I have used a more accurate string representation instead. <lang go>package main

import (

   "fmt"
   "github.com/ALTree/bigfloat"
   "math/big"

)

const (

   prec = 256 // say
   ps   = "3.1415926535897932384626433832795028841971693993751058209749445923078164"

)

func q(d int64) *big.Float {

   pi, _ := new(big.Float).SetPrec(prec).SetString(ps)
   t := new(big.Float).SetPrec(prec).SetInt64(d)
   t.Sqrt(t)
   t.Mul(pi, t)
   return bigfloat.Exp(t)

}

func main() {

   fmt.Println("Ramanujan's constant to 32 decimal places is:")
   fmt.Printf("%.32f\n", q(163))
   heegners := [4][2]int64{
       {19, 96},
       {43, 960},
       {67, 5280},
       {163, 640320},
   }
   fmt.Println("\nHeegner numbers yielding 'almost' integers:")
   t := new(big.Float).SetPrec(prec)
   for _, h := range heegners {
       qh := q(h[0])
       c := h[1]*h[1]*h[1] + 744
       t.SetInt64(c)
       t.Sub(t, qh)
       fmt.Printf("%3d: %51.32f ≈ %18d (diff: %.32f)\n", h[0], qh, c, t)
   }

}</lang>

Output:

Ramanujan's constant to 32 decimal places is:
262537412640768743.99999999999925007259719818568888

Heegner numbers yielding 'almost' integers:
 19:             885479.77768015431949753789348171962682 ≈             885480 (diff: 0.22231984568050246210651828037318)
 43:          884736743.99977746603490666193746207858538 ≈          884736744 (diff: 0.00022253396509333806253792141462)
 67:       147197952743.99999866245422450682926131257863 ≈       147197952744 (diff: 0.00000133754577549317073868742137)
163: 262537412640768743.99999999999925007259719818568888 ≈ 262537412640768744 (diff: 0.00000000000074992740280181431112)

Perl

Translation of: Sidef

<lang perl>use strict; use warnings; use Math::AnyNum;

sub ramanujan_const {

   my ($x, $decimals) = @_;

   $x = Math::AnyNum->new($x);
   my $prec = 4 * ((Math::AnyNum->pi * $x->sqrt) / log(2) + $decimals);
   local $Math::AnyNum::PREC = $prec->round->numify;

   my $e  = Math::AnyNum->e;
   my $pi = Math::AnyNum->pi;
   my $c  = $e**($pi * $x->sqrt);
   $c->round(-$decimals)->stringify;

}

my $decimals = 100; printf("Ramanujan's constant to $decimals decimals:\n%s\n\n",

   ramanujan_const(163, $decimals));

print "Heegner numbers yielding 'almost' integers:\n"; my @tests = (19, 96, 43, 960, 67, 5280, 163, 640320);

while (@tests) {

   my ($h, $x) = splice(@tests, 0, 2);
   my $c = ramanujan_const($h, 32);
   my $n = Math::AnyNum::ipow($x, 3) + 744;
   printf("%3s: %51s ≈ %18s (diff: %s)\n", $h, $c, $n, ($n - $c)->round(-32));

}</lang>

Output:

Ramanujan's constant to 100 decimals:
262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129511813461860645042

Heegner numbers yielding 'almost' integers:
 19:             885479.77768015431949753789348171962682 ≈             885480 (diff: 0.22231984568050246210651828037318)
 43:          884736743.99977746603490666193746207858538 ≈          884736744 (diff: 0.00022253396509333806253792141462)
 67:       147197952743.99999866245422450682926131257863 ≈       147197952744 (diff: 0.00000133754577549317073868742137)
163: 262537412640768743.99999999999925007259719818568888 ≈ 262537412640768744 (diff: 0.00000000000074992740280181431112)

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

Sidef

<lang ruby>func ramanujan_const(x, decimals=32) {

   local Num!PREC = *"#{4*round((Num.pi*√x)/log(10) + decimals + 1)}"
   exp(Num.pi * √x) -> round(-decimals).to_s

}

var decimals = 100 printf("Ramanujan's constant to #{decimals} decimals:\n%s\n\n",

    ramanujan_const(163, decimals))

say "Heegner numbers yielding 'almost' integers:" [19, 96, 43, 960, 67, 5280, 163, 640320].each_slice(2, {|h,x|

   var c = ramanujan_const(h, 32)
   var n = (x**3 + 744)
   printf("%3s: %51s ≈ %18s (diff: %s)\n", h, c, n, n-Num(c))

})</lang>

Output:

Ramanujan's constant to 100 decimals:
262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129511813461860645042

Heegner numbers yielding 'almost' integers:
 19:             885479.77768015431949753789348171962682 ≈             885480 (diff: 0.22231984568050246210651828037318)
 43:          884736743.99977746603490666193746207858538 ≈          884736744 (diff: 0.00022253396509333806253792141462)
 67:       147197952743.99999866245422450682926131257863 ≈       147197952744 (diff: 0.00000133754577549317073868742137)
163: 262537412640768743.99999999999925007259719818568888 ≈ 262537412640768744 (diff: 0.00000000000074992740280181431112)