Ramanujan's constant

From Rosetta Code
Ramanujan's constant is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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[edit]

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.

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)
}
}
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)


Julia[edit]

 
 
julia> a = BigFloat(MathConstants.e^(BigFloat(pi)))^(BigFloat(163.0)^0.5)
2.625374126407687439999999999992500725971981856888793538563373369908627075373427e+17
 
julia> 262537412640768744 - a
7.499274028018143111206461436626630091372924626572825942241598957614307213309258e-13
 
 

Perl[edit]

Direct calculation[edit]

Translation of: Sidef
use strict;
use warnings;
use Math::AnyNum;
 
sub ramanujan_const {
my ($x, $decimals) = @_;
 
$x = Math::AnyNum->new($x);
my $prec = (Math::AnyNum->pi * $x->sqrt)/log(10) + $decimals + 1;
local $Math::AnyNum::PREC = 4*$prec->round->numify;
 
exp(Math::AnyNum->pi * $x->sqrt)->round(-$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));
}
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)

Continued fractions[edit]

Translation of: Perl 6
use strict;
use Math::AnyNum <as_dec rat>;
 
sub continued_fr {
my ($a, $b, $n) = (@_[0,1], $_[2] // 100);
$a->() + ($n && $b->() / continued_fr($a, $b, $n-1));
}
 
my $r163 = continued_fr do {my $n; sub {$n++ ? 2*12 : 12 }}, do {my $n; sub { rat 19 }}, 40;
my $pi = continued_fr do {my $n; sub {$n++ ? 1 + 2*($n-2) : 0 }}, do {my $n; sub { rat($n++ ? ($n>2 ? ($n-1)**2 : 1) : 4)}}, 140;
my $p = $pi * $r163;
my $R = 1 + $p / continued_fr do { my $n; sub { $n++ ? $p+($n+0) : 1 } }, do {my $n; sub { $n++; -1*$n*$p }}, 180;
 
printf "Ramanujan's constant\n%s\n", as_dec($R,58);
 
Output:
Ramanujan's constant
262537412640768743.9999999999992500725971981856888793538563

Perl 6[edit]

Iterative calculations[edit]

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 = $x**3 + 744;
say format($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[edit]

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

Phix[edit]

Translation of: Go
Library: mpfr
include mpfr.e
 
constant dp_rqd = 18+32+2, -- (18 before, 32 after, plus 2 for kicks.)
precision_rqd = mpz_sizeinbase(mpz_init(repeat('9',dp_rqd)),2)
 
function q(integer d)
mpfr pi = mpfr_init(precision:=precision_rqd)
mpfr_const_pi(pi)
mpfr t = mpfr_init(d,precision:=precision_rqd)
mpfr_sqrt(t,t)
mpfr_mul(t,pi,t)
mpfr_exp(t,t)
return t
end function
 
printf(1,"Ramanujan's constant to 32 decimal places is:\n")
mpfr_printf(1, "%.32Rf\n", q(163))
sequence heegners = {{19, 96},
{43, 960},
{67, 5280},
{163, 640320},
}
printf(1,"\nHeegner numbers yielding 'almost' integers:\n")
mpfr t = mpfr_init(precision:=precision_rqd), qh
mpz c = mpz_init()
for i=1 to length(heegners) do
integer {h0,h1} = heegners[i]
qh = q(h0)
mpz_ui_pow_ui(c,h1,3)
mpz_add_ui(c,c,744)
mpfr_set_z(t,c)
mpfr_sub(t,t,qh)
string qhs = mpfr_sprintf("%51.32Rf",qh),
cs = mpz_get_str(c),
ts = mpfr_sprintf("%.32Rf",t)
printf(1,"%3d: %s ~= %18s (diff: %s)\n", {h0, qhs, cs, ts})
end for
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)

REXX[edit]

Instead of calculating   e   and     to some arbitrary length,   it was easier to just include those two constants with   201   decimal digits   (which is the amount of decimal digits used for the calculations).   The results are displayed   (right justified)   with one half of that number of decimal digits past the decimal point.

/*REXX pgm displays Ramanujan's constant to at least  100  decimal digits of precision. */
d= min( length(pi()), length(e()) ) - length(.) /*calculate max #decimal digs supported*/
parse arg digs sDigs . 1 . . $ /*obtain optional arguments from the CL*/
if digs=='' | digs=="," then digs= d /*Not specified? Then use the default.*/
if sDigs=='' | sDigs=="," then sDigs= d % 2 /* " " " " " " */
if $='' | $="," then $= 19 43 67 163 /* " " " " " " */
digs= min( digs, d) /*the minimum decimal digs for calc. */
sDigs= min(sDigs, d) /* " " " " display.*/
numeric digits digs /*inform REXX how many dec digs to use.*/
say "The value of Ramanujan's constant calculated with " d ' decimal digits of precision.'
say "shown with " sDigs ' decimal digits past the decimal point:'
say
do j=1 for words($); #= word($, j) /*process each of the Heegner numbers. */
say 'When using the Heegner number: ' # /*display which Heegner # is being used*/
z= exp(pi * sqrt(#) ) /*perform some heavy lifting here. */
say format(z, 25, sDigs); say /*display a limited amount of dec digs.*/
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi: pi= 3.1415926535897932384626433832795028841971693993751058209749445923078164062862,
|| 089986280348253421170679821480865132823066470938446095505822317253594081284,
|| 8111745028410270193852110555964462294895493038196; return pi
/*──────────────────────────────────────────────────────────────────────────────────────*/
e: e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535,
|| 475945713821785251664274274663919320030599218174135966290435729003342952605,
|| 9563073813232862794349076323382988075319525101901; return e
/*──────────────────────────────────────────────────────────────────────────────────────*/
exp: procedure; parse arg x; ix= x%1; if abs(x-ix)>.5 then ix= ix + sign(x); x= x-ix
z=1; _=1; w=z; do j=1; _= _*x/j; z=(z+_)/1; if z==w then leave; w=z; end
if z\==0 then z= z * e() ** ix; return z/1
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); h=d+6; numeric digits
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_%2
do j=0 while h>9; m.j=h; h=h % 2 + 1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g) * .5; end /*k*/; return g
output   when using the default inputs:
The value of Ramanujan's constant calculated with  201  decimal digits of precision.
shown with  100  decimal digits past the decimal point:

When using the Heegner number:  19
                   885479.7776801543194975378934817196268207142865018553571526577110128809842286637202423189990118182067775711

When using the Heegner number:  43
                884736743.9997774660349066619374620785853768473991271391609175146278344881148747592189635643106023717101372606

When using the Heegner number:  67
             147197952743.9999986624542245068292613125786285081833125038167126333712821051229509988315235020413792423533706290

When using the Heegner number:  163
       262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129511813461860645042

Sidef[edit]

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))
})
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)