Ramanujan's constant: Difference between revisions
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{{
Calculate Ramanujan's constant (as described on the [http://oeis.org/wiki/Ramanujan%27s_constant 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
the result is ''almost'' an integer.
=={{header|C++}}==
{{libheader|Boost}}
<syntaxhighlight lang="cpp">#include <iomanip>
#include <iostream>
#include <boost/math/constants/constants.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
using big_float = boost::multiprecision::cpp_dec_float_100;
big_float f(unsigned int n) {
big_float pi(boost::math::constants::pi<big_float>());
return exp(sqrt(big_float(n)) * pi);
}
int main() {
std::cout << "Ramanujan's constant using formula f(N) = exp(pi*sqrt(N)):\n"
<< std::setprecision(80) << f(163) << '\n';
std::cout << "\nResult with last four Heegner numbers:\n";
std::cout << std::setprecision(30);
for (unsigned int n : {19, 43, 67, 163}) {
auto x = f(n);
auto c = ceil(x);
auto pc = 100.0 * (x/c);
std::cout << "f(" << n << ") = " << x << " = "
<< pc << "% of " << c << '\n';
}
return 0;
}</syntaxhighlight>
{{out}}
<pre>
Ramanujan's constant using formula f(N) = exp(pi*sqrt(N)):
262537412640768743.99999999999925007259719818568887935385633733699086270753741038
Result with last four Heegner numbers:
f(19) = 885479.777680154319497537893482 = 99.9999748927309842681413350366% of 885480
f(43) = 884736743.999777466034906661937 = 99.9999999999748474372063224648% of 884736744
f(67) = 147197952743.999998662454224507 = 99.9999999999999990913285473342% of 147197952744
f(163) = 262537412640768743.999999999999 = 99.9999999999999999999999999997% of 262537412640768744
</pre>
=={{header|Fōrmulæ}}==
{{FormulaeEntry|page=https://formulae.org/?script=examples/Ramanujan%27s_constant}}
'''Solution'''
'''Case 1. Calculate Ramanujan's constant with at least 32 digits of precision'''
[[File:Fōrmulæ - Ramanujan's constant 01.png]]
[[File:Fōrmulæ - Ramanujan's constant 02.png]]
'''Case 2. Show that when evaluated with the last four Heegner numbers the result is almost an integer'''
[[File:Fōrmulæ - Ramanujan's constant 03.png]]
[[File:Fōrmulæ - Ramanujan's constant 04.png]]
=={{header|Go}}==
{{libheader|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.
<syntaxhighlight 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)
}
}</syntaxhighlight>
{{out}}
<pre>
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)
</pre>
=={{header|Haskell}}==
Calculations are done using an arbitrary precision "constructive reals" type (CReal).
<syntaxhighlight lang="haskell">import Control.Monad (forM_)
import Data.Number.CReal (CReal, showCReal)
import Text.Printf (printf)
ramfun :: CReal -> CReal
ramfun x = exp (pi * sqrt x)
-- Ramanujan's constant.
ramanujan :: CReal
ramanujan = ramfun 163
-- The last four Heegner numbers.
heegners :: [Int]
heegners = [19, 43, 67, 163]
-- The absolute distance to the nearest integer.
intDist :: CReal -> CReal
intDist x = abs (x - fromIntegral (round x))
main :: IO ()
main = do
let n = 35
printf "Ramanujan's constant: %s\n\n" (showCReal n ramanujan)
printf "%3s %34s%20s%s\n\n" " h " "e^(pi*sqrt(h))" "" " Dist. to integer"
forM_ heegners $ \h ->
let r = ramfun (fromIntegral h)
d = intDist r
in printf "%3d %54s %s\n" h (showCReal n r) (showCReal 15 d)</syntaxhighlight>
{{out}}
<pre>
Ramanujan's constant: 262537412640768743.99999999999925007259719818568887935
h e^(pi*sqrt(h)) Dist. to integer
19 885479.77768015431949753789348171962682071 0.222319845680502
43 884736743.99977746603490666193746207858537685 0.000222533965093
67 147197952743.99999866245422450682926131257862851 0.000001337545775
163 262537412640768743.99999999999925007259719818568887935 0.00000000000075
</pre>
=={{header|J}}==
Project: compute, expressed in mathematica then j notation, <pre>Exp[Pi*Sqrt[163]] ^ o. %: 163</pre> .
J natively supports arithmetic types Boolean 0 1, integer 00 01 2 3 9, extended integer<ref>Taking a rational power of an extended integer produces a floating-point result whenever the denominator of the power is not 1.</ref> 9x, rational 1r2, floating point as c double, and complex numbers 2ad90 (radius 2, 90 angle in degrees). J does not natively support arbitrary precision decimal, or ternary. J can format a rational as arbitrary precision base 10 literals.
Rational arithmetic with series expansion therefor serves to compute Ramanujan's constant. We test for convergence in the base 10 literal expression over the required length. Exponential expansion of "long" rational numbers and the number of terms needed for "large" numbers is unwieldly. We divide by 8x, then raise the series sum to the 8th power. We also convert the rational exponent to a rational number of sufficient digits to reduce size. Tolerant continued fraction expansion reduces the magnitude of the exponent's numerator and denominator.
{{reflist}}
===continued fraction===
<pre>
NB. approximation as a rational number
]RC=: +`%/ }: 1j1 (#!.1) 262537412640768743x 1 1333462407511 1 8 1 1 5 1 4 1 7 1 1 1 9 1 1 2 12 4 1 15 4 299 3 5 1 4 5 5 1 28 3 1 9 4 1 6 1 1 1 1 1 1 51 11 5 3 2 1 1 1 1 2 1 5 1 9 1x
45120712325606158012363304056579024470785114628332049030433r171863933112440071790625667019825998411698
NB. expressed in decimal
59j40 ": RC
262537412640768743.9999999999992500725971981856888793538563
</pre>
=== 𝑒**(π*√x) ===
<pre>
Note 'citation for pi computation'
@MISC {3129700,
TITLE = {Series that converge to $\pi$ quickly},
AUTHOR = {El Ectric (https://math.stackexchange.com/users/301661/el-ectric)},
HOWPUBLISHED = {Mathematics Stack Exchange},
NOTE = {URL:https://math.stackexchange.com/q/3129700 (version: 2019-02-28)},
EPRINT = {https://math.stackexchange.com/q/3129700},
URL = {https://math.stackexchange.com/q/3129700}
}
)
Digits=: adverb define NB. u Digits y u y is less accurate than u y+1
NB. returns u to at least y significant digits
format=. ' _.' -.~ ((j.~ 50&+) y)&":
i =. 5
current=. format u i
whilst. last -.@-: current do.
last =. current
i =. i + 2
current=. format result =. u i
end.
result
)
cf=: 0.1&$: :(4 :0) NB. tolerance cf value -> continued fraction approximation of value to tolerance
Y =. y
X =. 0 >. x
terms =. 0 $ 0x
whilst. X < | approximation - y do.
'term Y' =. <.`([:%1&|)`:0 Y
terms =. terms , term
approximation =. +`%/ }: 1j1 #!.1 terms
end.
)
assert (-: 0&cf) 649r200
assert 13r4 (-: cf) 649r200
NB. pi is sufficiently fast for partial series recomputation.
numerator=: (*&! +:) * _3 25&p.
denominator=: 2&^ * !@:(3&*)
pi=: (2 * [: +/ numerator % denominator)@:i.@:x: NB. use: pi TERMS
cf_sqrt=: 10&$: :(4 :0) NB. continued fraction approximation to square root. x sqrt y then x is the depth, y the square
a =. x: <. %: y NB. estimate
r =. y - *: a NB. remainder
a + %`+/ (+: x) $ r , +: a
)
exp=: (1x"_)`((($:~<:)+^%!@x:@])~)@.(0<[) NB. recursive Taylor series x exp y recursively sums x terms of Taylor series for Exp[y], memoization candidate.
</pre>
<syntaxhighlight lang="text">
NB. takes the constant beyond the repeat 9s.
S=: cf_sqrt&163 Digits 34
P=: pi Digits 34
Y=: 1e_36 cf 1r8*P*S
f=: exp&Y M. NB. memoize
59j40 ": 8 ^~ f Digits 34
262537412640768743.9999999999992500711164316586918409066184
NB. ^
</syntaxhighlight>
=={{header|Java}}==
Very interesting. Compute Pi, E, and square root to arbitrary precision.
<syntaxhighlight lang="java">
import java.math.BigDecimal;
import java.math.MathContext;
import java.util.Arrays;
import java.util.List;
public class RamanujanConstant {
public static void main(String[] args) {
System.out.printf("Ramanujan's Constant to 100 digits = %s%n%n", ramanujanConstant(163, 100));
System.out.printf("Heegner numbers yielding 'almost' integers:%n");
List<Integer> heegnerNumbers = Arrays.asList(19, 43, 67, 163);
List<Integer> heegnerVals = Arrays.asList(96, 960, 5280, 640320);
for ( int i = 0 ; i < heegnerNumbers.size() ; i++ ) {
int heegnerNumber = heegnerNumbers.get(i);
int heegnerVal = heegnerVals.get(i);
BigDecimal integer = BigDecimal.valueOf(heegnerVal).pow(3).add(BigDecimal.valueOf(744));
BigDecimal compute = ramanujanConstant(heegnerNumber, 50);
System.out.printf("%3d : %50s ~ %18s (diff ~ %s)%n", heegnerNumber, compute, integer, integer.subtract(compute, new MathContext(30)).toPlainString());
}
}
public static BigDecimal ramanujanConstant(int sqrt, int digits) {
// For accuracy on lat digit, computations with a few extra digits
MathContext mc = new MathContext(digits + 5);
return bigE(bigPi(mc).multiply(bigSquareRoot(BigDecimal.valueOf(sqrt), mc), mc), mc).round(new MathContext(digits));
}
// e = 1 + x/1! + x^2/2! + x^3/3! + ...
public static BigDecimal bigE(BigDecimal exponent, MathContext mc) {
BigDecimal e = BigDecimal.ONE;
BigDecimal ak = e;
int k = 0;
BigDecimal min = BigDecimal.ONE.divide(BigDecimal.TEN.pow(mc.getPrecision()));
while ( true ) {
k++;
ak = ak.multiply(exponent).divide(BigDecimal.valueOf(k), mc);
e = e.add(ak, mc);
if ( ak.compareTo(min) < 0 ) {
break;
}
}
return e;
}
// See : https://www.craig-wood.com/nick/articles/pi-chudnovsky/
public static BigDecimal bigPi(MathContext mc) {
int k = 0;
BigDecimal ak = BigDecimal.ONE;
BigDecimal a = ak;
BigDecimal b = BigDecimal.ZERO;
BigDecimal c = BigDecimal.valueOf(640320);
BigDecimal c3 = c.pow(3);
double digitePerTerm = Math.log10(c.pow(3).divide(BigDecimal.valueOf(24), mc).doubleValue()) - Math.log10(72);
double digits = 0;
while ( digits < mc.getPrecision() ) {
k++;
digits += digitePerTerm;
BigDecimal top = BigDecimal.valueOf(-24).multiply(BigDecimal.valueOf(6*k-5)).multiply(BigDecimal.valueOf(2*k-1)).multiply(BigDecimal.valueOf(6*k-1));
BigDecimal term = top.divide(BigDecimal.valueOf(k*k*k).multiply(c3), mc);
ak = ak.multiply(term, mc);
a = a.add(ak, mc);
b = b.add(BigDecimal.valueOf(k).multiply(ak, mc), mc);
}
BigDecimal total = BigDecimal.valueOf(13591409).multiply(a, mc).add(BigDecimal.valueOf(545140134).multiply(b, mc), mc);
return BigDecimal.valueOf(426880).multiply(bigSquareRoot(BigDecimal.valueOf(10005), mc), mc).divide(total, mc);
}
// See : https://en.wikipedia.org/wiki/Newton's_method#Square_root_of_a_number
public static BigDecimal bigSquareRoot(BigDecimal squareDecimal, MathContext mc) {
// Estimate
double sqrt = Math.sqrt(squareDecimal.doubleValue());
BigDecimal x0 = new BigDecimal(sqrt, mc);
BigDecimal two = BigDecimal.valueOf(2);
while ( true ) {
BigDecimal x1 = x0.subtract(x0.multiply(x0, mc).subtract(squareDecimal).divide(two.multiply(x0, mc), mc), mc);
String x1String = x1.toPlainString();
String x0String = x0.toPlainString();
if ( x1String.substring(0, x1String.length()-1).compareTo(x0String.substring(0, x0String.length()-1)) == 0 ) {
break;
}
x0 = x1;
}
return x0;
}
}
</syntaxhighlight>
{{out}}
<pre>
Ramanujan's Constant to 100 digits = 262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073130
Heegner numbers yielding 'almost' integers:
19 : 885479.77768015431949753789348171962682071428650186 ~ 885480 (diff ~ 0.222319845680502462106518280373)
43 : 884736743.99977746603490666193746207858537684739913 ~ 884736744 (diff ~ 0.000222533965093338062537921414623)
67 : 147197952743.99999866245422450682926131257862850818 ~ 147197952744 (diff ~ 0.00000133754577549317073868742137149)
163 : 262537412640768743.99999999999925007259719818568888 ~ 262537412640768744 (diff ~ 0.00000000000074992740280181431112)
</pre>
=={{header|Julia}}==
<syntaxhighlight lang="julia">
julia> a = BigFloat(MathConstants.e^(BigFloat(pi)))^(BigFloat(163.0)^0.5)
2.625374126407687439999999999992500725971981856888793538563373369908627075373427e+17
julia> 262537412640768744 - a
7.499274028018143111206461436626630091372924626572825942241598957614307213309258e-13
</syntaxhighlight>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">First[RealDigits[N[Exp[Pi Sqrt[163]], 200]]]
Table[
c = N[Exp[Pi Sqrt[h]], 40];
Log10[1 - FractionalPart[c]]
,
{h, {19, 43, 67, 163}}
]</syntaxhighlight>
{{out}}
<pre>{2,6,2,5,3,7,4,1,2,6,4,0,7,6,8,7,4,3,9,9,9,9,9,9,9,9,9,9,9,9,2,5,0,0,7,2,5,9,7,1,9,8,1,8,5,6,8,8,8,7,9,3,5,3,8,5,6,3,3,7,3,3,6,9,9,0,8,6,2,7,0,7,5,3,7,4,1,0,3,7,8,2,1,0,6,4,7,9,1,0,1,1,8,6,0,7,3,1,2,9,5,1,1,8,1,3,4,6,1,8,6,0,6,4,5,0,4,1,9,3,0,8,3,8,8,7,9,4,9,7,5,3,8,6,4,0,4,4,9,0,5,7,2,8,7,1,4,4,7,7,1,9,6,8,1,4,8,5,2,3,2,2,4,3,2,0,3,9,1,1,6,4,7,8,2,9,1,4,8,8,6,4,2,2,8,2,7,2,0,1,3,1,1,7,8,3,1,7,0,6}
(*Log10 of the difference between the number and an integer*)
{-0.653021767688625734085368753068345, -3.652603693775839429642336360, -5.87369134597671206721205, -12.1249807767}</pre>
=={{header|Nim}}==
{{libheader|nim-decimal}}
<syntaxhighlight lang="nim">import strformat, strutils
import decimal
setPrec(75)
let pi = newDecimal("3.1415926535897932384626433832795028841971693993751058209749445923078164")
proc eval(n: int): DecimalType =
result = exp(pi * sqrt(newDecimal(n)))
func format(n: DecimalType; d: Positive): string =
## Return the representation of "n" with "d" digits of precision.
let parts = ($n).split('.')
result = parts[0] & '.' & parts[1][0..<d]
echo "Ramanujan’s constant with 50 digits of precision:"
echo eval(163).format(50)
setPrec(50)
echo()
echo "Heegner numbers yielding 'almost' integers:"
for n in [19, 43, 67, 163]:
let x = eval(n)
let k = x.roundToInt
let d = x - k
let s = if d > 0: "+ " & $d else: "- " & $(-d)
echo &"{n:3}: {x}... = {k:>18} {s}..."</syntaxhighlight>
{{out}}
<pre>Ramanujan’s constant with 50 digits of precision:
262537412640768743.99999999999925007259719818568887935385633733699086
Heegner numbers yielding 'almost' integers:
19: 885479.77768015431949753789348171962682071428650216... = 885480 - 0.22231984568050246210651828037317928571349784...
43: 884736743.99977746603490666193746207858537684739914... = 884736744 - 0.00022253396509333806253792141462315260086...
67: 147197952743.99999866245422450682926131257862850810... = 147197952744 - 0.00000133754577549317073868742137149190...
163: 262537412640768743.99999999999925007259719818568865... = 262537412640768744 - 7.4992740280181431135e-13...</pre>
=={{header|Pari/GP}}==
<syntaxhighlight lang="parigp">\p 50
exp(Pi*sqrt(163))</syntaxhighlight>
{{out}}
<pre>
262537412640768743.99999999999925007259719818568888
</pre>
=={{header|Perl}}==
===Direct calculation===
{{trans|Sidef}}
<
use warnings;
use Math::AnyNum;
Line 16 ⟶ 438:
$x = Math::AnyNum->new($x);
my $prec =
local $Math::AnyNum::PREC = 4*$prec->round->numify;
}
Line 37 ⟶ 456:
my $n = Math::AnyNum::ipow($x, 3) + 744;
printf("%3s: %51s ≈ %18s (diff: %s)\n", $h, $c, $n, ($n - $c)->round(-32));
}</
{{out}}
<pre>
Line 50 ⟶ 469:
</pre>
===Continued fractions===
{{trans|Raku}}
<syntaxhighlight lang="perl">use strict;
use Math::AnyNum <as_dec rat>;
sub continued_fr {
my ($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);
</syntaxhighlight>
{{out}}
<pre>Ramanujan's constant
262537412640768743.9999999999992500725971981856888793538563</pre>
=={{header|Phix}}==
{{trans|Go}}
{{libheader|Phix/mpfr}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">without</span> <span style="color: #008080;">javascript_semantics</span> <span style="color: #000080;font-style:italic;">-- no mpfr_exp() under p2js (yet), sorry</span>
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.0"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (mpfr_set_default_prec[ision] has been renamed)</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #7060A8;">mpfr_set_default_precision</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">120</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (18 before, 100 after, plus 2 for kicks.)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">pi</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">mpfr_const_pi</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pi</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpfr_sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pi</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">mpfr_exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">t</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Ramanujan's constant to 100 decimal places is:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">(</span><span style="color: #000000;">163</span><span style="color: #0000FF;">),</span><span style="color: #000000;">100</span><span style="color: #0000FF;">))</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">heegners</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">19</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">96</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">43</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">960</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">67</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5280</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">163</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">640320</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">}</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nHeegner numbers yielding 'almost' integers:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(),</span> <span style="color: #000000;">qh</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">heegners</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">h0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">h1</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">heegners</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">qh</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">(</span><span style="color: #000000;">h0</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_ui_pow_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">h1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_add_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">744</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpfr_set_z</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpfr_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qh</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">qhs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qh</span><span style="color: #0000FF;">,</span><span style="color: #000000;">32</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">cs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">ts</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">32</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%3d: %51s ~= %18s (diff: %s)\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">h0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">qhs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">cs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ts</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Ramanujan's constant to 100 decimal places is:
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)
</pre>
=={{header|Python}}==
{{libheader|mpmath}}
<syntaxhighlight lang="python">from mpmath import mp
heegner = [19,43,67,163]
mp.dps = 50
x = mp.exp(mp.pi*mp.sqrt(163))
print("calculated Ramanujan's constant: {}".format(x))
print("Heegner numbers yielding 'almost' integers:")
for i in heegner:
print(" for {}: {} ~ {} error: {}".format(str(i),mp.exp(mp.pi*mp.sqrt(i)),round(mp.exp(mp.pi*mp.sqrt(i))),(mp.pi*mp.sqrt(i)) - round(mp.pi*mp.sqrt(i))))
</syntaxhighlight>
{{out}}
<pre>
calculated Ramanujan's constant: 262537412640768743.99999999999925007259719818568888
Heegner numbers yielding 'almost' integers:
for 19: 885479.77768015431949753789348171962682071428650187 ~ 885480 error: 0.30611510123230903757863689092534707729405221250933
for 43: 884736743.99977746603490666193746207858537684739915 ~ 884736744 error: -0.39919930568613989412676260444831671571796782935998
for 67: 147197952743.99999866245422450682926131257862850819 ~ 147197952744 error: -0.28495586484466040200154673774982799575003729030943
for 163: 262537412640768743.99999999999925007259719818568888 ~ 262537412640768736 error: 0.10916999113251975535008362290414005390053481224586
</pre>
=={{header|Raku}}==
(formerly Perl 6)
===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
Line 56 ⟶ 573:
[http://rosettacode.org/wiki/Calculating_the_value_of_e Euler's number], and
[http://rosettacode.org/wiki/Arithmetic-geometric_mean/Integer_roots integer roots]. Additional custom routines for exponentiation are used to ensure all computations are done with rationals, specifically <tt>FatRat</tt>s (rational numbers stored with arbitrary size numerator and denominator). The module <tt>Rat::Precise</tt> makes it simple to display these to a configurable precision.
<syntaxhighlight lang="raku"
# set the degree of precision for calculations
Line 76 ⟶ 593:
while (abs($mid - $exp) > ε) {
$sqr = sqrt($sqr);
if ($mid <= $exp) { $low = $mid; $acc
else { $high = $mid; $acc
$mid = ($low + $high) / 2;
}
Line 91 ⟶ 608:
for ^d {
given [ ($a + $g)/2, sqrt $a
$z -= (.[0] - $a)**2
$n += $n;
($a, $g) = @$_;
Line 102 ⟶ 619:
multi sqrt(FatRat $r --> FatRat) {
FatRat.new: sqrt($r.nude[0]
}
Line 117 ⟶ 634:
# calculation of 𝑒
sub postfix:<!> (Int $n) { (constant f = 1, |[\
sub 𝑒 (--> FatRat) { sum map { FatRat.new(1,.!) }, ^D }
Line 135 ⟶ 652:
constant 𝑒 = &𝑒();
my $Ramanujan = 𝑒**(π
say "Ramanujan's constant to 32 decimal places:\nActual: " ~
"262537412640768743.99999999999925007259719818568888\n" ~
Line 142 ⟶ 659:
say "Heegner numbers yielding 'almost' integers";
for 19, 96, 43, 960, 67, 5280, 163, 640320 -> $heegner, $x {
my $almost = 𝑒**(π
my $exact = $x
say format($exact, $almost);
}</
{{out}}
<pre>Ramanujan's constant to 32 decimal places:
Line 168 ⟶ 685:
Difference 0.000000000000750</pre>
===Continued fractions===
Ramanujan's constant can also be generated to an arbitrary precision using standard [https://en.wikipedia.org/wiki/Generalized_continued_fraction continued fraction formulas] for each component of the 𝑒**(π*√163) expression. Substantially slower than the first method.
<syntaxhighlight lang="raku" line>use Rat::Precise;
sub continued-fraction($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);</syntaxhighlight>
{{out}}
<pre>Ramanujan's constant to 32 decimal places:
262537412640768743.99999999999925007259719818568888</pre>
=={{header|REXX}}==
Instead of calculating <big> '''e''' </big> and <big><big><math>\pi</math></big></big> 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.
<syntaxhighlight lang="rexx">/*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</syntaxhighlight>
{{out|output|text= when using the default inputs:}}
<pre>
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
</pre>
=={{header|Ruby}}==
<syntaxhighlight lang="ruby">require "bigdecimal/math"
include BigMath
e, pi = E(200), PI(200)
[19, 43, 67, 163].each do |x|
puts "#{x}: #{(e ** (pi * BigMath.sqrt(BigDecimal(x), 200))).round(100).to_s("F")}"
end
</syntaxhighlight>
{{out}}
<pre>19: 885479.7776801543194975378934817196268207142865018553571526577110128809842286637202423189990118182067775711
43: 884736743.9997774660349066619374620785853768473991271391609175146278344881148747592189635643106023717101372606
67: 147197952743.999998662454224506829261312578628508183312503816712633371282105122950998831523502041379242353370629
163: 262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129511813461860645042
</pre>
=={{header|Sidef}}==
<
local Num!PREC = *"#{4*round((Num.pi*√x)/log(
}
Line 183 ⟶ 789:
var n = (x**3 + 744)
printf("%3s: %51s ≈ %18s (diff: %s)\n", h, c, n, n-Num(c))
})</
{{out}}
<pre>
Line 190 ⟶ 796:
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)
</pre>
=={{header|Wren}}==
{{libheader|Wren-big}}
{{libheader|Wren-fmt}}
Wren has BigRat but not BigFloat which means we are lacking both a 'big' value for pi and an arbitrary precision exponential method.
I've therefore hard-coded a value for pi with 70 decimal places (more than enough for this task) and written a 'big' exponential function using the Taylor series for e(x). The latter requires a lot of iterations and is therefore quite slow (takes about 5 seconds to calculate the Ramanujan constant). However, this is acceptable for a scripting language such as Wren.
<syntaxhighlight lang="wren">import "./big" for BigRat
import "./fmt" for Fmt
var pi = "3.1415926535897932384626433832795028841971693993751058209749445923078164"
var bigPi = BigRat.fromDecimal(pi)
var exp = Fn.new { |x, p|
var sum = x + 1
var prevTerm = x
var k = 2
var eps = BigRat.fromDecimal("0.5e-%(p)")
while (true) {
var nextTerm = prevTerm * x / k
sum = sum + nextTerm
if (nextTerm < eps) break
// speed up calculations by limiting precision to 'p' places
prevTerm = BigRat.fromDecimal(nextTerm.toDecimal(p))
k = k + 1
}
return sum
}
var ramanujan = Fn.new { |n, dp|
var e = bigPi * BigRat.new(n, 1).sqrt(70)
return exp.call(e, 70)
}
System.print("Ramanujan's constant to 32 decimal places is:")
System.print(ramanujan.call(163, 32).toDecimal(32))
var heegner = [19, 43, 67, 163]
System.print("\nHeegner numbers yielding almost integers:")
for (h in heegner) {
var r = ramanujan.call(h, 32)
var rc = r.ceil
var diff = (rc - r).toDecimal(32)
r = r.toDecimal(32)
rc = rc.toDecimal(32)
Fmt.print("$3d: $51s ≈ $18s (diff: $s)", h, r, rc, diff)
}</syntaxhighlight>
{{out}}
<pre>
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)
| |||