Almkvist-Giullera formula for pi

The Almkvist-Giullera formula for calculating 1/π² is based on the Calabi-Yau differential equations of order 4 and 5, which were originally used to describe certain manifolds in string theory.

The formula is:

1/π² = (2⁵/3) ∑₀^∞ ((6n)! / (n!⁶))(532n² + 126n + 9) / 1000²ⁿ⁺¹

This formula can be used to calculate the constant π^-2, and thus to calculate π.

Note that, because the product of all terms but the power of 1000 can be calculated as an integer, the terms in the series can be separated into a large integer term:

(2⁵) (6n)! (532n² + 126n + 9) / (3(n!)⁶) (***)

multiplied by a negative integer power of 10:

10^{-(6n + 3)}

Task

Print the integer portions (the starred formula, which is without the power of 1000 divisor) of the first 10 terms of the series.
Use the complete formula to calculate and print π to 70 decimal digits of precision.

Reference

Gert Almkvist and Jesús Guillera, Ramanujan-like series for 1/π² and string theory, Experimental Mathematics, 21 (2012), page 2, formula 1.

11l[edit]

Translation of: C#

F isqrt(BigInt =x)
   BigInt q = 1
   BigInt r = 0
   BigInt t
   L q <= x
      q *= 4
   L q > 1
      q I/= 4
      t = x - r - q
      r I/= 2
      I t >= 0
         x = t
         r += q
   R r
 
F dump(=digs, show)
   V gb = 1
   digs++
   V dg = digs + gb
   BigInt t1 = 1
   BigInt t2 = 9
   BigInt t3 = 1
   BigInt te
   BigInt su = 0
   V t = BigInt(10) ^ (I dg <= 60 {0} E dg - 60)
   BigInt d = -1
   BigInt _fn_ = 1
 
   V n = 0
   L n < dg
      I n > 0
         t3 *= BigInt(n) ^ 6
      te = t1 * t2 I/ t3
      V z = dg - 1 - n * 6
      I z > 0
         te *= BigInt(10) ^ z
      E
         te I/= BigInt(10) ^ -z
      I show & n < 10
         print(‘#2 #62’.format(n, te * 32 I/ 3 I/ t))
      su += te
 
      I te < 10
         I show
            digs--
            print("\n#. iterations required for #. digits after the decimal point.\n".format(n, digs))
         L.break
 
      L(j) n * 6 + 1 .. n * 6 + 6
         t1 *= j
      d += 2
      t2 += 126 + 532 * d
 
      n++
 
   V s = String(isqrt(BigInt(10) ^ (dg * 2 + 3) I/ su I/ 32 * 3 * BigInt(10) ^ (dg + 5)))
   R s[0]‘.’s[1 .+ digs]
 
print(dump(70, 1B))

Output:

 0  9600000000000000000000000000000000000000000000000000000000000
 1   512256000000000000000000000000000000000000000000000000000000
 2    19072247040000000000000000000000000000000000000000000000000
 3      757482485760000000000000000000000000000000000000000000000
 4       31254615037245600000000000000000000000000000000000000000
 5        1320787470322549142065152000000000000000000000000000000
 6          56727391979308908329225994240000000000000000000000000
 7           2465060024817298714011276371558400000000000000000000
 8            108065785435463945367040747443956640000000000000000
 9              4770177939159496628747057049083997888000000000000

53 iterations required for 70 digits after the decimal point.

3.1415926535897932384626433832795028841971693993751058209749445923078164

A little challenging due to lack of BigFloat or BigRational. Note the extended precision integers displayed for each term, not extended precision floats. Also features the next term based on the last term, rather than computing each term from scratch. And the multiply by 32, divide by 3 is reserved for final sum, instead of each term (except for the 0..9th displayed terms).

using System;
using BI = System.Numerics.BigInteger;
using static System.Console;
 
class Program {
 
  static BI isqrt(BI x) { BI q = 1, r = 0, t; while (q <= x) q <<= 2; while (q > 1) {
    q >>= 2; t = x - r - q; r >>= 1; if (t >= 0) { x = t; r += q; } } return r; }
 
  static string dump(int digs, bool show = false) {
    int gb = 1, dg = ++digs + gb, z;
    BI t1 = 1, t2 = 9, t3 = 1, te, su = 0,
       t = BI.Pow(10, dg <= 60 ? 0 : dg - 60), d = -1, fn = 1;
    for (BI n = 0; n < dg; n++) {
      if (n > 0) t3 *= BI.Pow(n, 6);
      te = t1 * t2 / t3;
      if ((z = dg - 1 - (int)n * 6) > 0) te *= BI.Pow (10, z);
      else te /= BI.Pow (10, -z);
      if (show && n < 10)
        WriteLine("{0,2} {1,62}", n, te * 32 / 3 / t);
      su += te; if (te < 10) {
        if (show) WriteLine("\n{0} iterations required for {1} digits " +
        "after the decimal point.\n", n, --digs); break; }
      for (BI j = n * 6 + 1; j <= n * 6 + 6; j++) t1 *= j;
      t2 += 126 + 532 * (d += 2);
    }
    string s = string.Format("{0}", isqrt(BI.Pow(10, dg * 2 + 3) /
      su / 32 * 3 * BI.Pow((BI)10, dg + 5)));
    return s[0] + "." + s.Substring(1, digs); }
 
  static void Main(string[] args) {
    WriteLine(dump(70, true)); }
}

Output:

 0  9600000000000000000000000000000000000000000000000000000000000
 1   512256000000000000000000000000000000000000000000000000000000
 2    19072247040000000000000000000000000000000000000000000000000
 3      757482485760000000000000000000000000000000000000000000000
 4       31254615037245600000000000000000000000000000000000000000
 5        1320787470322549142065152000000000000000000000000000000
 6          56727391979308908329225994240000000000000000000000000
 7           2465060024817298714011276371558400000000000000000000
 8            108065785435463945367040747443956640000000000000000
 9              4770177939159496628747057049083997888000000000000

53 iterations required for 70 digits after the decimal point.

3.1415926535897932384626433832795028841971693993751058209749445923078164

C++[edit]

Library: Boost

Library: GMP

#include <boost/multiprecision/cpp_dec_float.hpp>
#include <boost/multiprecision/gmp.hpp>
#include <iomanip>
#include <iostream>
 
namespace mp = boost::multiprecision;
using big_int = mp::mpz_int;
using big_float = mp::cpp_dec_float_100;
using rational = mp::mpq_rational;
 
big_int factorial(int n) {
    big_int result = 1;
    for (int i = 2; i <= n; ++i)
        result *= i;
    return result;
}
 
// Return the integer portion of the nth term of Almkvist-Giullera sequence.
big_int almkvist_giullera(int n) {
    return factorial(6 * n) * 32 * (532 * n * n + 126 * n + 9) /
           (pow(factorial(n), 6) * 3);
}
 
int main() {
    std::cout << "n |                  Integer portion of nth term\n"
              << "------------------------------------------------\n";
    for (int n = 0; n < 10; ++n)
        std::cout << n << " | " << std::setw(44) << almkvist_giullera(n)
                  << '\n';
 
    big_float epsilon(pow(big_float(10), -70));
    big_float prev = 0, pi = 0;
    rational sum = 0;
    for (int n = 0;; ++n) {
        rational term(almkvist_giullera(n), pow(big_int(10), 6 * n + 3));
        sum += term;
        pi = sqrt(big_float(1 / sum));
        if (abs(pi - prev) < epsilon)
            break;
        prev = pi;
    }
    std::cout << "\nPi to 70 decimal places is:\n"
              << std::fixed << std::setprecision(70) << pi << '\n';
}

Output:

n |                  Integer portion of nth term
------------------------------------------------
0 |                                           96
1 |                                      5122560
2 |                                 190722470400
3 |                             7574824857600000
4 |                        312546150372456000000
5 |                   13207874703225491420651520
6 |               567273919793089083292259942400
7 |          24650600248172987140112763715584000
8 |     1080657854354639453670407474439566400000
9 | 47701779391594966287470570490839978880000000

Pi to 70 decimal places is:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Common Lisp[edit]

Library: computable-reals

Translation of: Raku

(ql:quickload :computable-reals :silent t)
(use-package :computable-reals)
(setq *print-prec* 70)
(defparameter *iterations* 52)
 
; factorial using computable-reals multiplication op to keep precision
(defun !r (n)
  (let ((p 1))
    (loop for i from 2 to n doing (setq p (*r p i)))
    p))
 
; the nth integer term
(defun integral (n)
   (let* ((polynomial (+r (*r 532 n n) (*r 126 n) 9))
          (numer  (*r 32 (!r (*r 6 n)) polynomial))
          (denom  (*r 3 (expt-r (!r n) 6))))
    (/r  numer denom)))
 
 
; the exponent for 10 in the nth term of the series
(defun power (n) (- 3 (* 6 (1+ n))))
 
; the nth term of the series
(defun almkvist-giullera (n)
  (/r (integral n) (expt-r 10 (abs (power n)))))
 
; the sum of the first n terms
(defun almkvist-giullera-sigma (n)
  (let ((s 0)) 
    (loop for i from 0 to n doing (setq s (+r s (almkvist-giullera i))))
    s))
 
; the approximation to pi after n terms
(defun almkvist-giullera-pi (n)
  (sqrt-r (/r 1 (almkvist-giullera-sigma n))))
 
(format t "~A. ~44A~4A ~A~%" "N" "Integral part of Nth term" "×10^" "=Actual value of Nth term")
(loop for i from 0 to 9 doing
  (format t "~&~a. ~44d ~3d " i (integral i) (power i))
  (finish-output *standard-output*)
  (print-r (almkvist-giullera i) 50 nil))
 
(format t "~%~%Pi after ~a iterations: " *iterations*)
(print-r (almkvist-giullera-pi *iterations*) *print-prec*)

Output:

N. Integral part of Nth term                   ×10^ =Actual value of Nth term
0.                                           96  -3 +0.09600000000000000000000000000000000000000000000000...
1.                                      5122560  -9 +0.00512256000000000000000000000000000000000000000000...
2.                                 190722470400 -15 +0.00019072247040000000000000000000000000000000000000...
3.                             7574824857600000 -21 +0.00000757482485760000000000000000000000000000000000...
4.                        312546150372456000000 -27 +0.00000031254615037245600000000000000000000000000000...
5.                   13207874703225491420651520 -33 +0.00000001320787470322549142065152000000000000000000...
6.               567273919793089083292259942400 -39 +0.00000000056727391979308908329225994240000000000000...
7.          24650600248172987140112763715584000 -45 +0.00000000002465060024817298714011276371558400000000...
8.     1080657854354639453670407474439566400000 -51 +0.00000000000108065785435463945367040747443956640000...
9. 47701779391594966287470570490839978880000000 -57 +0.00000000000004770177939159496628747057049083997888...

Pi after 52 iterations: 
+3.1415926535897932384626433832795028841971693993751058209749445923078164...

dc[edit]

Translation of: Common Lisp

[* factorial *]sz
[ 1 Sp [ d lp * sp 1 - d 1 <f ]Sf d 1 <f Lfsz sz Lp ]sF
 
[* nth integral term *]sz
[ sn 32 6 ln * lFx 532 ln * ln * 126 ln * + 9 + * * 3 ln lFx 6 ^ * / ]sI
 
[* nth exponent of 10 *]sz
[ 1 + 6 * 3 r - ]sE
 
[* nth term in series *]sz
[ d lIx r 10 r lEx _1 * ^ / ]sA
 
[* sum of the first n terms *]sz
[ [li lAx ls + ss li 1 - d si 0 r !<L]sL si 0ss lLx ls]sS
 
[* approximation of pi after n terms *]sz
[ lSx 1 r / v ]sP
 
[* count digits in a number *]sz
[sn 0 sd lCx ld]sD
[ld 1 + sd ln 10 0k / d sn 0 !=C]sC
 
[* print a number in a given column width *]sz
[sw d lDx si lw li <T n]sW
[[ ]n li 1 + si lw li <T]sT
 
[* main loop: print values for first 10 terms *]sz
[N. Integral part of Nth term .................. × 10^ =Actual value of Nth term]p
0 sj
[
  lj n [. ]n
  lj lIx 0k 1 / 44 lWx [ ]n
  lj lEx 4 lWx [ ]n
  lj 99k lAx 50k 1 / p
  lj 1 + d sj 10 >M
] sM
lMx
 
[]p
 
[* print resulting value of pi to 70 places *]sz
[Pi after ]n 52n [ iterations:]p
99k 52 lPx 70k 1 / p

Output:

N. Integral part of Nth term .................. × 10^ =Actual value of Nth term
0.                                           96    -3 .09600000000000000000000000000000000000000000000000
1.                                      5122560    -9 .00512256000000000000000000000000000000000000000000
2.                                 190722470400   -15 .00019072247040000000000000000000000000000000000000
3.                             7574824857600000   -21 .00000757482485760000000000000000000000000000000000
4.                        312546150372456000000   -27 .00000031254615037245600000000000000000000000000000
5.                   13207874703225491420651520   -33 .00000001320787470322549142065152000000000000000000
6.               567273919793089083292259942400   -39 .00000000056727391979308908329225994240000000000000
7.          24650600248172987140112763715584000   -45 .00000000002465060024817298714011276371558400000000
8.     1080657854354639453670407474439566400000   -51 .00000000000108065785435463945367040747443956640000
9. 47701779391594966287470570490839978880000000   -57 .00000000000004770177939159496628747057049083997888

3.1415926535897932384626433832795028841971693993751058209749445923078\
164

Erlang[edit]

This version uses integer math only (does not resort to a rational number package) Since the denominator is always a power of 10, it's possible to just keep track of the log of the denominator and scale the numerator accordingly; to keep track of the accuracy we get the order of magnitude of the difference between terms by subtracting the log of the numerator from the log of the denominator, so again, no rational arithmetic is needed.

However, Erlang does not have much in the way of calculating with large integers beyond basic arithmetic, so this version implements integer powers, logs, square roots, as well as the factorial function.

 
-mode(compile).
 
% Integer math routines: factorial, power, square root, integer logarithm.
%
fac(N) -> fac(N, 1).
fac(N, A) when N < 2 -> A;
fac(N, A) -> fac(N - 1, N*A).
 
 
pow(_, N) when N < 0 -> pow_domain_error;
pow(2, N) -> 1 bsl N;
pow(A, N) -> ipow(A, N).
 
ipow(_, 0) -> 1;
ipow(A, 1) -> A;
ipow(A, 2) -> A*A;
ipow(A, N) ->
    case N band 1 of
        0 -> X = ipow(A, N bsr 1), X*X;
        1 -> A * ipow(A, N - 1)
    end.
 
% integer logarithm, based on Zeckendorf representations of integers.
%    https://www.keithschwarz.com/interesting/code/?dir=zeckendorf-logarithm 
%    we need this, since the exponents get larger than IEEE-754 double can handle.
 
lognext({A, B, S, T}) -> {B, A+B, T, S*T}.
logprev({A, B, S, T}) -> {B-A, A, T div S, S}.
 
ilog(A, B) when (A =< 0) or (B < 2) -> ilog_domain_error;
ilog(A, B) ->
    UBound = bracket(A, {0, 1, 1, B}),
    backlog(A, UBound, 0).
 
bracket(A, State = {_, _, _, T}) when T > A -> State;
bracket(A, State) -> bracket(A, lognext(State)).
 
backlog(_, {0, _, 1, _}, Log) -> Log;
backlog(N, State = {A, _, S, _}, Log) when S =< N ->
    backlog(N div S, logprev(State), Log + A);
backlog(N, State, Log) -> backlog(N, logprev(State), Log).
 
 
isqrt(N) when N < 0 -> isqrt_domain_error;
isqrt(N) ->
    X0 = pow(2, ilog(N, 2) div 2),
    iterate(N, newton(X0, N), N).
 
iterate(A, B, _) when A =< B -> A;
iterate(_, B, N) -> iterate(B, newton(B, N), N).
 
newton(X, N) -> (X + N div X) div 2.
 
 
% With this out of the way, we can get down to some serious calculation.
%
term(N) -> {  % returns numerator and log10 of the denominator.
    (fac(6*N)*(N*(532*N + 126) + 9) bsl 5) div (3*pow(fac(N), 6)),
    6*N + 3
    }.
 
neg_term({N, D}) -> {-N, D}.
abs_term({N, D}) -> {abs(N), D}.
 
add_term(T1 = {_, D1}, T2 = {_, D2}) when D1 > D2 -> add_term(T2, T1);
add_term({N1, D1}, {N2, D2}) ->
    Scale = pow(10, D2 - D1),
    {N1*Scale + N2, D2}.
 
calculate(Prec) -> calculate(Prec, {0, 0}, 0).
calculate(Prec, T0, K) ->
    T1 = add_term(T0, term(K)),
    {N, D} = abs_term(add_term(neg_term(T1), T0)),
    Accuracy = D - ilog(N, 10),
    if
        Accuracy < Prec -> calculate(Prec, T1, K + 1);
        true -> T1
    end.
 
get_pi(Prec) ->
    {N0, D0} = calculate(Prec),
    % from the term, t = n0/10^d0, calculate 1/√t
    % if the denominator is an odd power of 10, add 1 to the denominator and multiply the numerator by 10.
    {N, D} = case D0 band 1 of
        0 -> {N0, D0};
        1 -> {10*N0, D0 + 1}
    end,
    [Three|Rest] = lists:sublist(
            integer_to_list(pow(10, D) div isqrt(N)), Prec),
    [Three, $. | Rest].
 
show_term({A, Decimals}) ->
    Str = integer_to_list(A),
    [$0, $.] ++ lists:duplicate(Decimals - length(Str), $0) ++ Str.
 
main(_) ->
    Terms = [term(N) || N <- lists:seq(0, 9)],
    io:format("The first 10 terms as scaled decimals are:~n"),
    [io:format("    ~s~n", [show_term(T)]) || T <- Terms],
    io:format("~nThe sum of these terms (pi^-2) is ~s~n",
                [show_term(lists:foldl(fun add_term/2, {0, 0}, Terms))]),
    Pi70 = get_pi(71),
    io:format("~npi to 70 decimal places:~n"),
    io:format("~s~n", [Pi70]).

Output:

The first 10 terms as scaled decimals are:
    0.096
    0.005122560
    0.000190722470400
    0.000007574824857600000
    0.000000312546150372456000000
    0.000000013207874703225491420651520
    0.000000000567273919793089083292259942400
    0.000000000024650600248172987140112763715584000
    0.000000000001080657854354639453670407474439566400000
    0.000000000000047701779391594966287470570490839978880000000

The sum of these terms (pi^-2) is 0.101321183642335555356499725503850584160514406378880000000

pi to 70 decimal places:
3.1415926535897932384626433832795028841971693993751058209749445923078164

F#[edit]

This task uses Isqrt_(integer_square_root)_of_X#F.23

 
// Almkvist-Giullera formula for pi. Nigel Galloway: August 17th., 2021
let factorial(n:bigint)=MathNet.Numerics.SpecialFunctions.Factorial n
let fN g=(532I*g*g+126I*g+9I)*(factorial(6I*g))/(3I*(factorial g)**6)
[0..9]|>Seq.iter(bigint>>fN>>(*)32I>>printfn "%A\n")
let _,n=Seq.unfold(fun(n,g)->let n=n*(10I**6)+fN g in Some(Isqrt((10I**(145+6*(int g)))/(32I*n)),(n,g+1I)))(0I,0I)|>Seq.pairwise|>Seq.find(fun(n,g)->n=g)
printfn $"""pi to 70 decimal places is %s{(n.ToString()).Insert(1,".")}"""

Output:

96
5122560
190722470400
7574824857600000
312546150372456000000
13207874703225491420651520
567273919793089083292259942400
24650600248172987140112763715584000
1080657854354639453670407474439566400000
47701779391594966287470570490839978880000000

pi to 70 decimal places is 3.14159265358979323846264338327950288419716939937510582097494459230781640

Factor[edit]

Works with: Factor version 0.99 2020-08-14

USING: continuations formatting io kernel locals math
math.factorials math.functions sequences ;
 
:: integer-term ( n -- m )
    32 6 n * factorial * 532 n sq * 126 n * + 9 + *
    n factorial 6 ^ 3 * / ;
 
: exponent-term ( n -- m ) 6 * 3 + neg ;
 
: nth-term ( n -- x )
    [ integer-term ] [ exponent-term 10^ * ] bi ;
 
! Factor doesn't have an arbitrary-precision square root afaik,
! so make one using Heron's method.
 
: sqrt-approx ( r x -- r' x ) [ over / + 2 / ] keep ;
 
:: almkvist-guillera ( precision -- x )
    0 0 :> ( summed! next-add! )
    [
        100,000,000 <iota> [| n |
            summed n nth-term + next-add!
            next-add summed - abs precision neg 10^ <
            [ return ] when
            next-add summed!
        ] each
    ] with-return
    next-add ;
 
CONSTANT: 1/pi 113/355  ! Use as initial guess for square root approximation
 
: pi ( -- )
    1/pi 70 almkvist-guillera 5 [ sqrt-approx ] times
    drop recip "%.70f\n" printf ;
 
! Task
"N                               Integer Portion  Pow  Nth Term (33 dp)" print
89 CHAR: - <repetition> print
10 [
    dup [ integer-term ] [ exponent-term ] [ nth-term ] tri
    "%d  %44d  %3d  %.33f\n" printf
] each-integer nl
"Pi to 70 decimal places:" print pi

Output:

N                               Integer Portion  Pow  Nth Term (33 dp)
-----------------------------------------------------------------------------------------
0                                            96   -3  0.096000000000000000000000000000000
1                                       5122560   -9  0.005122560000000000000000000000000
2                                  190722470400  -15  0.000190722470400000000000000000000
3                              7574824857600000  -21  0.000007574824857600000000000000000
4                         312546150372456000000  -27  0.000000312546150372456000000000000
5                    13207874703225491420651520  -33  0.000000013207874703225491420651520
6                567273919793089083292259942400  -39  0.000000000567273919793089083292260
7           24650600248172987140112763715584000  -45  0.000000000024650600248172987140113
8      1080657854354639453670407474439566400000  -51  0.000000000001080657854354639453670
9  47701779391594966287470570490839978880000000  -57  0.000000000000047701779391594966287

Pi to 70 decimal places:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Go[edit]

Translation of: Wren

package main
 
import (
    "fmt"
    "math/big"
    "strings"
)
 
func factorial(n int64) *big.Int {
    var z big.Int
    return z.MulRange(1, n)
}
 
var one = big.NewInt(1)
var three = big.NewInt(3)
var six = big.NewInt(6)
var ten = big.NewInt(10)
var seventy = big.NewInt(70)
 
func almkvistGiullera(n int64, print bool) *big.Rat {
    t1 := big.NewInt(32)
    t1.Mul(factorial(6*n), t1)
    t2 := big.NewInt(532*n*n + 126*n + 9)
    t3 := new(big.Int)
    t3.Exp(factorial(n), six, nil)
    t3.Mul(t3, three)
    ip := new(big.Int)
    ip.Mul(t1, t2)
    ip.Quo(ip, t3)
    pw := 6*n + 3
    t1.SetInt64(pw)
    tm := new(big.Rat).SetFrac(ip, t1.Exp(ten, t1, nil))
    if print {
        fmt.Printf("%d  %44d  %3d  %-35s\n", n, ip, -pw, tm.FloatString(33))
    }
    return tm
}
 
func main() {
    fmt.Println("N                               Integer Portion  Pow  Nth Term (33 dp)")
    fmt.Println(strings.Repeat("-", 89))
    for n := int64(0); n < 10; n++ {
        almkvistGiullera(n, true)
    }
 
    sum := new(big.Rat)
    prev := new(big.Rat)
    pow70 := new(big.Int).Exp(ten, seventy, nil)
    prec := new(big.Rat).SetFrac(one, pow70)
    n := int64(0)
    for {
        term := almkvistGiullera(n, false)
        sum.Add(sum, term)
        z := new(big.Rat).Sub(sum, prev)
        z.Abs(z)
        if z.Cmp(prec) < 0 {
            break
        }
        prev.Set(sum)
        n++
    }
    sum.Inv(sum)
    pi := new(big.Float).SetPrec(256).SetRat(sum)
    pi.Sqrt(pi)
    fmt.Println("\nPi to 70 decimal places is:")
    fmt.Println(pi.Text('f', 70))
}

Output:

N                               Integer Portion  Pow  Nth Term (33 dp)
-----------------------------------------------------------------------------------------
0                                            96   -3  0.096000000000000000000000000000000
1                                       5122560   -9  0.005122560000000000000000000000000
2                                  190722470400  -15  0.000190722470400000000000000000000
3                              7574824857600000  -21  0.000007574824857600000000000000000
4                         312546150372456000000  -27  0.000000312546150372456000000000000
5                    13207874703225491420651520  -33  0.000000013207874703225491420651520
6                567273919793089083292259942400  -39  0.000000000567273919793089083292260
7           24650600248172987140112763715584000  -45  0.000000000024650600248172987140113
8      1080657854354639453670407474439566400000  -51  0.000000000001080657854354639453670
9  47701779391594966287470570490839978880000000  -57  0.000000000000047701779391594966287

Pi to 70 decimal places is:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Haskell[edit]

Library: numbers

Translation of: Common Lisp

import Control.Monad
import Data.Number.CReal
import GHC.Integer
import Text.Printf
 
iterations = 52
main = do
  printf "N. %44s %4s %s\n" 
          "Integral part of Nth term" "×10^" "=Actual value of Nth term"
 
  forM_ [0..9] $ \n ->
    printf "%d. %44d %4d %s\n" n
                               (almkvistGiulleraIntegral n)
                               (tenExponent n)
                               (showCReal 50 (almkvistGiullera n))
 
  printf "\nPi after %d iterations:\n" iterations
  putStrLn $ showCReal 70 $ almkvistGiulleraPi iterations
 
-- The integral part of the Nth term in the Almkvist-Giullera series
almkvistGiulleraIntegral n =
  let polynomial  = (532 `timesInteger` n `timesInteger` n) `plusInteger` (126 `timesInteger` n) `plusInteger` 9
      numerator   = 32 `timesInteger` (facInteger (6 `timesInteger` n)) `timesInteger` polynomial
      denominator = 3 `timesInteger` (powInteger (facInteger n) 6)
   in numerator `divInteger` denominator
 
-- The exponent for 10 in the Nth term of the series
tenExponent n = 3 `minusInteger` (6 `timesInteger` (1 `plusInteger` n))
 
-- The Nth term in the series (integral * 10^tenExponent)
almkvistGiullera n = fromInteger (almkvistGiulleraIntegral n) / fromInteger (powInteger 10 (abs (tenExponent n)))
 
-- The sum of the first N terms
almkvistGiulleraSum n = sum $ map almkvistGiullera [0 .. n]
 
-- The approximation of pi from the first N terms
almkvistGiulleraPi n = sqrt $ 1 / almkvistGiulleraSum n
 
-- Utility: factorial for arbitrary-precision integers
facInteger n = if n `leInteger` 1 then 1 else n `timesInteger` facInteger (n `minusInteger` 1)
 
-- Utility: exponentiation for arbitrary-precision integers
powInteger 1 _ = 1
powInteger _ 0 = 1
powInteger b 1 = b
powInteger b e = b `timesInteger` powInteger b (e `minusInteger` 1)

Output:

N.                    Integral part of Nth term ×10^ =Actual value of Nth term
0.                                           96   -3 0.096
1.                                      5122560   -9 0.00512256
2.                                 190722470400  -15 0.0001907224704
3.                             7574824857600000  -21 0.0000075748248576
4.                        312546150372456000000  -27 0.000000312546150372456
5.                   13207874703225491420651520  -33 0.00000001320787470322549142065152
6.               567273919793089083292259942400  -39 0.0000000005672739197930890832922599424
7.          24650600248172987140112763715584000  -45 0.000000000024650600248172987140112763715584
8.     1080657854354639453670407474439566400000  -51 0.0000000000010806578543546394536704074744395664
9. 47701779391594966287470570490839978880000000  -57 0.00000000000004770177939159496628747057049083997888

Pi after 52 iterations:
3.1415926535897932384626433832795028841971693993751058209749445923078164

J[edit]

This solution just has it hard-coded that 53 iterations is necessary for 70 decimals. It would be possible to write a loop with a test, though in practice it would also be acceptable to just experiment to find the number of iterations.

sqrt is noticeably slow, bringing execution time to over 1 second. I'm not sure if it's because it's coded imperatively using traditional loops vs. J point-free style, or if it's due to the fact that the numbers are very large. I suspect the latter since it only takes 4 iterations of Heron's method to get the square root.

 
numerator =: monad define "0
    (3 * (! x: y)^6) %~ 32 * (!6x*y) * (y*(126 + 532*y)) + 9x
)
 
term =: numerator % 10x ^ 3 + 6&*
 
echo 'The first 10 numerators are:'
echo ,. numerator i.10
 
echo ''
echo 'The sum of the first 10 terms (pi^-2) is ', 0j15 ": +/ term i.10
 
heron =: [: -: ] + %
 
sqrt =: dyad define NB. usage: x0 tolerance sqrt x
                    NB. e.g.: (1, %10^100x) sqrt 2 -> √2 to 100 decimals as a ratio p/q
    x0  =. }: x
    eps =. }. x
    x1  =. y heron x0
    while. (| x1 - x0) > eps do.
        x2 =. y heron x1
        x0 =. x1
        x1 =. x2
    end.
    x1
)
 
pi70 =. (355r113, %10^70x) sqrt % +/ term i.53
echo ''
echo 'pi to 70 decimals: ', 0j70 ": pi70
exit ''

Output:

The first 10 numerators are:
                                          96
                                     5122560
                                190722470400
                            7574824857600000
                       312546150372456000000
                  13207874703225491420651520
              567273919793089083292259942400
         24650600248172987140112763715584000
    1080657854354639453670407474439566400000
47701779391594966287470570490839978880000000

The sum of the first 10 terms (pi^-2) is 0.101321183642336

pi to 70 decimals: 3.1415926535897932384626433832795028841971693993751058209749445923078164

JavaScript[edit]

Translation of: Common Lisp

Works with: Node.js version 13+

Library: bigfloat-esnext

Library: es-main

to support use of module as main code

import esMain from 'es-main';
import { BigFloat, set_precision as SetPrecision } from 'bigfloat-esnext';
 
const Iterations = 52;
 
export const demo = function() {
  SetPrecision(-75);
  console.log("N." + "Integral part of Nth term".padStart(45) + " ×10^ =Actual value of Nth term");
  for (let i=0; i<10; i++) {
    let line = `${i}. `;
    line += `${integral(i)} `.padStart(45);
    line += `${tenExponent(i)} `.padStart(5);
    line += nthTerm(i);
    console.log(line);
  }
 
  let pi = approximatePi(Iterations);
  SetPrecision(-70);
  pi = pi.dividedBy(100000).times(100000);
  console.log(`\nPi after ${Iterations} iterations: ${pi}`)
}
 
export const bigFactorial = n => n <= 1n ? 1n : n * bigFactorial(n-1n);
 
// the nth integer term
export const integral = function(i) {
  let n = BigInt(i);
  const polynomial  = 532n * n * n + 126n * n + 9n;
  const numerator   = 32n * bigFactorial(6n * n) * polynomial;
  const denominator = 3n * bigFactorial(n) ** 6n;
  return numerator / denominator;
}
 
// the exponent for 10 in the nth term of the series
export const tenExponent = n => 3n - 6n * (BigInt(n) + 1n);
 
// the nth term of the series
export const nthTerm = n =>
  new BigFloat(integral(n)).dividedBy(new BigFloat(10n ** -tenExponent(n)))
 
// the sum of the first n terms
export const sumThrough = function(n) {
  let sum = new BigFloat(0);
  for (let i=0; i<=n; ++i) {
    sum = sum.plus(nthTerm(i));
  }
  return sum;
}
 
// the approximation to pi after n terms
export const approximatePi  = n =>
   new BigFloat(1).dividedBy(sumThrough(n)).sqrt();
 
if (esMain(import.meta))
   demo();

Output:

N.                    Integral part of Nth term ×10^ =Actual value of Nth term
0.                                           96   -3 0.096
1.                                      5122560   -9 0.00512256
2.                                 190722470400  -15 0.0001907224704
3.                             7574824857600000  -21 0.0000075748248576
4.                        312546150372456000000  -27 0.000000312546150372456
5.                   13207874703225491420651520  -33 0.00000001320787470322549142065152
6.               567273919793089083292259942400  -39 0.0000000005672739197930890832922599424
7.          24650600248172987140112763715584000  -45 0.000000000024650600248172987140112763715584
8.     1080657854354639453670407474439566400000  -51 0.0000000000010806578543546394536704074744395664
9. 47701779391594966287470570490839978880000000  -57 0.00000000000004770177939159496628747057049083997888

Pi after 52 iterations: 3.1415926535897932384626433832795028841971693993751058209749445923078164

Julia[edit]

using Formatting
 
setprecision(BigFloat, 300)
 
function integerterm(n)
    p = BigInt(532) * n * n + BigInt(126) * n + 9
    return (p * BigInt(2)^5 * factorial(BigInt(6) * n)) ÷ (3 * factorial(BigInt(n))^6)
end
 
exponentterm(n) = -(6n + 3)
 
nthterm(n) = integerterm(n) * big"10.0"^exponentterm(n)
 
println("  N                       Integer Term              Power of 10     Nth Term")
println("-"^90)
for n in 0:9
    println(lpad(n, 3), lpad(integerterm(n), 48), lpad(exponentterm(n), 4),
        lpad(format("{1:22.19e}", nthterm(n)), 35))
end
 
function AlmkvistGuillera(floatprecision)
    summed = nthterm(0)
    for n in 1:10000000
        next = summed + nthterm(n)
        if abs(next - summed) < big"10.0"^(-floatprecision)
            return next
        end
        summed = next
    end
end
 
println("\nπ to 70 digits is ", format(big"1.0" / sqrt(AlmkvistGuillera(70)), precision=70))
 
println("Computer π is     ", format(π + big"0.0", precision=70))

Output:

  N                       Integer Term              Power of 10     Nth Term
------------------------------------------------------------------------------------------
  0                                              96  -3          9.6000000000000000000e-02
  1                                         5122560  -9          5.1225600000000000000e-03
  2                                    190722470400 -15          1.9072247040000000000e-04
  3                                7574824857600000 -21          7.5748248576000000000e-06
  4                           312546150372456000000 -27          3.1254615037245600000e-07
  5                      13207874703225491420651520 -33          1.3207874703225491421e-08
  6                  567273919793089083292259942400 -39          5.6727391979308908329e-10
  7             24650600248172987140112763715584000 -45          2.4650600248172987140e-11
  8        1080657854354639453670407474439566400000 -51          1.0806578543546394537e-12
  9    47701779391594966287470570490839978880000000 -57          4.7701779391594966287e-14

π to 70 digits is 3.1415926535897932384626433832795028841971693993751058209749445923078164
Computer π is     3.1415926535897932384626433832795028841971693993751058209749445923078164

Mathematica/Wolfram Language[edit]

ClearAll[numerator, denominator]
numerator[n_] := (2^5) ((6 n)!) (532 n^2 + 126 n + 9)/(3 (n!)^6)
denominator[n_] := 10^(6 n + 3)
numerator /@ Range[0, 9]
val = 1/Sqrt[Total[numerator[#]/denominator[#] & /@ Range[0, 100]]];
N[val, 70]

Output:

{96,5122560,190722470400,7574824857600000,312546150372456000000,13207874703225491420651520,567273919793089083292259942400,24650600248172987140112763715584000,1080657854354639453670407474439566400000,47701779391594966287470570490839978880000000}
3.141592653589793238462643383279502884197169399375105820974944592307816

Nim[edit]

Library: nim-decimal

Derived from Wren version with some simplifications.

import strformat, strutils
import decimal
 
proc fact(n: int): DecimalType =
  result = newDecimal(1)
  if n < 2: return
  for i in 2..n:
    result *= i
 
proc almkvistGiullera(n: int): DecimalType =
  ## Return the integer portion of the nth term of Almkvist-Giullera sequence.
  let t1 = fact(6 * n) * 32
  let t2 = 532 * n * n + 126 * n + 9
  let t3 = fact(n) ^ 6 * 3
  result = t1 * t2 / t3
 
let One = newDecimal(1)
 
setPrec(78)
echo "N                               Integer portion"
echo repeat('-', 47)
for n in 0..9:
  echo &"{n}  {almkvistGiullera(n):>44}"
echo()
 
echo "Pi to 70 decimal places:"
var
  sum = newDecimal(0)
  prev = newDecimal(0)
  prec = One.scaleb(newDecimal(-70))
  n = 0
while true:
  sum += almkvistGiullera(n) / One.scaleb(newDecimal(6 * n + 3))
  if abs(sum - prev) < prec: break
  prev = sum.clone
  inc n
let pi = 1 / sqrt(sum)
echo ($pi)[0..71]

Output:

N                               Integer portion
-----------------------------------------------
0                                            96
1                                       5122560
2                                  190722470400
3                              7574824857600000
4                         312546150372456000000
5                    13207874703225491420651520
6                567273919793089083292259942400
7           24650600248172987140112763715584000
8      1080657854354639453670407474439566400000
9  47701779391594966287470570490839978880000000

Pi to 70 decimal places:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Perl[edit]

Translation of: Raku

use strict;
use warnings;
use feature qw(say);
use Math::AnyNum qw(:overload factorial);
 
sub almkvist_giullera_integral {
    my($n) = @_;
    (32 * (14*$n * (38*$n + 9) + 9) * factorial(6*$n)) / (3*factorial($n)**6);
}
 
sub almkvist_giullera {
    my($n) = @_;
    almkvist_giullera_integral($n) / (10**(6*$n + 3));
}
 
sub almkvist_giullera_pi {
    my ($prec) = @_;
 
    local $Math::AnyNum::PREC = 4*($prec+1);
 
    my $sum = 0;
    my $target = '';
 
    for (my $n = 0; ; ++$n) {
        $sum += almkvist_giullera($n);
        my $curr = ($sum**-.5)->as_dec;
        return $target if ($curr eq $target);
        $target = $curr;
    }
}
 
say 'First 10 integer portions: ';
say "$_  " . almkvist_giullera_integral($_) for 0..9;
 
my $precision = 70;
 
printf("π to %s decimal places is:\n%s\n",
    $precision, almkvist_giullera_pi($precision));

Output:

First 10 integer portions:
0  96
1  5122560
2  190722470400
3  7574824857600000
4  312546150372456000000
5  13207874703225491420651520
6  567273919793089083292259942400
7  24650600248172987140112763715584000
8  1080657854354639453670407474439566400000
9  47701779391594966287470570490839978880000000
π to 70 decimal places is:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Phix[edit]

with javascript_semantics
requires("1.0.0") 
include mpfr.e
mpfr_set_default_precision(-70)
 
function almkvistGiullera(integer n, bool bPrint)
    mpz {t1,t2,ip} = mpz_inits(3)
    mpz_fac_ui(t1,6*n) 
    mpz_mul_si(t1,t1,32)                -- t1:=2^5*(6n)!
    mpz_fac_ui(t2,n)
    mpz_pow_ui(t2,t2,6)
    mpz_mul_si(t2,t2,3)                 -- t2:=3*(n!)^6
    mpz_mul_si(ip,t1,532*n*n+126*n+9)   -- ip:=t1*(532n^2+126n+9)
    mpz_fdiv_q(ip,ip,t2)                -- ip:=ip/t2
    integer pw := 6*n+3
    mpz_ui_pow_ui(t1,10,pw)             -- t1 := 10^(6n+3)
    mpq tm = mpq_init_set_z(ip,t1)      -- tm := rat(ip/t1)
    if bPrint then
        string ips = mpz_get_str(ip),
               tms = mpfr_get_fixed(mpfr_init_set_q(tm),50)
        tms = trim_tail(tms,"0")
        printf(1,"%d  %44s  %3d  %s\n", {n, ips, -pw, tms})
    end if
    return tm
end function
 
constant hdr = "N --------------- Integer portion -------------  Pow  ----------------- Nth term (50 dp) -----------------"
printf(1,"%s\n%s\n",{hdr,repeat('-',length(hdr))})
for n=0 to 9 do
    {} = almkvistGiullera(n, true)
end for
 
mpq {res,prev,z} = mpq_inits(3),
    prec = mpq_init_set_str(sprintf("1/1%s",repeat('0',70)))
integer n = 0
while true do
    mpq term := almkvistGiullera(n, false)
    mpq_add(res,res,term)
    mpq_sub(z,res,prev)
    mpq_abs(z,z)
    if mpq_cmp(z,prec) < 0 then exit end if
    mpq_set(prev,res)
    n += 1
end while
mpq_inv(res,res)
mpfr pi = mpfr_init_set_q(res)
mpfr_sqrt(pi,pi)
printf(1,"\nCalculation of pi took %d iterations using the Almkvist-Giullera formula.\n\n",n)
printf(1,"Pi to 70 d.p.: %s\n",mpfr_get_fixed(pi,70))
mpfr_const_pi(pi)
printf(1,"Pi (builtin) : %s\n",mpfr_get_fixed(pi,70))

Output:

N --------------- Integer portion -------------  Pow  ----------------- Nth term (50 dp) -----------------
----------------------------------------------------------------------------------------------------------
0                                            96   -3  0.096
1                                       5122560   -9  0.00512256
2                                  190722470400  -15  0.0001907224704
3                              7574824857600000  -21  0.0000075748248576
4                         312546150372456000000  -27  0.000000312546150372456
5                    13207874703225491420651520  -33  0.00000001320787470322549142065152
6                567273919793089083292259942400  -39  0.0000000005672739197930890832922599424
7           24650600248172987140112763715584000  -45  0.000000000024650600248172987140112763715584
8      1080657854354639453670407474439566400000  -51  0.0000000000010806578543546394536704074744395664
9  47701779391594966287470570490839978880000000  -57  0.00000000000004770177939159496628747057049083997888

Calculation of pi took 52 iterations using the Almkvist-Giullera formula.

Pi to 70 d.p.: 3.1415926535897932384626433832795028841971693993751058209749445923078164
Pi (builtin) : 3.1415926535897932384626433832795028841971693993751058209749445923078164

Python[edit]

import mpmath as mp
 
with mp.workdps(72):
 
    def integer_term(n):
        p = 532 * n * n + 126 * n + 9
        return (p * 2**5 * mp.factorial(6 * n)) / (3 * mp.factorial(n)**6)
 
    def exponent_term(n):
        return -(mp.mpf("6.0") * n + 3)
 
    def nthterm(n):
        return integer_term(n) * mp.mpf("10.0")**exponent_term(n)
 
 
    for n in range(10):
        print("Term ", n, '  ', int(integer_term(n)))
 
 
    def almkvist_guillera(floatprecision):
        summed, nextadd = mp.mpf('0.0'), mp.mpf('0.0')
        for n in range(100000000):
            nextadd = summed + nthterm(n)
            if abs(nextadd - summed) < 10.0**(-floatprecision):
                break
 
            summed = nextadd
 
        return nextadd
 
 
    print('\nπ to 70 digits is ', end='')
    mp.nprint(mp.mpf(1.0 / mp.sqrt(almkvist_guillera(70))), 71)
    print('mpmath π is       ', end='')
    mp.nprint(mp.pi, 71)

Output:

Term  0    96
Term  1    5122560
Term  2    190722470400
Term  3    7574824857600000
Term  4    312546150372456000000
Term  5    13207874703225491420651520
Term  6    567273919793089083292259942400
Term  7    24650600248172987140112763715584000
Term  8    1080657854354639453670407474439566400000
Term  9    47701779391594966287470570490839978880000000

π to 70 digits is 3.1415926535897932384626433832795028841971693993751058209749445923078164
mpmath π is       3.1415926535897932384626433832795028841971693993751058209749445923078164

Quackery[edit]

  [ $ "bigrat.qky" loadfile ] now!
 
  [ 1 swap times [ i^ 1+ * ] ] is !       ( n --> n   )
 
  [ dup dup 2 ** 532 *
    over 126 * + 9 +
    swap 6 * ! * 32 *
    swap ! 6 ** 3 * / ]        is intterm ( n --> n   )
 
  [ dup intterm 
    10 rot 6 * 3 + ** 
    reduce ]                   is vterm   ( n --> n/d )
 
  10 times [ i^ intterm echo cr ] cr
 
  0 n->v 
  53 times [ i^ vterm v+ ]
  1/v 70 vsqrt drop 
  70 point$ echo$ cr

Output:

96
5122560
190722470400
7574824857600000
312546150372456000000
13207874703225491420651520
567273919793089083292259942400
24650600248172987140112763715584000
1080657854354639453670407474439566400000
47701779391594966287470570490839978880000000

3.1415926535897932384626433832795028841971693993751058209749445923078164

Raku[edit]

# 20201013 Raku programming solution
 
use BigRoot;
use Rat::Precise;
use experimental :cached;
 
BigRoot.precision = 75 ;
my $Precision     = 70 ;
my $AGcache       =  0 ;
 
sub postfix:<!>(Int $n --> Int) is cached { [*] 1 .. $n }
 
sub Integral(Int $n --> Int) is cached {
   (2⁵*(6*$n)! * (532*$n² + 126*$n + 9)) div (3*($n!)⁶)
}
 
sub A-G(Int $n --> FatRat) is cached { # Almkvist-Giullera
   Integral($n).FatRat / (10**(6*$n + 3)).FatRat
}
 
sub Pi(Int $n --> Str) {
   (1/(BigRoot.newton's-sqrt: $AGcache += A-G $n)).precise($Precision)
}
 
say "First 10 integer portions : ";
say $_, "\t", Integral $_ for ^10;
 
my $target = Pi my $Nth = 0;
 
loop { $target eq ( my $next = Pi ++$Nth ) ?? ( last ) !! $target = $next }
 
say "π to $Precision decimal places is :\n$target"

Output:

First 10 integer portions :
0       96
1       5122560
2       190722470400
3       7574824857600000
4       312546150372456000000
5       13207874703225491420651520
6       567273919793089083292259942400
7       24650600248172987140112763715584000
8       1080657854354639453670407474439566400000
9       47701779391594966287470570490839978880000000
π to 70 decimal places is :
3.1415926535897932384626433832795028841971693993751058209749445923078164

REXX[edit]

/*REXX program uses the Almkvist─Giullera formula for   1 / (pi**2)     [or  pi ** -2]. */
numeric digits length( pi() )  +  length(.);                                       w= 102
say $(   , 3)       $(              , w%2)       $('power', 5)       $(          , w)
say $('N', 3)       $('integer term', w%2)       $('of 10', 5)       $('Nth term', w)
say $(   , 3, "─")  $(              , w%2, "─")  $(       , 5, "─")  $(          , w, "─")
                                  s= 0           /*initialize   S   (the sum)  to zero. */
     do n=0  until old=s;    old= s              /*use the "older" value of  S  for OLD.*/
     a= 2**5  *  !(6*n)  *  (532 * n**2  +  126*n  +  9)  /  (3 * !(n)**6)
     z= 10 ** (- (6*n + 3) )
     s= s     +   a * z
     if n>10  then do;  do 3*(n==11);  say ' .';  end;  iterate;  end
     say $(n, 3) right(a, w%2)  $(powX(z), 5)  right( lowE( format(a*z, 1, w-6, 2, 0)), w)
     end   /*n*/
say
say 'The calculation of pi took '       n       " iterations with "         digits() ,
    " decimal digits precision using"   subword( sourceLine(1), 4, 3).
say
numeric digits length( pi() ) - length(.);  d= digits() - length(.);          @= ' ↓↓↓ '
say center(@ 'calculated pi to '  d   " fractional decimal digits (below) is "@, d+4, '─')
say ' 'sqrt(1/s);          say
say ' 'pi();  @= ' ↑↑↑ '
say center(@ 'the  true  pi to '  d   " fractional decimal digits (above) is" @, d+4, '─')
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
$:    procedure; parse arg text,width,fill;     return center(text, width, left(fill, 1) )
!:    procedure; parse arg x; !=1;;      do j=2  to x;    != !*j;    end;         return !
lowE: procedure; parse arg x; return translate(x, 'e', "E")
powX: procedure; parse arg p; return right( format( p, 1, 3, 2, 0),  3)   +   0
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi:   pi=3.141592653589793238462643383279502884197169399375105820974944592307816406286208,
      ||9986280348253421170679821480865132823066470938446095505822317253594081284811174503
      return pi
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x;  if x=0  then return 0;  d=digits();  numeric digits;  h=d+6
      m.=9; numeric form; 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*/
      numeric digits d;           return g/1

output when using the internal default input:

(Shown at two─thirds size.)

                                                        power
 N                     integer term                     of 10                                                Nth term
─── ─────────────────────────────────────────────────── ───── ──────────────────────────────────────────────────────────────────────────────────────────────────────
 0                                                   96  -3   9.600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-02
 1                                              5122560  -9   5.122560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-03
 2                                         190722470400  -15  1.907224704000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-04
 3                                     7574824857600000  -21  7.574824857600000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-06
 4                                312546150372456000000  -27  3.125461503724560000000000000000000000000000000000000000000000000000000000000000000000000000000000e-07
 5                           13207874703225491420651520  -33  1.320787470322549142065152000000000000000000000000000000000000000000000000000000000000000000000000e-08
 6                       567273919793089083292259942400  -39  5.672739197930890832922599424000000000000000000000000000000000000000000000000000000000000000000000e-10
 7                  24650600248172987140112763715584000  -45  2.465060024817298714011276371558400000000000000000000000000000000000000000000000000000000000000000e-11
 8             1080657854354639453670407474439566400000  -51  1.080657854354639453670407474439566400000000000000000000000000000000000000000000000000000000000000e-12
 9         47701779391594966287470570490839978880000000  -57  4.770177939159496628747057049083997888000000000000000000000000000000000000000000000000000000000000e-14
10    2117262852373157207626265529989139651218848358400  -63  2.117262852373157207626265529989139651218848358400000000000000000000000000000000000000000000000000e-15
 .
 .
 .

The calculation of pi took  122  iterations with  163  decimal digits precision using the Almkvist─Giullera formula.

────────────────────────────────────────────── ↓↓↓  calculated pi to  160  fractional decimal digits (below) is  ↓↓↓ ───────────────────────────────────────────────
 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174503

 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174503
────────────────────────────────────────────── ↑↑↑  the  true  pi to  160  fractional decimal digits (above) is  ↑↑↑ ───────────────────────────────────────────────

Sidef[edit]

func almkvist_giullera(n) {
    (32 * (14*n * (38*n + 9) + 9) * (6*n)!) / (3 * n!**6)
}
 
func almkvist_giullera_pi(prec = 70) {
 
    local Num!PREC = (4*(prec+1)).numify
 
    var sum = 0
    var target = -1
 
    for n in (0..Inf) {
        sum += (almkvist_giullera(n) / (10**(6*n + 3)))
        var curr = (sum**-.5).as_dec
        return target if (target == curr)
        target = curr
    }
}
 
say 'First 10 integer portions: '
 
10.of {|n|
    say "#{n} #{almkvist_giullera(n)}"
}
 
with(70) {|n|
    say "π to #{n} decimal places is:"
    say almkvist_giullera_pi(n)
}

Output:

First 10 integer portions: 
0 96
1 5122560
2 190722470400
3 7574824857600000
4 312546150372456000000
5 13207874703225491420651520
6 567273919793089083292259942400
7 24650600248172987140112763715584000
8 1080657854354639453670407474439566400000
9 47701779391594966287470570490839978880000000
π to 70 decimal places is:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Visual Basic .NET[edit]

Library: System.Numerics

Translation of: C#

Imports System, BI = System.Numerics.BigInteger, System.Console
 
Module Module1
 
    Function isqrt(ByVal x As BI) As BI
        Dim t As BI, q As BI = 1, r As BI = 0
        While q <= x : q <<= 2 : End While
        While q > 1 : q >>= 2 : t = x - r - q : r >>= 1
            If t >= 0 Then x = t : r += q
        End While : Return r
    End Function
 
    Function dump(ByVal digs As Integer, ByVal Optional show As Boolean = False) As String
        digs += 1
        Dim z As Integer, gb As Integer = 1, dg As Integer = digs + gb
        Dim te As BI, t1 As BI = 1, t2 As BI = 9, t3 As BI = 1, su As BI = 0, t As BI = BI.Pow(10, If(dg <= 60, 0, dg - 60)), d As BI = -1, fn As BI = 1
        For n As BI = 0 To dg - 1
            If n > 0 Then t3 = t3 * BI.Pow(n, 6)
            te = t1 * t2 / t3 : z = dg - 1 - CInt(n) * 6
            If z > 0 Then te = te * BI.Pow(10, z) Else te = te / BI.Pow(10, -z)
            If show AndAlso n < 10 Then WriteLine("{0,2} {1,62}", n, te * 32 / 3 / t)
            su += te : If te < 10 Then
                digs -= 1
                If show Then WriteLine(vbLf & "{0} iterations required for {1} digits " & _
                    "after the decimal point." & vbLf, n, digs)
                Exit For
            End If
            For j As BI = n * 6 + 1 To n * 6 + 6
                t1 = t1 * j : Next
            d += 2 : t2 += 126 + 532 * d
        Next
        Dim s As String = String.Format("{0}", isqrt(BI.Pow(10, dg * 2 + 3) _
            / su / 32 * 3 * BI.Pow(CType(10, BI), dg + 5)))
        Return s(0) & "." & s.Substring(1, digs)
    End Function
 
    Sub Main(ByVal args As String())
        WriteLine(dump(70, true))
    End Sub
 
End Module

Output:

 0  9600000000000000000000000000000000000000000000000000000000000
 1   512256000000000000000000000000000000000000000000000000000000
 2    19072247040000000000000000000000000000000000000000000000000
 3      757482485760000000000000000000000000000000000000000000000
 4       31254615037245600000000000000000000000000000000000000000
 5        1320787470322549142065152000000000000000000000000000000
 6          56727391979308908329225994240000000000000000000000000
 7           2465060024817298714011276371558400000000000000000000
 8            108065785435463945367040747443956640000000000000000
 9              4770177939159496628747057049083997888000000000000

53 iterations required for 70 digits after the decimal point.

3.1415926535897932384626433832795028841971693993751058209749445923078164

Wren[edit]

Library: Wren-big

Library: Wren-fmt

import "/big" for BigInt, BigRat
import "/fmt" for Fmt
 
var factorial = Fn.new { |n|
    if (n < 2) return BigInt.one
    var fact = BigInt.one
    for (i in 2..n) fact = fact * i
    return fact
}
 
var almkvistGiullera = Fn.new { |n, print|
    var t1 = factorial.call(6*n) * 32
    var t2 = 532*n*n + 126*n + 9
    var t3 = factorial.call(n).pow(6)*3
    var ip = t1 * t2 / t3
    var pw = 6*n + 3
    var tm = BigRat.new(ip, BigInt.ten.pow(pw))
    if (print) {
        Fmt.print("$d  $44i  $3d  $-35s", n, ip, -pw, tm.toDecimal(33))
    } else {
        return tm
    }
}
 
System.print("N                               Integer Portion  Pow  Nth Term (33 dp)")
System.print("-" * 89)
for (n in 0..9) {
    almkvistGiullera.call(n, true)
}
 
var sum  = BigRat.zero
var prev = BigRat.zero
var prec = BigRat.new(BigInt.one, BigInt.ten.pow(70))
var n = 0
while(true) {
    var term = almkvistGiullera.call(n, false)
    sum = sum + term
    if ((sum-prev).abs < prec) break
    prev = sum
    n = n + 1
}
var pi = BigRat.one/sum.sqrt(70)
System.print("\nPi to 70 decimal places is:")
System.print(pi.toDecimal(70))

Output:

N                               Integer Portion  Pow  Nth Term (33 dp)
-----------------------------------------------------------------------------------------
0                                            96   -3  0.096                              
1                                       5122560   -9  0.00512256                         
2                                  190722470400  -15  0.0001907224704                    
3                              7574824857600000  -21  0.0000075748248576                 
4                         312546150372456000000  -27  0.000000312546150372456            
5                    13207874703225491420651520  -33  0.00000001320787470322549142065152 
6                567273919793089083292259942400  -39  0.000000000567273919793089083292260
7           24650600248172987140112763715584000  -45  0.000000000024650600248172987140113
8      1080657854354639453670407474439566400000  -51  0.000000000001080657854354639453670
9  47701779391594966287470570490839978880000000  -57  0.000000000000047701779391594966287

Pi to 70 decimal places is:
3.1415926535897932384626433832795028841971693993751058209749445923078164

Almkvist-Giullera formula for pi

Contents

11l[edit]

C#[edit]

C++[edit]

Common Lisp[edit]

dc[edit]

Erlang[edit]

F#[edit]

Factor[edit]

Go[edit]

Haskell[edit]

J[edit]

JavaScript[edit]

Julia[edit]

Mathematica/Wolfram Language[edit]

Nim[edit]

Perl[edit]

Phix[edit]

Python[edit]

Quackery[edit]

Raku[edit]

REXX[edit]

Sidef[edit]

Visual Basic .NET[edit]

Wren[edit]