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# Almkvist-Giullera formula for pi

Almkvist-Giullera formula for pi
You are encouraged to solve this task according to the task description, using any language you may know.

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)

• 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

## 11l

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
```

## C#

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++

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

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

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]p0 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] sMlMx []p [* print resulting value of pi to 70 places *]sz[Pi after ]n 52n [ iterations:]p99k 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

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#

` // Almkvist-Giullera formula for pi. Nigel Galloway: August 17th., 2021let factorial(n:bigint)=MathNet.Numerics.SpecialFunctions.Factorial nlet 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

Works with: Factor version 0.99 2020-08-14
`USING: continuations formatting io kernel locals mathmath.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)" print89 CHAR: - <repetition> print10 [    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

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
```

Library: numbers
Translation of: Common Lisp
`import Control.Monadimport Data.Number.CRealimport GHC.Integerimport Text.Printf iterations = 52main = 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 seriesalmkvistGiulleraIntegral 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 seriestenExponent 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 termsalmkvistGiulleraSum n = sum \$ map almkvistGiullera [0 .. n] -- The approximation of pi from the first N termsalmkvistGiulleraPi n = sqrt \$ 1 / almkvistGiulleraSum n -- Utility: factorial for arbitrary-precision integersfacInteger n = if n `leInteger` 1 then 1 else n `timesInteger` facInteger (n `minusInteger` 1) -- Utility: exponentiation for arbitrary-precision integerspowInteger 1 _ = 1powInteger _ 0 = 1powInteger b 1 = bpowInteger 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

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.53echo ''echo 'pi to 70 decimals: ', 0j70 ": pi70exit '' `
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

Translation of: Common Lisp
Works with: Node.js version 13+
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 termexport 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 seriesexport const tenExponent = n => 3n - 6n * (BigInt(n) + 1n); // the nth term of the seriesexport const nthTerm = n =>  new BigFloat(integral(n)).dividedBy(new BigFloat(10n ** -tenExponent(n))) // the sum of the first n termsexport 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 termsexport 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```

## jq

Works with gojq, the Go implementation of jq

This entry uses the "rational" module, which can be found at Arithmetic/Rational#jq.

Preliminaries

`# A reminder to include the "rational" module:# include "rational"; def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l)[:\$l] + .; # To take advantage of gojq's arbitrary-precision integer arithmetic:def power(\$b): . as \$in | reduce range(0;\$b) as \$i (1; . * \$in); def factorial:    if . < 2 then 1    else reduce range(2;.+1) as \$i (1; .*\$i)    end; `

Almkvist-Giullera Formula

` def almkvistGiullera(print):  . as \$n  | ((6*\$n) | factorial * 32) as \$t1  | (532*\$n*\$n + 126*\$n + 9) as \$t2  | ((\$n | factorial | power(6))*3) as \$t3  | (\$t1 * \$t2 / \$t3) as \$ip  | ( 6*\$n + 3) as \$pw  | r(\$ip; 10 | power(\$pw)) as \$tm  | if print    then "\(\$n|lpad(2)) \(\$ip|lpad(44)) \(-\$pw|lpad(3)), \(\$tm|r_to_decimal(100))"    else \$tm    end; `

` def task1:  "N                               Integer Portion  Pow  Nth Term",  ("-" * 89),  (range(0;10) | almkvistGiullera(true)) ; def task2(\$precision):  r(1; 10 | power(\$precision)) as \$p  | {sum: r(0;1), prev: r(0;1), n:  0 }  | until(.stop;    .sum = radd(.sum; .n | almkvistGiullera(false))    | if rminus(.sum; .prev) | rabs | rlessthan(\$p)      then .stop = true      else .prev = .sum      | .n += 1      end)   | .sum | rinv   | rsqrt(\$precision)   | "\nPi to \(\$precision) decimal places is:",    "\(r_to_decimal(\$precision))" ; task1,""task2(70)`
Output:
```N                               Integer Portion  Pow  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 to 70 decimal places is:
3.1415926535897932384626433832795028841971693993751058209749445923078164
```

## Julia

`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    endend 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

`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

Library: nim-decimal

Derived from Wren version with some simplifications.

`import strformat, strutilsimport 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 = 0while true:  sum += almkvistGiullera(n) / One.scaleb(newDecimal(6 * n + 3))  if abs(sum - prev) < prec: break  prev = sum.clone  inc nlet 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

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

```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_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

`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

`  [ \$ "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

`# 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

`/*REXX program uses the Almkvist─Giullera formula for   1 / (pi**2)     [or  pi ** -2]. */numeric digits length( pi() )  +  length(.);                                       w= 102say \$(   , 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*/saysay 'The calculation of pi took '       n       " iterations with "         digits() ,    " decimal digits precision using"   subword( sourceLine(1), 4, 3).saynumeric digits length( pi() ) - length(.);  d= digits() - length(.);          @= ' ↓↓↓ 'say center(@ 'calculated pi to '  d   " fractional decimal digits (below) is "@, d+4, '─')say ' 'sqrt(1/s);          saysay ' '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

`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

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

Library: Wren-big
Library: Wren-fmt
`import "/big" for BigInt, BigRatimport "/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.zerovar prev = BigRat.zerovar prec = BigRat.new(BigInt.one, BigInt.ten.pow(70))var n = 0while(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
```