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Almkvist-Giullera formula for pi

From Rosetta Code
Task
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)


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



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

C#[edit]

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

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

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 $n --> Int) is cached {
(2*(6*$n)! * (532*$n² + 126*$n + 9)) div (3*($n!))
}
 
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]

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