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The purpose of this task is to demonstrate how to use this approximation in order to compute a large number of decimals of <math>\pi</math>.
 
=={{header|BASIC}}==
==={{header|BASIC256}}===
<syntaxhighlight lang="basic">digits = 500
an = 1.0
bn = sqr(0.5)
tn = 0.5 ^ 2
pn = 1.0
 
while pn <= digits
prevAn = an
an = (bn + an) / 2
bn = sqr(bn * prevAn)
prevAn -= an
tn -= (pn * prevAn ^ 2)
pn *= 2
end while
print ((an + bn) ^ 2) / (tn * 4)</syntaxhighlight>
 
==={{header|FreeBASIC}}===
<syntaxhighlight lang="vb">Dim As Short digits = 500
Dim As Double an = 1
Dim As Double bn = Sqr(0.5)
Dim As Double tn = 0.5^2
Dim As Double pn = 1
Dim As Double prevAn
 
While pn <= digits
prevAn = an
an = (bn + an) / 2
bn = Sqr(bn * prevAn)
prevAn -= an
tn -= (pn * prevAn^2)
pn *= 2
Wend
Dim As Double pi = ((an + bn)^2) / (tn * 4)
Print pi
 
Sleep</syntaxhighlight>
{{out}}
<pre>3.141592653589794</pre>
 
==={{header|IS-BASIC}}===
<syntaxhighlight lang="is-basic">100 PROGRAM "PI.bas"
110 LET DIGITS=10
120 LET AN,PN=1
130 LET BN=SQR(.5)
140 LET TN=.5^2
150 DO WHILE PN<=DIGITS
160 LET PREVAN=AN
170 LET AN=(BN+AN)/2
180 LET BN=SQR(BN*PREVAN)
190 LET PREVAN=PREVAN-AN
200 LET TN=TN-(PN*PREVAN^2)
210 LET PN=PN+PN
220 LOOP
230 PRINT (AN+BN)^2/(TN*4)</syntaxhighlight>
 
==={{header|True BASIC}}===
{{works with|QBasic}}
<syntaxhighlight lang="qbasic">LET digits = 500
LET an = 1.0
LET bn = SQR(0.5)
LET tn = 0.5 ^ 2
LET pn = 1.0
 
DO WHILE pn <= digits
LET prevAn = an
LET an = (bn + an) / 2
LET bn = SQR(bn * prevAn)
LET prevAn = prevAn - an
LET tn = tn - (pn * prevAn ^ 2)
LET pn = pn + pn
LOOP
PRINT ((an + bn) ^ 2) / (tn * 4)
END</syntaxhighlight>
 
==={{header|Yabasic}}===
<syntaxhighlight lang="basic">digits = 500
an = 1.0
bn = sqrt(0.5)
tn = 0.5 ^ 2
pn = 1.0
 
while pn <= digits
prevAn = an
an = (bn + an) / 2
bn = sqrt(bn * prevAn)
prevAn = prevAn - an
tn = tn - (pn * prevAn ^ 2)
pn = pn + pn
wend
print ((an + bn) ^ 2) / (tn * 4)</syntaxhighlight>
 
=={{header|C}}==
Line 489 ⟶ 582:
3.141592653589793238...81377399510065895288
</pre>
=={{header|EasyLang}}==
{{trans|FreeBASIC}}
<syntaxhighlight lang=easylang>
an = 1
bn = sqrt 0.5
tn = 0.25
pn = 1
while pn <= 5
prevAn = an
an = (bn + an) / 2
bn = sqrt (bn * prevAn)
prevAn -= an
tn -= (pn * prevAn * prevAn)
pn *= 2
.
mypi = (an + bn) * (an + bn) / (tn * 4)
numfmt 15 0
print mypi
</syntaxhighlight>
 
=={{header|Erlang}}==
{{trans|python}}
Line 569 ⟶ 682:
Iteration: 4 Diff: 3.05653257536554156111405386493810661E-0010 Pi: 3.14159265358979323846636060270664556
Iteration: 5 Diff: 3.71721942712928151094186846648146566E-0021 Pi: 3.14159265358979323846264338327951628
</pre>
 
=={{header|FutureBasic}}==
<syntaxhighlight lang="futurebasic">
 
double a, c, g, t, p
double apprpi
short i
 
// Initial values
a = 1
g = sqr(0.5)
p = 1
t = 0.25
 
//Iterate just 3 times
for i = 1 to 3
c = a
a = ( a + g ) / 2
g = sqr( c * g )
c -= a
t -= ( p * c^2 )
p *= 2
apprpi = (( a + g )^2) / ( t * 4 )
print "Iteration "i": ", apprpi
next
 
print "Actual value:",pi
 
handleevents
 
</syntaxhighlight>
{{output}}
<pre>
Iteration 1: 3.140579250522169
Iteration 2: 3.141592646213543
Iteration 3: 3.141592653589794
Actual value: 3.141592653589793
</pre>
 
Line 948 ⟶ 1,099:
pi[7, 100]
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628046852228654</syntaxhighlight>
 
=={{header|MATLAB}}==
{{trans|Julia}}
<syntaxhighlight lang="MATLAB">
 
clear all;close all;clc;
testMakePi();
 
 
function [a, g] = agm1step(x, y)
a = (x + y) / 2;
g = sqrt(x * y);
end
 
function [a, g, s, k] = approxPiStep(x, y, z, n)
[a, g] = agm1step(x, y);
k = n + 1;
s = z + 2^(k + 1) * (a^2 - g^2);
end
 
function pi_approx = approxPi(a, g, s)
pi_approx = 4 * a^2 / (1 - s);
end
 
function testMakePi()
digits(512); % Set the precision for variable-precision arithmetic
a = vpa(1.0);
g = 1 / sqrt(vpa(2.0));
s = vpa(0.0);
k = 0;
oldPi = vpa(0.0);
% Define a small value as a threshold for convergence
convergence_threshold = vpa(10)^(-digits);
 
fprintf(' k Error Result\n');
for i = 1:100
[a, g, s, k] = approxPiStep(a, g, s, k);
estPi = approxPi(a, g, s);
if abs(estPi - oldPi) < convergence_threshold
break;
end
oldPi = estPi;
err = abs(vpa(pi) - estPi);
fprintf('%4d%10.1e', i, double(err));
fprintf('%70.60f\n', double(estPi));
end
end
</syntaxhighlight>
{{out}}
<pre>
k Error Result
1 4.6e-02 3.187672642712108483920019352808594703674316406250000000000000
2 8.8e-05 3.141680293297653303596916884998790919780731201171875000000000
3 3.1e-10 3.141592653895446396461466065375134348869323730468750000000000
4 3.7e-21 3.141592653589793115997963468544185161590576171875000000000000
5 5.5e-43 3.141592653589793115997963468544185161590576171875000000000000
6 1.2e-86 3.141592653589793115997963468544185161590576171875000000000000
7 5.8e-174 3.141592653589793115997963468544185161590576171875000000000000
8 0.0e+00 3.141592653589793115997963468544185161590576171875000000000000
9 0.0e+00 3.141592653589793115997963468544185161590576171875000000000000
</pre>
 
=={{header|МК-61/52}}==
{{Output?}}
<syntaxhighlight lang="text">3 П0 1 П1 П4 2 КвКор 1/x П2 1
^ 4 / П3 ИП3 ИП1 ИП2 + 2 /
Line 956 ⟶ 1,167:
* КвКор П2 ИП5 П1 КИП4 L0 14 ИП1 x^2
ИП3 / С/П</syntaxhighlight>
 
{{out}}
<pre>3.1415927</pre>
 
=={{header|Nim}}==
{{libheader|bignum}}
{{Trans|DelphyDelphi}}
<syntaxhighlight lang="nim">from math import sqrt
import times
Line 1,085 ⟶ 1,299:
{{out}}
<pre>%1 = 3.1415926535897932384626433832795028841971693993751058209749445923078164062878</pre>
 
=={{header|Pascal}}==
{{works with|FPC}}
{{libheader|GMP}}
The program expects as an input parameter the required number of decimal digits of '''''π''''' (32 ≤ N ≤ 1000000), default is 256 digits.
<syntaxhighlight lang="pascal">
program AgmForPi;
{$mode objfpc}{$h+}{$b-}{$warn 5091 off}
uses
SysUtils, Math, GMP;
 
const
MIN_DIGITS = 32;
MAX_DIGITS = 1000000;
 
var
Digits: Cardinal = 256;
 
procedure ReadInput;
var
UserDigits: Cardinal;
begin
if (ParamCount > 0) and TryStrToDWord(ParamStr(1), UserDigits) then
Digits := Min(MAX_DIGITS, Max(UserDigits, MIN_DIGITS));
f_set_default_prec(Ceil((Digits + 1)/LOG_10_2));
end;
 
function Sqrt(a: MpFloat): MpFloat;
begin
Result := f_sqrt(a);
end;
 
function Sqr(a: MpFloat): MpFloat;
begin
Result := a * a;
end;
 
function PiDigits: string;
var
a0, b0, an, bn, tn: MpFloat;
n: Cardinal;
begin
n := 1;
an := 1;
bn := Sqrt(MpFloat(0.5));
tn := 0.25;
while n < Digits do begin
a0 := an;
b0 := bn;
an := (a0 + b0)/2;
bn := Sqrt(a0 * b0);
tn := tn - Sqr(an - a0) * n;
n := n + n;
end;
Result := Sqr(an + bn)/(tn * 4);
SetLength(Result, Succ(Digits));
end;
 
begin
ReadInput;
WriteLn(PiDigits);
end.
</syntaxhighlight>
{{out}}
<pre>
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648
</pre>
 
=={{header|Perl}}==
Line 1,328 ⟶ 1,609:
} = \frac{\sqrt{n 10^{2N} / d}}{10^N}</math>
 
so that what we need is one square root of a big number that we'll truncate to its integer part. We'll computeuse the squaremethod rootdescribed ofin this[[Integer bigroots]] integerto by usingcompute the convergencesquare root of thethis recursivebig integer. sequence:
 
<math>u_{n+1} = \frac{1}{2}(u_n + \frac{x}{u_n})</math>
 
It's not too hard to see that such a sequence converges towards <math>\sqrt x</math>.
 
Notice that we don't get the exact number of decimals required : the last two decimals or so can be wrong. This is because we don't need <math>a_n</math>, but rather <math>a_n^2</math>. Elevating to the square makes us lose a bit of precision. It could be compensated by choosing a slightly higher value of N (in a way that could be precisely calculated), but that would probably be overkill.
Line 1,338 ⟶ 1,615:
 
multi sqrt(Int $n) {
my $guess = (10**($n.chars div 2);, { ($_ + $n div $_) div 2 } ... * == *).tail
my $iterator = { ( $^x + $n div ($^x) ) div 2 };
my $endpoint = { $^x == $^y|$^z };
return min (+$guess, $iterator … $endpoint)[*-1, *-2];
}
 
multi sqrt(FatRat $r --> FatRat) {
return FatRat.new:
sqrt($r.nude[0]numerator * 10**(number-of-decimals*2) div $r.nude[1]denominator),
10**number-of-decimals;
}
Line 1,353 ⟶ 1,627:
my FatRat $g = sqrt(1/2.FatRat);
my $z = .25;
 
for ^10 {
given [ ($a + $g)/2, sqrt($a * $g) ] {
$z -= (.[0] - $a)**2 * $n;
$n += $n;
($a, $g) = @$_;
say ($a ** 2 / $z).substr: 0, 2 + number-of-decimals;
}
}</syntaxhighlight>
{{out}}
Line 1,543 ⟶ 1,817:
1004 3.141...201989381
1005 3.141...2019893810</pre>
 
=={{header|RPL}}==
{{trans|BASIC}}
{{works with|Halcyon Calc|4.2.7}}
{| class="wikitable"
! RPL code
! Comment
|-
|
≪ → digits
≪ 0.5 SQ 1 1 0.5 √
'''WHILE''' 3 PICK digits ≤ '''REPEAT'''
OVER
ROT 3 PICK + 2 / ROT ROT
SWAP OVER * √ SWAP
3 PICK -
5 ROLL SWAP SQ 5 PICK * - 4 ROLLD
ROT DUP + ROT ROT
'''END'''
+ SQ ROT 4 * /
SWAP DROP
≫ ≫ ‘'''AGMPI'''’ STO
≪ { } 1 5 FOR d d '''AGMPI''' NEXT
≫ ‘'''TASK'''’ STO
|
'''AGMPI''' ''( digits -- pi )''
tn = 0.5 ^ 2 : pn = 1.0 : an = 1.0 : bn = sqrt(0.5)
while pn <= digits
prevAn = an
an = (bn + an) / 2
bn = sqrt(bn * prevAn)
prevAn = prevAn - an
tn = tn - (pn * prevAn ^ 2)
pn = pn + pn
wend
print ((an + bn) ^ 2) / (tn * 4)
// clean stack
|}
{{out}}
<pre>
5: 2.91421356238
4: 3.14057925053
3: 3.14159264619
2: 3.14159264619
1: 3.14159265359
</pre>
 
=={{header|Ruby}}==
Line 1,794 ⟶ 2,119:
<pre style="height:64ex;white-space: pre-wrap;">Computation time: 4.1539 seconds
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</pre>
 
=={{header|TI SR-56}}==
{| class="wikitable"
|+ Texas Instruments SR-56 Program Listing for "Calculate Pi"
|-
! Display !! Key !! Display !! Key !! Display !! Key !! Display !! Key
|-
| 00 03 || 3 || 25 54 || / || 50 04 || 4 || 75 ||
|-
| 01 33 || STO || 26 02 || 2 || 51 34 || RCL || 76 ||
|-
| 02 00 || 0 || 27 94 || = || 52 01 || 1 || 77 ||
|-
| 03 01 || 1 || 28 32 || x><t|| 53 43 || x² || 78 ||
|-
| 04 33 || STO || 29 64 || * || 54 54 || / || 79 ||
|-
| 05 01 || 1 || 30 34 || RCL || 55 34 || RCL || 80 ||
|-
| 06 33 || STO || 31 02 || 2 || 56 04 || 4 || 81 ||
|-
| 07 03 || 3 || 32 94 || = || 57 94 || = || 82 ||
|-
| 08 02 || 2 || 33 48 || *√x || 58 59 || *pause || 83 ||
|-
| 09 48 || *√x || 34 33 || STO || 59 27 || *dsz|| 84 ||
|-
| 10 20 || *1/x || 35 02 || 2 || 60 01 || 1 || 85 ||
|-
| 11 33 || STO || 36 32 || x><t|| 61 08 || 8 || 86 ||
|-
| 12 02 || 2 || 37 74 || - || 62 41 || R/S || 87 ||
|-
| 13 92 || . || 38 39 || *EXC|| 63 || || 88 ||
|-
| 14 02 || 2 || 39 01 || 1 || 64 || || 89 ||
|-
| 15 05 || 5 || 40 94 || = || 65 || || 90 ||
|-
| 16 33 || STO || 41 43 || x² || 66 || || 91 ||
|-
| 17 04 || 4 || 42 64 || * || 67 || || 92 ||
|-
| 18 34 || RCL || 43 34 || RCL || 68 || || 93 ||
|-
| 19 01 || 1 || 44 03 || 3 || 69 || || 94 ||
|-
| 20 84 || + || 45 35 || SUM || 70 || || 95 ||
|-
| 21 32 || x><t || 46 03 || 3 || 71 || || 96 ||
|-
| 22 34 || RCL || 47 94 || = || 72 || || 97 ||
|-
| 23 02 || 2 || 48 12 || INV || 73 || || 98 ||
|-
| 24 94 || = || 49 35 || SUM || 74 || || 99 ||
|}
 
Asterisk denotes 2nd function key.
 
{| class="wikitable"
|+ Register allocation
|-
| 0: Loop count || 1: Arithmetic Term || 2: Geometric Term || 3: Power of Two || 4: Divisor Term
|-
| 5: Unused || 6: Unused || 7: Unused || 8: Unused || 9: Unused
|}
 
Annotated listing:
<syntaxhighlight lang="text">
3 STO 0 // r0 = 3 (loop count)
1 STO 1 STO 3 // r1 = a0, r3 = 1
2 √x 1/x STO 2 // r2 = g0
. 2 5 STO 4 // r4 = 0.25
RCL 1 + x><t RCL 2 = / 2 = // t = a0, x = a1
x><t * RCL 2 = √x STO 2 // t = a1, r2 = g1
x><t - EXC 1 = // x = (a1 - a0), r1 = a1
x² * RCL 3 SUM 3 = INV SUM 4 // r4 = r4-r3(a1-a0)^2, r3 = r3*2
RCL 1 x² / RCL 4 = pause
dsz 18
R/S
</syntaxhighlight>
 
'''Usage:'''
 
Press RST R/S.
 
{{out}}
 
Intermediate results flash on the screen, converging on the correct answer.
 
<pre>
3.187672643
</pre>
 
<pre>
3.141680293
</pre>
 
The third, final iteration yields:
 
<pre>
3.141592654
</pre>
 
=={{header|Wren}}==
{{trans|Sidef}}
{{libheader|Wren-big}}
<syntaxhighlight lang="ecmascriptwren">import "./big" for BigRat
 
var digits = 500
Aspen138
337

edits

Retrieved from "https://rosettacode.org/wiki/Special:MobileDiff/328967...352279"