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Write a function to compute the [[wp:Arithmetic-geometric mean|arithmetic-geometric mean]] of two numbers. [http://mathworld.wolfram.com/Arithmetic-GeometricMean.html] The arithmetic-geometric mean of two numbers can be (usefully) denoted as $\mathrm\left\{agm\right\}\left(a,g\right)$, and is equal to the limit of the sequence: : $a_0 = a; \qquad g_0 = g$ : $a_\left\{n+1\right\} = \tfrac\left\{1\right\}\left\{2\right\}\left(a_n + g_n\right); \quad g_\left\{n+1\right\} = \sqrt\left\{a_n g_n\right\}.$ Since the limit of $a_n-g_n$ tends (rapidly) to zero with iterations, this is an efficient method.

Demonstrate the function by calculating: :$\mathrm\left\{agm\right\}\left(1,1/\sqrt\left\{2\right\}\right)$

;Also see:

• [http://mathworld.wolfram.com/Arithmetic-GeometricMean.html mathworld.wolfram.com/Arithmetic-Geometric Mean]

## 11l

{{trans|Python}}

F agm(a0, g0, tolerance = 1e-10)
V an = (a0 + g0) / 2.0
V gn = sqrt(a0 * g0)
L abs(an - gn) > tolerance
(an, gn) = ((an + gn) / 2.0, sqrt(an * gn))
R an

print(agm(1, 1 / sqrt(2)))


{{out}}

0.847213


## 360 Assembly

For maximum compatibility, this program uses only the basic instruction set.

AGM      CSECT
USING  AGM,R13
SAVEAREA B      STM-SAVEAREA(R15)
DC     17F'0'
DC     CL8'AGM'
STM      STM    R14,R12,12(R13)
ST     R13,4(R15)
ST     R15,8(R13)
LR     R13,R15
ZAP    A,K                a=1
ZAP    PWL8,K
MP     PWL8,K
DP     PWL8,=P'2'
ZAP    PWL8,PWL8(7)
BAL    R14,SQRT
ZAP    G,PWL8             g=sqrt(1/2)
WHILE1   EQU    *                  while a!=g
ZAP    PWL8,A
SP     PWL8,G
CP     PWL8,=P'0'         (a-g)!=0
BE     EWHILE1
ZAP    PWL8,A
AP     PWL8,G
DP     PWL8,=P'2'
ZAP    AN,PWL8(7)         an=(a+g)/2
ZAP    PWL8,A
MP     PWL8,G
BAL    R14,SQRT
ZAP    G,PWL8             g=sqrt(a*g)
ZAP    A,AN               a=an
B      WHILE1
EWHILE1  EQU    *
ZAP    PWL8,A
UNPK   ZWL16,PWL8
MVC    CWL16,ZWL16
OI     CWL16+15,X'F0'
MVI    CWL16,C'+'
CP     PWL8,=P'0'
BNM    *+8
MVI    CWL16,C'-'
MVC    CWL80+0(15),CWL16
MVC    CWL80+9(1),=C'.'   /k  (15-6=9)
XPRNT  CWL80,80           display a
L      R13,4(0,R13)
LM     R14,R12,12(R13)
XR     R15,R15
BR     R14
DS     0F
K        DC     PL8'1000000'       10^6
A        DS     PL8
G        DS     PL8
AN       DS     PL8
* ****** SQRT   *******************
SQRT     CNOP   0,4                function sqrt(x)
ZAP    X,PWL8
ZAP    X0,=P'0'           x0=0
ZAP    X1,=P'1'           x1=1
WHILE2   EQU    *                  while x0!=x1
ZAP    PWL8,X0
SP     PWL8,X1
CP     PWL8,=P'0'         (x0-x1)!=0
BE     EWHILE2
ZAP    X0,X1              x0=x1
ZAP    PWL16,X
DP     PWL16,X1
ZAP    XW,PWL16(8)        xw=x/x1
ZAP    PWL8,X1
AP     PWL8,XW
DP     PWL8,=P'2'
ZAP    PWL8,PWL8(7)
ZAP    X2,PWL8            x2=(x1+xw)/2
ZAP    X1,X2              x1=x2
B      WHILE2
EWHILE2  EQU    *
ZAP    PWL8,X1            return x1
BR     R14
DS     0F
X        DS     PL8
X0       DS     PL8
X1       DS     PL8
X2       DS     PL8
XW       DS     PL8
* end SQRT
PWL8     DC     PL8'0'
PWL16    DC     PL16'0'
CWL80    DC     CL80' '
CWL16    DS     CL16
ZWL16    DS     ZL16
LTORG
YREGS
END    AGM


{{out}}


+00000000.84721



## 8th

: epsilon  1.0e-12 ;

with: n

: iter  \ n1 n2 -- n1 n2
2dup * sqrt >r + 2 / r> ;

: agn  \ n1 n2 -- n
repeat  iter  2dup epsilon ~= not while!  drop ;

"agn(1, 1/sqrt(2)) = " .  1  1 2 sqrt /  agn  "%.10f" s:strfmt . cr

;with
bye



{{out}}


agn(1, 1/sqrt(2)) = 0.8472130848



with Ada.Text_IO, Ada.Numerics.Generic_Elementary_Functions;

procedure Arith_Geom_Mean is

type Num is digits 18; -- the largest value gnat/gcc allows

function AGM(A, G: Num) return Num is
Old_G: Num;
New_G: Num := G;
New_A: Num := A;
begin
loop
Old_G := New_G;
New_G := Math.Sqrt(New_A*New_G);
New_A := (Old_G + New_A) * 0.5;
exit when (New_A - New_G) <= Num'Epsilon;
-- Num'Epsilon denotes the relative error when performing arithmetic over Num
end loop;
return New_G;
end AGM;

begin
N_IO.Put(AGM(1.0, 1.0/Math.Sqrt(2.0)), Fore => 1, Aft => 17, Exp => 0);
end Arith_Geom_Mean;


Output:

0.84721308479397909


## ALGOL 68

Algol 68 Genie gives IEEE double precision for REAL quantities, 28 decimal digits for LONG REALs and, by default, 63 decimal digits for LONG LONG REAL though this can be made arbitrarily greater with a pragmat.

Printing out the difference between the means at each iteration nicely demonstrates the quadratic convergence.


BEGIN
PROC agm = (LONG REAL x, y) LONG REAL :
BEGIN
IF x < LONG 0.0 OR y < LONG 0.0 THEN -LONG 1.0
ELIF x + y = LONG 0.0 THEN LONG 0.0		CO Edge cases CO
ELSE
LONG REAL a := x, g := y;
LONG REAL epsilon := a + g;
LONG REAL next a := (a + g) / LONG 2.0, next g := long sqrt (a * g);
LONG REAL next epsilon := ABS (a - g);
WHILE next epsilon < epsilon
DO
print ((epsilon, "   ", next epsilon, newline));
epsilon := next epsilon;
a := next a; g := next g;
next a := (a + g) / LONG 2.0; next g := long sqrt (a * g);
next epsilon := ABS (a - g)
OD;
a
FI
END;
printf (($l(-35,33)l$, agm (LONG 1.0, LONG 1.0 / long sqrt (LONG 2.0))))
END



Output:

+1.707106781186547524400844362e  +0   +2.928932188134524755991556379e  -1
+2.928932188134524755991556379e  -1   +1.265697533955921916929670477e  -2
+1.265697533955921916929670477e  -2   +2.363617660269221214237489508e  -5
+2.363617660269221214237489508e  -5   +8.242743980540458935740117000e -11
+8.242743980540458935740117000e -11   +1.002445937606580000000000000e -21
+1.002445937606580000000000000e -21   +4.595001000000000000000000000e -29
+4.595001000000000000000000000e -29   +4.595000000000000000000000000e -29

0.847213084793979086606499123550000



## APL


agd←{(⍺-⍵)<10*¯8:⍺⋄((⍺+⍵)÷2)∇(⍺×⍵)*÷2}
1 agd ÷2*÷2



Output:

0.8472130848


## AppleScript

By functional composition:

-- ARITHMETIC GEOMETRIC MEAN -------------------------------------------------

property tolerance : 1.0E-5

-- agm :: Num a => a -> a -> a
on agm(a, g)
script withinTolerance
on |λ|(m)
tell m to ((its an) - (its gn)) < tolerance
end |λ|
end script

script nextRefinement
on |λ|(m)
tell m
set {an, gn} to {its an, its gn}
{an:(an + gn) / 2, gn:(an * gn) ^ 0.5}
end tell
end |λ|
end script

an of |until|(withinTolerance, ¬
nextRefinement, {an:(a + g) / 2, gn:(a * g) ^ 0.5})
end agm

-- TEST ----------------------------------------------------------------------
on run

agm(1, 1 / (2 ^ 0.5))

--> 0.847213084835

end run

-- GENERIC FUNCTIONS ---------------------------------------------------------

-- until :: (a -> Bool) -> (a -> a) -> a -> a
on |until|(p, f, x)
set mp to mReturn(p)
set v to x
tell mReturn(f)
repeat until mp's |λ|(v)
set v to |λ|(v)
end repeat
end tell
return v
end |until|

-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn


{{Out}}

0.847213084835


## AutoHotkey

agm(a, g, tolerance=1.0e-15){
While abs(a-g) > tolerance
{
an := .5 * (a + g)
g  := sqrt(a*g)
a  := an
}
return a
}
SetFormat, FloatFast, 0.15
MsgBox % agm(1, 1/sqrt(2))


Output:

0.847213084793979


## AWK

#!/usr/bin/awk -f
BEGIN {
printf "%.16g\n", agm(1.0,sqrt(0.5))
}
function agm(a,g) {
while (1) {
a0=a
a=(a0+g)/2
g=sqrt(a0*g)
if (abs(a0-a) < abs(a)*1e-15) break
}
return a
}
function abs(x) {
return (x<0 ? -x : x)
}



Output

0.8472130847939792


=

## Commodore BASIC

=

10 A = 1
20 G = 1/SQR(2)
30 GOSUB 100
40 PRINT A
50 END
100 TA = A
110 A = (A+G)/2
120 G = SQR(TA*G)
130 IF A<TA THEN 100
140 RETURN


=

## BBC BASIC

= {{works with|BBC BASIC for Windows}}

      *FLOAT 64
@% = &1010
PRINT FNagm(1, 1/SQR(2))
END

DEF FNagm(a,g)
LOCAL ta
REPEAT
ta = a
a = (a+g)/2
g = SQR(ta*g)
UNTIL a = ta
= a



Produces this output:


0.8472130847939792



==={{header|IS-BASIC}}=== 100 PRINT AGM(1,1/SQR(2)) 110 DEF AGM(A,G) 120 DO 130 LET TA=A 140 LET A=(A+G)/2:LET G=SQR(TA*G) 150 LOOP UNTIL A=TA 160 LET AGM=A 170 END DEF



## bc

bc
/* Calculate the arithmethic-geometric mean of two positive
* numbers x and y.
* Result will have d digits after the decimal point.
*/
define m(x, y, d) {
auto a, g, o

o = scale
scale = d
d = 1 / 10 ^ d

a = (x + y) / 2
g = sqrt(x * y)
while ((a - g) > d) {
x = (a + g) / 2
g = sqrt(a * g)
a = x
}

scale = o
return(a)
}

scale = 20
m(1, 1 / sqrt(2), 20)


{{Out}}

.84721308479397908659


## C

### Basic

#include <math.h>
#include <stdio.h>
#include <stdlib.h>

double agm( double a, double g ) {
/* arithmetic-geometric mean */
double iota = 1.0E-16;
double a1, g1;

if( a*g < 0.0 ) {
printf( "arithmetic-geometric mean undefined when x*y<0\n" );
exit(1);
}

while( fabs(a-g)>iota ) {
a1 = (a + g) / 2.0;
g1 = sqrt(a * g);

a = a1;
g = g1;
}

return a;
}

int main( void ) {
double x, y;
printf( "Enter two numbers: " );
scanf( "%lf%lf", &x, &y );
printf( "The arithmetic-geometric mean is %lf\n", agm(x, y) );
return 0;
}



Original output:


Enter two numbers: 1.0 2.0
The arithmetic-geometric mean is 1.456791



Task output, the second input (0.707) is 1/sqrt(2) correct to 3 decimal places:


Enter two numbers: 1 0.707
The arithmetic-geometric mean is 0.847155



### GMP

/*Arithmetic Geometric Mean of 1 and 1/sqrt(2)

Nigel_Galloway
February 7th., 2012.
*/

#include "gmp.h"

void agm (const mpf_t in1, const mpf_t in2, mpf_t out1, mpf_t out2) {
mpf_div_ui (out1, out1, 2);
mpf_mul (out2, in1, in2);
mpf_sqrt (out2, out2);
}

int main (void) {
mpf_set_default_prec (65568);
mpf_t x0, y0, resA, resB;

mpf_init_set_ui (y0, 1);
mpf_init_set_d (x0, 0.5);
mpf_sqrt (x0, x0);
mpf_init (resA);
mpf_init (resB);

for(int i=0; i<7; i++){
agm(x0, y0, resA, resB);
agm(resA, resB, x0, y0);
}
gmp_printf ("%.20000Ff\n", x0);
gmp_printf ("%.20000Ff\n\n", y0);

return 0;
}


The first couple of iterations produces:


0.853
0.840



Then 7 iterations produces:


0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229942122285625233410963097962665830871059699713635983384251176326814289060389706768601616650048281188721897713309411767462019944392962902167289194499507231677897346863947606671057980557852173140349398304200422119216039839553595098193641293716340646029599967970599434351602031842648756950242174863855405981954581601742417887854192758804162719012085587685648326834140431218400804035809204559494313877815120926522254574397124286820766340954733674599621792665535348625686118543308626287287287563010835563193570668714785639088982115108836352147696979612621832943228417868113768445170018146021913694027020945996683513596327880804274345481744587363220025153952936265806614198365616491626259607434723706616902353080017375312847852558430631907454274934152685790655269406003147591020332746719686124796325510554648902820855297439651249940096625528660675804487353892185701401167716976535014084952476848993257321337028984668939194665861873752966387562266045914777044204681089256584408380320409106190031537067341195941010074743310599055058205243260099516927924174782169767810616836977141107392733439215501430220070873673659622721492587761928510523803670268904639096219076636442355380859029452340651900133423451058383417121805142550039237011113254111446126289062541335505266436535958245521562933975182514706501346410470569793556813066063293733450387109770972948759171790158173202815782884871499313408154933423677970447127859376185950851466773645546792016159342239971429840707888822790326567515965284358177957272848083564899635044041407342261101833835469759626633304220849998523007427039302772434749797179732645525465430198316949684610986907439050680137661192529197709384412997070158894931666611619945922650113111839663525025305616464315872084545229887754751772727476567216489829182392388952072076428397108847059603569219929218319015481412807665926982944644571492396663299730758139049576224389624231752095073190184244624423709864272811495111808228260538624846176751801409831274972576519837564923569028002161749055314272081534395405955635763711272816570597373374429700390560401563886630722257003892301591123769601215800817790778633512408624310735715837659265045466527873378744448344063102447570396812554539822664303534164130356138016341655752655897529445211668734512201912274667331915712407637538211069681410769263900748331757433967523196603308649735713838741960989838322028826948821913028193669499544222406972761686213695116578388850121990961606554546115432531481642493326947970041594914763231129205935165189979433500459762882172926259180894055084314663937825483351395501906533708720620640240770560758487964998436515927282645344286366154191425857771067561850172780332871751951893050318055052454260223355229007714181287986543511879180063562795936247682677864122494603381260826282540988953125276775346562432792145112295555160318184331336929617230417838551571255674049834166659269695800089537245730576945422753721602096871914703988784663672432627061911270717165908246400416799411204056571036408300024192943985530739946565396778104927010554103595133394321999250666762020783946955537605517964010097492188563113010178138885787938131720959480625392013009836502879176958279859052799477219417979970249430621584194688853281154977215799601944096234776861440850757392842988237593968232236705803341347746231128976258593243766317789749110772619097044895222045096307255155900938249040213648077920347672150485684460225544099928261631743126422857876289833806507220230103717531492635046310601885737725670066183812905806389545081270313113710437161358334880658339554312179013483988332164130576352447125115394720666703301013487165163241138288176398396295261211412632197959650986567867552507607604240959075175230219461045325643332496149012535333292237238689481278850201359663053760558493589283916304694038878549600274714871978014576595790495858022600660995249673643249668334617601066081567069751423818665036108388522097616550025160731149921612947757901997292486896382206038087602762816723701668191066335857751546503813342367223476420265585655884641601021054048985561871147358849763784064864267981865044863190774703822867114351511230036070865742988647714667473375011434581885279700605621172469217484718069486625119947289344427037830462070735493805287272062156063071882868580564521110696708028569906982576917722099867195996850779068144349493280497681154368046325993869307623507099951829512958112123570724538335482619075239515827309824818054966589790916886798407170779370595904577584091047341310960419411135775662072733779783320379730113767265853574771027978140972130961214239385473746276961504130795283737288205065871915225976508402779699176117539300672549249122984508236297556872271106584943553385049453263873648980460665597995436016950309279009245005785647723587619884898603441219534079536900299641197454906074160097885953766072290516077242859007090115663913836429904122082676962979786764903235649998199076599743987054864876909102491192709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00



The limit (19,740) is imposed by the accuracy (65568). Using 6 iterations would produce a less accurate result. At 7 iterations increasing the 65568 would mean we already have 38,000 or so digits accurate.

## C++


#include<bits/stdc++.h>
using namespace std;
#define _cin	ios_base::sync_with_stdio(0);	cin.tie(0);
#define rep(a, b)	for(ll i =a;i<=b;++i)

double agm(double a, double g)		//ARITHMETIC GEOMETRIC MEAN
{	double epsilon = 1.0E-16,a1,g1;
if(a*g<0.0)
{	cout<<"Couldn't find arithmetic-geometric mean of these numbers\n";
exit(1);
}
while(fabs(a-g)>epsilon)
{	a1 = (a+g)/2.0;
g1 = sqrt(a*g);
a = a1;
g = g1;
}
return a;
}

int main()
{	_cin;    //fast input-output
double x, y;
cout<<"Enter X and Y: ";	//Enter two numbers
cin>>x>>y;
cout<<"\nThe Arithmetic-Geometric Mean of "<<x<<" and "<<y<<" is "<<agm(x, y);
return 0;
}



Enter X and Y: 1.0 2.0
The Arithmetic-Geometric Mean of 1.0 and 2.0 is 1.45679103104690677028543177584651857614517211914062



## C#

namespace RosettaCode.ArithmeticGeometricMean
{
using System;
using System.Collections.Generic;
using System.Globalization;

internal static class Program
{
private static double ArithmeticGeometricMean(double number,
double otherNumber,
IEqualityComparer<double>
comparer)
{
return comparer.Equals(number, otherNumber)
? number
: ArithmeticGeometricMean(
ArithmeticMean(number, otherNumber),
GeometricMean(number, otherNumber), comparer);
}

private static double ArithmeticMean(double number, double otherNumber)
{
return 0.5 * (number + otherNumber);
}

private static double GeometricMean(double number, double otherNumber)
{
return Math.Sqrt(number * otherNumber);
}

private static void Main()
{
Console.WriteLine(
ArithmeticGeometricMean(1, 0.5 * Math.Sqrt(2),
new RelativeDifferenceComparer(1e-5)).
ToString(CultureInfo.InvariantCulture));
}

private class RelativeDifferenceComparer : IEqualityComparer<double>
{

internal RelativeDifferenceComparer(double maximumRelativeDifference)
{
_maximumRelativeDifference = maximumRelativeDifference;
}

public bool Equals(double number, double otherNumber)
{
return RelativeDifference(number, otherNumber) <=
_maximumRelativeDifference;
}

public int GetHashCode(double number)
{
return number.GetHashCode();
}

private static double RelativeDifference(double number,
double otherNumber)
{
return AbsoluteDifference(number, otherNumber) /
Norm(number, otherNumber);
}

private static double AbsoluteDifference(double number,
double otherNumber)
{
return Math.Abs(number - otherNumber);
}

private static double Norm(double number, double otherNumber)
{
return 0.5 * (Math.Abs(number) + Math.Abs(otherNumber));
}
}
}
}


Output:

0.847213084835193


### C# with System.Numerics

{{Libheader|System.Numerics}} Even though the System.Numerics library directly supports only '''BigInteger''' (and not big rationals or big floating point numbers), it can be coerced into making this calculation. One just has to keep track of the decimal place and multiply by a very large constant.

using System;
using System.Numerics;

namespace agm
{
class Program
{
static BigInteger BIP(char leadDig, int numDigs)
{
return BigInteger.Parse(leadDig + new string('0', numDigs));
}

static BigInteger IntSqRoot(BigInteger v)
{
int digs = Math.Max(0, v.ToString().Length / 2);
BigInteger res = BIP('3', digs), term;
while (true) {
term = v / res; if (Math.Abs((double)(term - res)) < 2) break;
res = (res + term) / 2; } return res;
}

static BigInteger CalcByAGM(int digits)
{
int digs = digits + (int)(Math.Log(digits) / 2), d2 = digs * 2;
BigInteger a = BIP('1', digs),              // initial value = 1
b = IntSqRoot(BIP('5', d2 - 1)), // initial value = square root of 0.5
c;
while (true) {
c = a; a = ((a + b) / 2); b = IntSqRoot(c * b);
if (Math.Abs((double)(a - b)) <= 1) break;
}
return b;
}

static void Main(string[] args)
{
int digits = 25000;
if (args.Length > 0)
{
int.TryParse(args[0], out digits);
if (digits < 1 || digits > 999999) digits = 25000;
}
Console.WriteLine("0.{0}", CalcByAGM(digits).ToString());
}
}
}


{{out}}

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4850842


## Clojure

lisp
(ns agmcompute
(:gen-class))

; Java Arbitray Precision Library
(import '(org.apfloat Apfloat ApfloatMath))

(def precision 70)
(def one (Apfloat. 1M precision))
(def two (Apfloat. 2M precision))
(def half (Apfloat. 0.5M precision))
(def isqrt2 (.divide one  (ApfloatMath/pow two half)))
(def TOLERANCE (Apfloat. 0.000000M precision))

(defn agm [a g]
" Simple AGM Loop calculation "
(let [THRESH 1e-65                 ; done when error less than threshold or we exceed max loops
MAX-LOOPS 1000000]
(loop [[an gn] [a g], cnt 0]
(if (or (< (ApfloatMath/abs (.subtract an gn)) THRESH)
(> cnt MAX-LOOPS))
an
(recur [(.multiply (.add an gn) half) (ApfloatMath/pow (.multiply an gn) half)]
(inc cnt))))))

(println  (agm one isqrt2))



{{Output}}

txt

8.47213084793979086606499123482191636481445910326942185060579372659734e-1



## COBOL

cobol
IDENTIFICATION DIVISION.
PROGRAM-ID. ARITHMETIC-GEOMETRIC-MEAN-PROG.
DATA DIVISION.
WORKING-STORAGE SECTION.
01  AGM-VARS.
05 A       PIC 9V9(16).
05 A-ZERO  PIC 9V9(16).
05 G       PIC 9V9(16).
05 DIFF    PIC 9V9(16) VALUE 1.
* Initialize DIFF with a non-zero value, otherwise AGM-PARAGRAPH
* is never performed at all.
PROCEDURE DIVISION.
TEST-PARAGRAPH.
MOVE    1 TO A.
COMPUTE G = 1 / FUNCTION SQRT(2).
* The program will run with the test values. If you would rather
* calculate the AGM of numbers input at the console, comment out
* TEST-PARAGRAPH and un-comment-out INPUT-A-AND-G-PARAGRAPH.
* INPUT-A-AND-G-PARAGRAPH.
*     DISPLAY 'Enter two numbers.'
*     ACCEPT  A.
*     ACCEPT  G.
CONTROL-PARAGRAPH.
PERFORM AGM-PARAGRAPH UNTIL DIFF IS LESS THAN 0.000000000000001.
DISPLAY A.
STOP RUN.
AGM-PARAGRAPH.
MOVE     A TO A-ZERO.
COMPUTE  A = (A-ZERO + G) / 2.
MULTIPLY A-ZERO BY G GIVING G.
COMPUTE  G = FUNCTION SQRT(G).
SUBTRACT A FROM G GIVING DIFF.
COMPUTE  DIFF = FUNCTION ABS(DIFF).


{{out}}

txt
0.8472130847939792


## Common Lisp

lisp
(defun agm (a0 g0 &optional (tolerance 1d-8))
(loop for a = a0 then (* (+ a g) 5d-1)
and g = g0 then (sqrt (* a g))
until (< (abs (- a g)) tolerance)
finally (return a)))



{{out}}

txt
CL-USER> (agm 1d0 (/ 1d0 (sqrt 2d0)))
0.8472130848351929d0
CL-USER> (agm 1d0 (/ 1d0 (sqrt 2d0)) 1d-10)
0.8472130848351929d0
CL-USER> (agm 1d0 (/ 1d0 (sqrt 2d0)) 1d-12)
0.8472130847939792d0


## D

d
import std.stdio, std.math, std.meta, std.typecons;

real agm(real a, real g, in int bitPrecision=60) pure nothrow @nogc @safe {
do {
//{a, g} = {(a + g) / 2.0, sqrt(a * g)};
AliasSeq!(a, g) = tuple((a + g) / 2.0, sqrt(a * g));
} while (feqrel(a, g) < bitPrecision);
return a;
}

void main() @safe {
writefln("%0.19f", agm(1, 1 / sqrt(2.0)));
}


{{out}}

txt
0.8472130847939790866


All the digits shown are exact.

## EchoLisp

We use the '''(~= a b)''' operator which tests for  |a - b| < ε = (math-precision).

scheme

(lib 'math)

(define (agm a g)
(if (~= a g) a
(agm (// (+ a g ) 2) (sqrt (* a g)))))

(math-precision)
→ 0.000001 ;; default
(agm 1 (/ 1 (sqrt 2)))
→ 0.8472130848351929
(math-precision 1.e-15)
→ 1e-15
(agm 1 (/ 1 (sqrt 2)))
→ 0.8472130847939792



## Elixir

Elixir
defmodule ArithhGeom do
def mean(a,g,tol) when abs(a-g) <= tol, do: a
def mean(a,g,tol) do
mean((a+g)/2,:math.pow(a*g, 0.5),tol)
end
end

IO.puts ArithhGeom.mean(1,1/:math.sqrt(2),0.0000000001)


{{out}}

txt

0.8472130848351929



## Erlang

Erlang
%% Arithmetic Geometric Mean of 1 and 1 / sqrt(2)
%% Author: Abhay Jain

-module(agm_calculator).
-export([find_agm/0]).
-define(TOLERANCE, 0.0000000001).

find_agm() ->
A = 1,
B = 1 / (math:pow(2, 0.5)),
AGM = agm(A, B),
io:format("AGM = ~p", [AGM]).

agm (A, B) when abs(A-B) =< ?TOLERANCE ->
A;
agm (A, B) ->
A1 = (A+B) / 2,
B1 = math:pow(A*B, 0.5),
agm(A1, B1).


Output:

Erlang>AGM = 0.8472130848351929
PROGRAM AGM

!
! for rosettacode.org
!

!$DOUBLE PROCEDURE AGM(A,G->A) LOCAL TA REPEAT TA=A A=(A+G)/2 G=SQR(TA*G) UNTIL A=TA END PROCEDURE BEGIN AGM(1.0,1/SQR(2)->A) PRINT(A) END PROGRAM  =={{header|F_Sharp|F#}}== {{trans|OCaml}} fsharp let rec agm a g precision = if precision > abs(a - g) then a else agm (0.5 * (a + g)) (sqrt (a * g)) precision printfn "%g" (agm 1. (sqrt(0.5)) 1e-15)  Output txt 0.847213  ## Factor factor USING: kernel math math.functions prettyprint ; IN: rosetta-code.arithmetic-geometric-mean : agm ( a g -- a' g' ) 2dup [ + 0.5 * ] 2dip * sqrt ; 1 1 2 sqrt / [ 2dup - 1e-15 > ] [ agm ] while drop .  {{out}} txt 0.8472130847939792  ## Forth forth : agm ( a g -- m ) begin fover fover f+ 2e f/ frot frot f* fsqrt fover fover 1e-15 f~ until fdrop ; 1e 2e -0.5e f** agm f. \ 0.847213084793979  ## Fortran A '''Fortran 77''' implementation fortran function agm(a,b) implicit none double precision agm,a,b,eps,c parameter(eps=1.0d-15) 10 c=0.5d0*(a+b) b=sqrt(a*b) a=c if(a-b.gt.eps*a) go to 10 agm=0.5d0*(a+b) end program test implicit none double precision agm print*,agm(1.0d0,1.0d0/sqrt(2.0d0)) end  ## FreeBASIC freebasic ' version 16-09-2015 ' compile with: fbc -s console Function agm(a As Double, g As Double) As Double Dim As Double t_a Do t_a = (a + g) / 2 g = Sqr(a * g) Swap a, t_a Loop Until a = t_a Return a End Function ' ------=< MAIN >=------ Print agm(1, 1 / Sqr(2) ) ' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End  {{out}} txt 0.8472130847939792  ## Futhark {{incorrect|Futhark|Futhark's syntax has changed, so this example will not compile}} Futhark import "futlib/math" fun agm(a: f64, g: f64): f64 = let eps = 1.0E-16 loop ((a,g)) = while f64.abs(a-g) > eps do ((a+g) / 2.0, f64.sqrt (a*g)) in a fun main(x: f64, y: f64): f64 = agm(x,y)  ## Go go package main import ( "fmt" "math" ) const ε = 1e-14 func agm(a, g float64) float64 { for math.Abs(a-g) > math.Abs(a)*ε { a, g = (a+g)*.5, math.Sqrt(a*g) } return a } func main() { fmt.Println(agm(1, 1/math.Sqrt2)) }  {{out}} txt 0.8472130847939792  ## Groovy {{trans|Java}} Solution: groovy double agm (double a, double g) { double an = a, gn = g while ((an-gn).abs() >= 10.0**-14) { (an, gn) = [(an+gn)*0.5, (an*gn)**0.5] } an }  Test: groovy println "agm(1, 0.5**0.5) = agm(1,${0.5**0.5}) = ${agm(1, 0.5**0.5)}" assert (0.8472130847939792 - agm(1, 0.5**0.5)).abs() <= 10.0**-14  Output: txt agm(1, 0.5**0.5) = agm(1, 0.7071067811865476) = 0.8472130847939792  ## Haskell haskell -- Return an approximation to the arithmetic-geometric mean of two numbers. -- The result is considered accurate when two successive approximations are -- sufficiently close, as determined by "eq". agm :: (Floating a) => a -> a -> ((a, a) -> Bool) -> a agm a g eq = snd . head . dropWhile (not . eq)$ iterate step (a, g)
where step (a, g) = ((a + g) / 2, sqrt (a * g))

-- Return the relative difference of the pair.  We assume that at least one of
-- the values is far enough from 0 to not cause problems.
relDiff :: (Fractional a) => (a, a) -> a
relDiff (x, y) = let n = abs (x - y)
d = ((abs x) + (abs y)) / 2
in n / d

main = do
let equal = (< 0.000000001) . relDiff
print $agm 1 (1 / sqrt 2) equal  {{out}} txt 0.8472130847527654  =={{header|Icon}} and {{header|Unicon}}== procedure main(A) a := real(A[1]) | 1.0 g := real(A[2]) | (1 / 2^0.5) epsilon := real(A[3]) write("agm(",a,",",g,") = ",agm(a,g,epsilon)) end procedure agm(an, gn, e) /e := 1e-15 while abs(an-gn) > e do { ap := (an+gn)/2.0 gn := (an*gn)^0.5 an := ap } return an end  Output: txt ->agm agm(1.0,0.7071067811865475) = 0.8472130847939792 ->  ## J This one is probably worth not naming, in J, because there are so many interesting variations. First, the basic approach (with display precision set to 16 digits, which slightly exceeds the accuracy of 64 bit IEEE floating point arithmetic): j mean=: +/ % # (mean , */ %:~ #)^:_] 1,%%:2 0.8472130847939792 0.8472130847939791  This is the limit -- it stops when values are within a small epsilon of previous calculations. We can ask J for unique values (which also means -- unless we specify otherwise -- values within a small epsilon of each other, for floating point values): j ~.(mean , */ %:~ #)^:_] 1,%%:2 0.8472130847939792  Another variation would be to show intermediate values, in the limit process: j (mean, */ %:~ #)^:a: 1,%%:2 1 0.7071067811865475 0.8535533905932737 0.8408964152537145 0.8472249029234942 0.8472012667468915 0.8472130848351929 0.8472130847527654 0.8472130847939792 0.8472130847939791  ### Arbitrary Precision Another variation would be to use [[j:Essays/Extended%20Precision%20Functions|arbitrary precision arithmetic]] in place of floating point arithmetic. Borrowing routines from that page, but going with a default of approximately 100 digits of precision: J DP=:101 round=: DP&$: : (4 : 0)
b %~ <.1r2+y*b=. 10x^x
)

sqrt=: DP&$: : (4 : 0) " 0 assert. 0<:y %/ <.@%: (2 x: (2*x) round y)*10x^2*x+0>.>.10^.y ) ln=: DP&$: : (4 : 0) " 0
assert. 0:) (x:!.0 y)%2x^m
if. x<-:#":t do. t=. (1+x) round t end.
ln2=. 2*+/1r3 (^%]) 1+2*i.>.0.5*(%3)^.0.5*0.1^x+>.10^.1>.m
lnr=. 2*+/t   (^%]) 1+2*i.>.0.5*(|t)^.0.5*0.1^x
lnr + m * ln2
)

exp=: DP&$: : (4 : 0) " 0 m=. <.0.5+y%^.2 xm=. x+>.m*10^.2 d=. (x:!.0 y)-m*xm ln 2 if. xm<-:#":d do. d=. xm round d end. e=. 0.1^xm n=. e (>i.1:) a (^%!@]) i.>.a^.e [ a=. |y-m*^.2 (2x^m) * 1++/*/\d%1+i.n )  We are also going to want a routine to display numbers with this precision, and we are going to need to manage epsilon manually, and we are going to need an arbitrary root routine: J fmt=:[: ;:inv DP&$: : (4 :0)&.>
x{.deb (x*2j1)":y
)

root=: ln@] exp@% [

epsilon=: 1r9^DP


Some example uses:

J
fmt sqrt 2
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572
fmt *~sqrt 2
2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
fmt epsilon
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000418
fmt 2 root 2
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572


Note that 2 root 2 is considerably slower than sqrt 2. The price of generality. So, while we could define geometric mean generally, a desire for good performance pushes us to use a routine specialized for two numbers:

J
geomean=: */ root~ #
geomean2=: [: sqrt */


A quick test to make sure these can be equivalent:

J
fmt geomean 3 5
3.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673517
fmt geomean2 3 5
3.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673517


J
fmt (mean, geomean2)^:(epsilon <&| -/)^:a: 1,%sqrt 2
1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0.707106781186547524400844362104849039284835937688474036588339868995366239231053519425193767163820786
0.853553390593273762200422181052424519642417968844237018294169934497683119615526759712596883581910393 0.840896415253714543031125476233214895040034262356784510813226085974924754953902239814324004199292536
0.847224902923494152615773828642819707341226115600510764553698010236303937284714499763460443890601464 0.847201266746891460403631453693352397963981013612000500823295747923488191871327668107581434542353536
0.847213084835192806509702641168086052652603564606255632688496879079896064578021083935520939216477500 0.847213084752765366704298051779902070392110656059452583317776227659438896688518556753569298762449381
0.847213084793979086607000346473994061522357110332854108003136553369667480633269820344545118989463440 0.847213084793979086605997900490389211440534858586261300461413929971399281619068666682569108141224710
0.847213084793979086606499123482191636481445984459557704232275241670533381126169243513557113565344075 0.847213084793979086606499123482191636481445836194326665888883503648934628542100275932846717790147361
0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723201915677745718 0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723198672311476741
0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229 0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229


We could of course extract out only a representative final value, but it's obvious enough, and showing how rapidly this converges is fun.

## Java

Java
/*
* Arithmetic-Geometric Mean of 1 & 1/sqrt(2)
* Brendan Shaklovitz
* 5/29/12
*/
public class ArithmeticGeometricMean {

public static double agm(double a, double g) {
double a1 = a;
double g1 = g;
while (Math.abs(a1 - g1) >= 1.0e-14) {
double arith = (a1 + g1) / 2.0;
double geom = Math.sqrt(a1 * g1);
a1 = arith;
g1 = geom;
}
return a1;
}

public static void main(String[] args) {
System.out.println(agm(1.0, 1.0 / Math.sqrt(2.0)));
}
}


{{out}}

txt
0.8472130847939792


## JavaScript

### ES5

JavaScript
function agm(a0, g0) {
var an = (a0 + g0) / 2,
gn = Math.sqrt(a0 * g0);
while (Math.abs(an - gn) > tolerance) {
an = (an + gn) / 2, gn = Math.sqrt(an * gn)
}
return an;
}

agm(1, 1 / Math.sqrt(2));


### ES6

JavaScript
(() => {
'use strict';

// ARITHMETIC-GEOMETRIC MEAN

// agm :: Num a => a -> a -> a
let agm = (a, g) => {
let abs = Math.abs,
sqrt = Math.sqrt;

return until(
m => abs(m.an - m.gn) < tolerance,
m => {
return {
an: (m.an + m.gn) / 2,
gn: sqrt(m.an * m.gn)
};
}, {
an: (a + g) / 2,
gn: sqrt(a * g)
}
)
.an;
},

// GENERIC

// until :: (a -> Bool) -> (a -> a) -> a -> a
until = (p, f, x) => {
let v = x;
while (!p(v)) v = f(v);
return v;
};

// TEST

let tolerance = 0.000001;

return agm(1, 1 / Math.sqrt(2));

})();


{{Out}}

JavaScript>0.8472130848351929 tolerance
then [add/2, ((.[0] * .[1])|sqrt)] | _agm
else .
end;
[a, g] | _agm | .[0] ;


This version avoids an infinite loop if the requested tolerance is too small:

jq
def agm(a; g; tolerance):
def abs: if . < 0 then -. else . end;
def _agm:
# state [an,gn, delta]
((.[0] - .[1])|abs) as $delta | if$delta == .[2] and $delta < 10e-16 then . elif$delta > tolerance
then [ .[0:2]|add / 2, ((.[0] * .[1])|sqrt), $delta] | _agm else . end; if tolerance <= 0 then error("specified tolerance must be > 0") else [a, g, 0] | _agm | .[0] end ; # Example: agm(1; 1/(2|sqrt); 1e-100)  {{Out}}$ jq -n -f Arithmetic-geometric_mean.jq
0.8472130847939792

## Julia

{{works with|Julia|1.2}}

Julia
function agm(x, y, e::Real = 5)
(x ≤ 0 || y ≤ 0 || e ≤ 0) && throw(DomainError("x, y must be strictly positive"))
g, a = minmax(x, y)
while e * eps(x) < a - g
a, g = (a + g) / 2, sqrt(a * g)
end
a
end

x, y = 1.0, 1 / √2
println("# Using literal-precision float numbers:")
@show agm(x, y)

println("# Using half-precision float numbers:")
x, y = Float32(x), Float32(y)
@show agm(x, y)

println("# Using ", precision(BigFloat), "-bit float numbers:")
x, y = big(1.0), 1 / √big(2.0)
@show agm(x, y)


The ε  for this calculation is given as a positive integer multiple of the machine ε for x.

{{out}}

txt
# Using literal-precision float numbers:
agm(x, y) = 0.8472130847939792
# Using half-precision float numbers:
agm(x, y) = 0.84721315f0
# Using 256-bit float numbers:
agm(x, y) = 8.472130847939790866064991234821916364814459103269421850605793726597340048341323e-01


## Kotlin

scala
// version 1.0.5-2

fun agm(a: Double, g: Double): Double {
var aa = a             // mutable 'a'
var gg = g             // mutable 'g'
var ta: Double         // temporary variable to hold next iteration of 'aa'
val epsilon = 1.0e-16  // tolerance for checking if limit has been reached

while (true) {
ta = (aa + gg) / 2.0
if (Math.abs(aa - ta) <= epsilon) return ta
gg = Math.sqrt(aa * gg)
aa = ta
}
}

fun main(args: Array) {
println(agm(1.0, 1.0 / Math.sqrt(2.0)))
}


{{out}}

txt

0.8472130847939792



## LFE

lisp

(defun agm (a g)
(agm a g 1.0e-15))

(defun agm (a g tol)
(if (=< (- a g) tol)
a
(agm (next-a a g)
(next-g a g)
tol)))

(defun next-a (a g)
(/ (+ a g) 2))

(defun next-g (a g)
(math:sqrt (* a g)))



Usage:

txt

> (agm 1 (/ 1 (math:sqrt 2)))
0.8472130847939792



## Liberty BASIC

lb

print agm(1, 1/sqr(2))
print using("#.#################",agm(1, 1/sqr(2)))

function agm(a,g)
do
absdiff = abs(a-g)
an=(a+g)/2
gn=sqr(a*g)
a=an
g=gn
loop while abs(an-gn)< absdiff
agm = a
end function



## LiveCode

LiveCode
function agm aa,g
put abs(aa-g) into absdiff
put (aa+g)/2 into aan
put sqrt(aa*g) into gn
repeat while abs(aan - gn) < absdiff
put abs(aa-g) into absdiff
put (aa+g)/2 into aan
put sqrt(aa*g) into gn
put aan into aa
put gn into g
end repeat
return aa
end agm


Example

LiveCode
put agm(1, 1/sqrt(2))
-- ouput
-- 0.847213


## LLVM

llvm
; This is not strictly LLVM, as it uses the C library function "printf".
; LLVM does not provide a way to print values, so the alternative would be
; to just load the string into memory, and that would be boring.

; Additional comments have been inserted, as well as changes made from the output produced by clang such as putting more meaningful labels for the jumps

$"ASSERTION" = comdat any$"OUTPUT" = comdat any

@"ASSERTION" = linkonce_odr unnamed_addr constant [48 x i8] c"arithmetic-geometric mean undefined when x*y<0\0A\00", comdat, align 1
@"OUTPUT" = linkonce_odr unnamed_addr constant [42 x i8] c"The arithmetic-geometric mean is %0.19lf\0A\00", comdat, align 1

;--- The declarations for the external C functions
declare i32 @printf(i8*, ...)
declare void @exit(i32) #1
declare double @sqrt(double) #1

declare double @llvm.fabs.f64(double) #2

;----------------------------------------------------------------
;-- arithmetic geometric mean
define double @agm(double, double) #0 {
%3 = alloca double, align 8                     ; allocate local g
%4 = alloca double, align 8                     ; allocate local a
%5 = alloca double, align 8                     ; allocate iota
%6 = alloca double, align 8                     ; allocate a1
%7 = alloca double, align 8                     ; allocate g1
store double %1, double* %3, align 8            ; store param g in local g
store double %0, double* %4, align 8            ; store param a in local a
store double 1.000000e-15, double* %5, align 8  ; store 1.0e-15 in iota (1.0e-16 was causing the program to hang)

%10 = fmul double %8, %9                        ; a * g
%11 = fcmp olt double %10, 0.000000e+00         ; a * g < 0.0
br i1 %11, label %enforce, label %loop

enforce:
%12 = call i32 (i8*, ...) @printf(i8* getelementptr inbounds ([48 x i8], [48 x i8]* @"ASSERTION", i32 0, i32 0))
call void @exit(i32 1) #6
unreachable

loop:
%15 = fsub double %13, %14                      ; a - g
%16 = call double @llvm.fabs.f64(double %15)    ; fabs(a - g)
%18 = fcmp ogt double %16, %17                  ; fabs(a - g) > iota
br i1 %18, label %loop_body, label %eom

loop_body:
%21 = fadd double %19, %20                      ; a + g
%22 = fdiv double %21, 2.000000e+00             ; (a + g) / 2.0
store double %22, double* %6, align 8           ; store %22 in a1

%25 = fmul double %23, %24                      ; a * g
%26 = call double @sqrt(double %25) #4          ; sqrt(a * g)
store double %26, double* %7, align 8           ; store %26 in g1

store double %27, double* %4, align 8           ; store a1 in a

store double %28, double* %3, align 8           ; store g1 in g

br label %loop

eom:
ret double %29                                  ; return a
}

;----------------------------------------------------------------
;-- main
define i32 @main() #0 {
%1 = alloca double, align 8                     ; allocate x
%2 = alloca double, align 8                     ; allocate y

store double 1.000000e+00, double* %1, align 8  ; store 1.0 in x

%3 = call double @sqrt(double 2.000000e+00) #4  ; calculate the square root of two
%4 = fdiv double 1.000000e+00, %3               ; divide 1.0 by %3
store double %4, double* %2, align 8            ; store %4 in y

%7 = call double @agm(double %6, double %5)     ; agm(x, y)

%8 = call i32 (i8*, ...) @printf(i8* getelementptr inbounds ([42 x i8], [42 x i8]* @"OUTPUT", i32 0, i32 0), double %7)

ret i32 0                                       ; finished
}

attributes #0 = { noinline nounwind optnone uwtable "correctly-rounded-divide-sqrt-fp-math"="false" "disable-tail-calls"="false" "less-precise-fpmad"="false" "no-frame-pointer-elim"="false" "no-infs-fp-math"="false" "no-jump-tables"="false" "no-nans-fp-math"="false" "no-signed-zeros-fp-math"="false" "no-trapping-math"="false" "stack-protector-buffer-size"="8" "target-cpu"="x86-64" "target-features"="+fxsr,+mmx,+sse,+sse2,+x87" "unsafe-fp-math"="false" "use-soft-float"="false" }
attributes #1 = { noreturn "correctly-rounded-divide-sqrt-fp-math"="false" "disable-tail-calls"="false" "less-precise-fpmad"="false" "no-frame-pointer-elim"="false" "no-infs-fp-math"="false" "no-nans-fp-math"="false" "no-signed-zeros-fp-math"="false" "no-trapping-math"="false" "stack-protector-buffer-size"="8" "target-cpu"="x86-64" "target-features"="+fxsr,+mmx,+sse,+sse2,+x87" "unsafe-fp-math"="false" "use-soft-float"="false" }
attributes #2 = { nounwind readnone speculatable }
attributes #4 = { nounwind }
attributes #6 = { noreturn }


{{out}}

txt
The arithmetic-geometric mean is 0.8472130847939791654


## Logo

logo
output and [:a - :b < 1e-15] [:a - :b > -1e-15]
end
to agm :arith :geom
if about :arith :geom [output :arith]
output agm (:arith + :geom)/2  sqrt (:arith * :geom)
end

show agm 1 1/sqrt 2



## Lua

lua
function agm(a, b, tolerance)
if not tolerance or tolerance < 1e-15 then
tolerance = 1e-15
end
repeat
a, b = (a + b) / 2, math.sqrt(a * b)
until math.abs(a-b) < tolerance
return a
end

print(string.format("%.15f", agm(1, 1 / math.sqrt(2))))


'''Output:'''

0.847213084793979

## M2000 Interpreter

M2000 Interpreter

Module Checkit {
Function Agm {
\\ new stack constructed at calling the Agm() with two values
Repeat {
Push  Sqrt(a0*b0), (a0+b0)/2
} Until Stackitem(1)==Stackitem(2)
=Stackitem(1)
\\ stack deconstructed at exit of function
}
Print Agm(1,1/Sqrt(2))
}
Checkit



## Maple

Maple provides this function under the name GaussAGM.  To compute a floating point approximation, use evalf.

Maple

> evalf( GaussAGM( 1, 1 / sqrt( 2 ) ) ); # default precision is 10 digits
0.8472130847

> evalf[100]( GaussAGM( 1, 1 / sqrt( 2 ) ) ); # to 100 digits
0.847213084793979086606499123482191636481445910326942185060579372659\
7340048341347597232002939946112300



Alternatively, if one or both arguments is already a float, Maple will compute a floating point approximation automatically.

Maple

> GaussAGM( 1.0, 1 / sqrt( 2 ) );
0.8472130847



## Mathematica

To any arbitrary precision, just increase PrecisionDigits

Mathematica
PrecisionDigits = 85;
AGMean[a_, b_] := FixedPoint[{ Tr@#/2, Sqrt[Times@@#] }&, N[{a,b}, PrecisionDigits]]〚1〛


txt
AGMean[1, 1/Sqrt[2]]
0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597232


MATLAB
function [a,g]=agm(a,g)
%%arithmetic_geometric_mean(a,g)
while (1)
a0=a;
a=(a0+g)/2;
g=sqrt(a0*g);
if (abs(a0-a) < a*eps) break; end;
end;
end


txt
octave:26> agm(1,1/sqrt(2))
ans =  0.84721



## Maxima

maxima
agm(a, b) := %pi/4*(a + b)/elliptic_kc(((a - b)/(a + b))^2)$agm(1, 1/sqrt(2)), bfloat, fpprec: 85; /* 8.472130847939790866064991234821916364814459103269421850605793726597340048341347597232b-1 */  =={{header|МК-61/52}}== П1 <-> П0 1 ВП 8 /-/ П2 ИП0 ИП1 - ИП2 - /-/ x<0 31 ИП1 П3 ИП0 ИП1 * КвКор П1 ИП0 ИП3 + 2 / П0 БП 08 ИП0 С/П  =={{header|Modula-2}}== {{trans|C}} modula2 MODULE AGM; FROM EXCEPTIONS IMPORT AllocateSource,ExceptionSource,GetMessage,RAISE; FROM LongConv IMPORT ValueReal; FROM LongMath IMPORT sqrt; FROM LongStr IMPORT RealToStr; FROM Terminal IMPORT ReadChar,Write,WriteString,WriteLn; VAR TextWinExSrc : ExceptionSource; PROCEDURE ReadReal() : LONGREAL; VAR buffer : ARRAY[0..63] OF CHAR; i : CARDINAL; c : CHAR; BEGIN i := 0; LOOP c := ReadChar(); IF ((c >= '0') AND (c <= '9')) OR (c = '.') THEN buffer[i] := c; Write(c); INC(i) ELSE WriteLn; EXIT END END; buffer[i] := 0C; RETURN ValueReal(buffer) END ReadReal; PROCEDURE WriteReal(r : LONGREAL); VAR buffer : ARRAY[0..63] OF CHAR; BEGIN RealToStr(r, buffer); WriteString(buffer) END WriteReal; PROCEDURE AGM(a,g : LONGREAL) : LONGREAL; CONST iota = 1.0E-16; VAR a1, g1 : LONGREAL; BEGIN IF a * g < 0.0 THEN RAISE(TextWinExSrc, 0, "arithmetic-geometric mean undefined when x*y<0") END; WHILE ABS(a - g) > iota DO a1 := (a + g) / 2.0; g1 := sqrt(a * g); a := a1; g := g1 END; RETURN a END AGM; VAR x, y, z: LONGREAL; BEGIN WriteString("Enter two numbers: "); x := ReadReal(); y := ReadReal(); WriteReal(AGM(x, y)); WriteLn END AGM.  {{out}} txt Enter two numbers: 1.0 2.0 1.456791031046900  txt Enter two numbers: 1.0 0.707 0.847154622368330  ## NetRexx {{trans|Java}} NetRexx /* NetRexx */ options replace format comments java crossref symbols nobinary numeric digits 18 parse arg a_ g_ . if a_ = '' | a_ = '.' then a0 = 1 else a0 = a_ if g_ = '' | g_ = '.' then g0 = 1 / Math.sqrt(2) else g0 = g_ say agm(a0, g0) return -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ method agm(a0, g0) public static returns Rexx a1 = a0 g1 = g0 loop while (a1 - g1).abs() >= Math.pow(10, -14) temp = (a1 + g1) / 2 g1 = Math.sqrt(a1 * g1) a1 = temp end return a1 + 0  '''Output:''' txt 0.8472130847939792  ## NewLISP NewLISP (define (a-next a g) (mul 0.5 (add a g))) (define (g-next a g) (sqrt (mul a g))) (define (amg a g tolerance) (if (<= (sub a g) tolerance) a (amg (a-next a g) (g-next a g) tolerance) ) ) (define quadrillionth 0.000000000000001) (define root-reciprocal-2 (div 1.0 (sqrt 2.0))) (println "To the nearest one-quadrillionth, " "the arithmetic-geometric mean of " "1 and the reciprocal of the square root of 2 is " (amg 1.0 root-reciprocal-2 quadrillionth) )  ## Nim nim import math proc agm(a, g: float,delta: float = 1.0e-15): float = var aNew: float = 0 aOld: float = a gOld: float = g while (abs(aOld - gOld) > delta): aNew = 0.5 * (aOld + gOld) gOld = sqrt(aOld * gOld) aOld = aNew result = aOld echo$agm(1.0,1.0/sqrt(2.0))


Output:

txt

8.4721308479397917e-01



See first 24 iterations:

nim
from math import sqrt
from strutils import parseFloat, formatFloat, ffDecimal

proc agm(x,y: float): tuple[resA,resG: float] =
var
a,g: array[0 .. 23,float]

a[0] = x
g[0] = y

for n in 1 .. 23:
a[n] = 0.5 * (a[n - 1] + g[n - 1])
g[n] = sqrt(a[n - 1] * g[n - 1])

(a[23], g[23])

var t = agm(1, 1/sqrt(2.0))

echo("Result A: " & formatFloat(t.resA, ffDecimal, 24))
echo("Result G: " & formatFloat(t.resG, ffDecimal, 24))


{{works with|oo2c}}

oberon2

MODULE Agm;
IMPORT
Math := LRealMath,
Out;

CONST
epsilon = 1.0E-15;

PROCEDURE Of*(a,g: LONGREAL): LONGREAL;
VAR
na,ng,og: LONGREAL;
BEGIN
na := a; ng := g;
LOOP
og := ng;
ng := Math.sqrt(na * ng);
na := (na + og) * 0.5;
IF na - ng <= epsilon THEN EXIT END
END;
RETURN ng;
END Of;

BEGIN
Out.LongReal(Of(1,1 / Math.sqrt(2)),0,0);Out.Ln
END Agm.



{{Out}}

txt

8.4721308479397905E-1



## Objeck

{{trans|Java}}

objeck

class ArithmeticMean {
function : Amg(a : Float, g : Float) ~ Nil {
a1 := a;
g1 := g;
while((a1-g1)->Abs() >= Float->Power(10, -14)) {
tmp := (a1+g1)/2.0;
g1 := Float->SquareRoot(a1*g1);
a1 := tmp;
};
a1->PrintLine();
}

function : Main(args : String[]) ~ Nil {
Amg(1,1/Float->SquareRoot(2));
}
}



Output:

txt
0.847213085


## OCaml

ocaml
let rec agm a g tol =
if tol > abs_float (a -. g) then a else
agm (0.5*.(a+.g)) (sqrt (a*.g)) tol

let _ = Printf.printf "%.16f\n" (agm 1.0 (sqrt 0.5) 1e-15)


Output

txt
0.8472130847939792


## Oforth

Oforth
: agm   \ a b -- m
while( 2dup <> ) [ 2dup + 2 / -rot * sqrt ] drop ;


Usage :

Oforth>1 2 sqrt inv agm Double {
(x + y) / 2.
}
gmean: func (x: Double, y: Double) -> Double {
sqrt(x * y)
}
agm: func (a: Double, g: Double) -> Double {
while ((a - g) abs() > pow(10, -12)) {
(a1, g1) := (amean(a, g), gmean(a, g))
(a, g) = (a1, g1)
}
a
}

main: func {
"%.16f" printfln(agm(1., sqrt(0.5)))
}



Output

txt
0.8472130847939792


## ooRexx

ooRexx
numeric digits 20
say agm(1, 1/rxcalcsqrt(2,16))

::routine agm
use strict arg a, g
numeric digits 20

a1 = a
g1 = g

loop while abs(a1 - g1) >= 1e-14
temp = (a1 + g1)/2
g1 = rxcalcsqrt(a1*g1,16)
a1 = temp
end
return a1+0

::requires rxmath LIBRARY


{{out}}

txt
0.8472130847939791968


## PARI/GP

Built-in:

parigp
agm(1,1/sqrt(2))


Iteration:

parigp
agm2(x,y)=if(x==y,x,agm2((x+y)/2,sqrt(x*y))


## Pascal

{{works with|Free_Pascal}}
Port of the C example:

pascal
Program ArithmeticGeometricMean;

uses
gmp;

procedure agm (in1, in2: mpf_t; var out1, out2: mpf_t);
begin
mpf_div_ui (out1, out1, 2);
mpf_mul (out2, in1, in2);
mpf_sqrt (out2, out2);
end;

const
nl = chr(13)+chr(10);
var
x0, y0, resA, resB: mpf_t;
i: integer;
begin
mpf_set_default_prec (65568);

mpf_init_set_ui (y0, 1);
mpf_init_set_d (x0, 0.5);
mpf_sqrt (x0, x0);
mpf_init (resA);
mpf_init (resB);

for i := 0 to 6 do
begin
agm(x0, y0, resA, resB);
agm(resA, resB, x0, y0);
end;
mp_printf ('%.20000Ff'+nl, @x0);
mp_printf ('%.20000Ff'+nl+nl, @y0);
end.


Output is as long as the C example.

## Perl

perl
#!/usr/bin/perl -w

my ($a0,$g0, $a1,$g1);

sub agm() {
$a0 = shift;$g0 = shift;
do {
$a1 = ($a0 + $g0)/2;$g1 = sqrt($a0 *$g0);
$a0 = ($a1 + $g1)/2;$g0 = sqrt($a1 *$g1);
} while ($a0 !=$a1);
return $a0; } print agm(1, 1/sqrt(2))."\n";  Output: txt 0.847213084793979  ## Perl 6 perl6 sub agm($a is copy, $g is copy ) { ($a, $g) = ($a + $g)/2, sqrt$a * $g until$a ≅ $g; return$a;
}

say agm 1, 1/sqrt 2;


{{out}}

txt
0.84721308479397917


It's also possible to write it recursively:

Perl 6
sub agm( $a,$g ) {
$a ≅$g ?? $a !! agm(|@$_)
given ($a +$g)/2, sqrt $a *$g;
}

say agm 1, 1/sqrt 2;


## Phix

Phix
function agm(atom a, atom g, atom tolerance=1.0e-15)
while abs(a-g)>tolerance do
{a,g} = {(a + g)/2,sqrt(a*g)}
printf(1,"%0.15g\n",a)
end while
return a
end function
?agm(1,1/sqrt(2))   -- (rounds to 10 d.p.)


{{out}}

txt

0.853553390593274
0.847224902923494
0.847213084835193
0.847213084793979
0.8472130848



## PHP

php

define('PRECISION', 13);

function agm($a0,$g0, $tolerance = 1e-10) { // the bc extension deals in strings and cannot convert // floats in scientific notation by itself - hence // this manual conversion to a string$limit = number_format($tolerance, PRECISION, '.', '');$an    = $a0;$gn    = $g0; do { list($an, $gn) = array( bcdiv(bcadd($an, $gn), 2), bcsqrt(bcmul($an, $gn)), ); } while (bccomp(bcsub($an, $gn),$limit) > 0);

return $an; } bcscale(PRECISION); echo agm(1, 1 / bcsqrt(2));  {{out}} txt 0.8472130848350  ## PicoLisp PicoLisp (scl 80) (de agm (A G) (do 7 (prog1 (/ (+ A G) 2) (setq G (sqrt A G) A @) ) ) ) (round (agm 1.0 (*/ 1.0 1.0 (sqrt 2.0 1.0))) 70 )  Output: txt -> "0.8472130847939790866064991234821916364814459103269421850605793726597340"  ## PL/I PL/I arithmetic_geometric_mean: /* 31 August 2012 */ procedure options (main); declare (a, g, t) float (18); a = 1; g = 1/sqrt(2.0q0); put skip list ('The arithmetic-geometric mean of ' || a || ' and ' || g || ':'); do until (abs(a-g) < 1e-15*a); t = (a + g)/2; g = sqrt(a*g); a = t; put skip data (a, g); end; put skip list ('The result is:', a); end arithmetic_geometric_mean;  Results: txt The arithmetic-geometric mean of 1.00000000000000000E+0000 and 7.07106781186547524E-0001: A= 8.53553390593273762E-0001 G= 8.40896415253714543E-0001; A= 8.47224902923494153E-0001 G= 8.47201266746891460E-0001; A= 8.47213084835192807E-0001 G= 8.47213084752765367E-0001; A= 8.47213084793979087E-0001 G= 8.47213084793979087E-0001; The result is: 8.47213084793979087E-0001  ## Potion Input values should be floating point potion sqrt = (x) : xi = 1 7 times : xi = (xi + x / xi) / 2 . xi . agm = (x, y) : 7 times : a = (x + y) / 2 g = sqrt(x * y) x = a y = g . x .  ## PowerShell PowerShell function agm ([Double]$a, [Double]$g) { [Double]$eps = 1E-15
[Double]$a1 = [Double]$g1 = 0
while([Math]::Abs($a -$g) -gt $eps) {$a1, $g1 =$a, $g$a = ($a1 +$g1)/2
$g = [Math]::Sqrt($a1*$g1) } [pscustomobject]@{ a = "$a"
g = "$g" } } agm 1 (1/[Math]::Sqrt(2))  Output: txt a g - - 0.847213084793979 0.847213084793979  ## Prolog Prolog agm(A,G,A) :- abs(A-G) < 1.0e-15, !. agm(A,G,Res) :- A1 is (A+G)/2.0, G1 is sqrt(A*G),!, agm(A1,G1,Res). ?- agm(1,1/sqrt(2),Res). Res = 0.8472130847939792.  ## PureBasic purebasic Procedure.d AGM(a.d, g.d, ErrLim.d=1e-15) Protected.d ta=a+1, tg While ta <> a ta=a: tg=g a=(ta+tg)*0.5 g=Sqr(ta*tg) Wend ProcedureReturn a EndProcedure If OpenConsole() PrintN(StrD(AGM(1, 1/Sqr(2)), 16)) Input() CloseConsole() EndIf  0.8472130847939792 ## Python The calculation generates two new values from two existing values which is the classic example for the use of [https://docs.python.org/3/reference/simple_stmts.html#grammar-token-target_list assignment to a list of values in the one statement], so ensuring an gn are only calculated from an-1 gn-1. ### Basic Version python from math import sqrt def agm(a0, g0, tolerance=1e-10): """ Calculating the arithmetic-geometric mean of two numbers a0, g0. tolerance the tolerance for the converged value of the arithmetic-geometric mean (default value = 1e-10) """ an, gn = (a0 + g0) / 2.0, sqrt(a0 * g0) while abs(an - gn) > tolerance: an, gn = (an + gn) / 2.0, sqrt(an * gn) return an print agm(1, 1 / sqrt(2))  {{out}} txt 0.847213084835  ===Multi-Precision Version=== python from decimal import Decimal, getcontext def agm(a, g, tolerance=Decimal("1e-65")): while True: a, g = (a + g) / 2, (a * g).sqrt() if abs(a - g) < tolerance: return a getcontext().prec = 70 print agm(Decimal(1), 1 / Decimal(2).sqrt())  {{out}} txt 0.847213084793979086606499123482191636481445910326942185060579372659734  All the digits shown are correct. ## R r arithmeticMean <- function(a, b) { (a + b)/2 } geometricMean <- function(a, b) { sqrt(a * b) } arithmeticGeometricMean <- function(a, b) { rel_error <- abs(a - b) / pmax(a, b) if (all(rel_error < .Machine$double.eps, na.rm=TRUE)) {
agm <- a
return(data.frame(agm, rel_error));
}
Recall(arithmeticMean(a, b), geometricMean(a, b))
}

agm <- arithmeticGeometricMean(1, 1/sqrt(2))
print(format(agm, digits=16))


{{out}}

txt
agm             rel_error
1 0.8472130847939792 1.310441309927519e-16


This function also works on vectors a and b (following the spirit of R):

r
a <- c(1, 1, 1)
b <- c(1/sqrt(2), 1/sqrt(3), 1/2)
agm <- arithmeticGeometricMean(a, b)
print(format(agm, digits=16))


{{out}}

txt
agm             rel_error
1 0.8472130847939792 1.310441309927519e-16
2 0.7741882646460426 0.000000000000000e+00
3 0.7283955155234534 0.000000000000000e+00


## Racket

This version uses Racket's normal numbers:

racket

#lang racket
(define (agm a g [ε 1e-15])
(if (<= (- a g) ε)
a
(agm (/ (+ a g) 2) (sqrt (* a g)) ε)))

(agm 1 (/ 1 (sqrt 2)))



Output:

txt

0.8472130847939792



This alternative version uses arbitrary precision floats:

racket

#lang racket
(require math/bigfloat)
(bf-precision 200)
(bfagm 1.bf (bf/ (bfsqrt 2.bf)))



Output:

txt

(bf #e0.84721308479397908660649912348219163648144591032694218506057918)



## Raven

Raven
define agm  use  $a,$g, $errlim #$errlim $g$a "%d %g %d\n" print
$a 1.0 + as$t
repeat $a 1.0 *$g - abs -15 exp10 $a * > while$a $g + 2 / as$t
$a$g * sqrt  as $g$t as $a$g $a$t  "t: %g a: %g g: %g\n" print
$a 16 1 2 sqrt / 1 agm "agm: %.15g\n" print  {{out}} txt t: 0.853553 a: 0.853553 g: 0.840896 t: 0.847225 a: 0.847225 g: 0.847201 t: 0.847213 a: 0.847213 g: 0.847213 t: 0.847213 a: 0.847213 g: 0.847213 agm: 0.847213084793979  ## REXX Also, this version of the AGM REXX program has three ''short circuits'' within it for an equality case and for two zero cases. REXX supports arbitrary precision, so the default digits can be changed if desired. rexx /*REXX program calculates the AGM (arithmetic─geometric mean) of two (real) numbers. */ parse arg a b digs . /*obtain optional numbers from the C.L.*/ if digs=='' | digs=="," then digs=110 /*No DIGS specified? Then use default.*/ numeric digits digs /*REXX will use lots of decimal digits.*/ if a=='' | a=="," then a=1 /*No A specified? Then use the default*/ if b=='' | b=="," then b=1 / sqrt(2) /* " B " " " " " */ call AGM a,b /*invoke the AGM function. */ say '1st # =' a /*display the A value. */ say '2nd # =' b /* " " B " */ say ' AGM =' agm(a, b) /* " " AGM " */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ agm: procedure: parse arg x,y; if x=y then return x /*is this an equality case?*/ if y=0 then return 0 /*is Y equal to zero ? */ if x=0 then return y/2 /* " X " " " */ d= digits() /*obtain the current decimal digits. */ numeric digits d + 5 /*add 5 more digs to ensure convergence*/ tiny= '1e-' || (digits() - 1) /*construct a pretty tiny REXX number. */ ox= x + 1 /*ensure that the old X ¬= new X. */ do while ox\=x & abs(ox)>tiny /*compute until the old X ≡ new X. */ ox= x /*save the old value of X. */ oy= y /* " " " " " Y. */ x= (ox + oy) * .5 /*compute " new " " X. */ y= sqrt(ox * oy) /* " " " " " Y. */ end /*while*/ numeric digits d /*restore the original decimal digits. */ return x / 1 /*normalize X to new " " */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; numeric form; h=d+6 numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g *.5'e'_ % 2 do j=0 while h>9; m.j=h; h=h % 2 + 1; end /*j*/ do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g  {{out|output|text= when using the default input:}} txt 1st # = 1 2nd # = 0.70710678118654752440084436210484903928483593768847403658833986899536623923105351942519376716382078636750692312 AGM = 0.84721308479397908660649912348219163648144591032694218506057937265973400483413475972320029399461122994212228563  ## Ring ring decimals(9) see agm(1, 1/sqrt(2)) + nl see agm(1,1/pow(2,0.5)) + nl func agm agm,g while agm an = (agm + g)/2 gn = sqrt(agm*g) if fabs(agm-g) <= fabs(an-gn) exit ok agm = an g = gn end return gn  ## Ruby ### Flt Version The thing to note about this implementation is that it uses the [http://flt.rubyforge.org/ Flt] library for high-precision math. This lets you adapt context (including precision and epsilon) to a ridiculous-in-real-life degree. ruby # The flt package (http://flt.rubyforge.org/) is useful for high-precision floating-point math. # It lets us control 'context' of numbers, individually or collectively -- including precision # (which adjusts the context's value of epsilon accordingly). require 'flt' include Flt BinNum.Context.precision = 512 # default 53 (bits) def agm(a,g) new_a = BinNum a new_g = BinNum g while new_a - new_g > new_a.class.Context.epsilon do old_g = new_g new_g = (new_a * new_g).sqrt new_a = (old_g + new_a) * 0.5 end new_g end puts agm(1, 1 / BinNum(2).sqrt)  {{out}} txt 0.84721308479397908660649912348219163648144591032694218506057937265973400483413475972320029399461122994212228562523341096309796266583087105969971363598338426  Adjusting the precision setting (at about line 9) will of course affect this. :-) ### BigDecimal Version Ruby has a BigDecimal class in standard library ruby require 'bigdecimal' PRECISION = 100 EPSILON = 0.1 ** (PRECISION/2) BigDecimal::limit(PRECISION) def agm(a,g) while a - g > EPSILON a, g = (a+g)/2, (a*g).sqrt(PRECISION) end [a, g] end a = BigDecimal(1) g = 1 / BigDecimal(2).sqrt(PRECISION) puts agm(a, g)  {{out}} txt 0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723201915677745718E0 0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597231986723114767413E0  ## Run BASIC runbasic print agm(1, 1/sqr(2)) print agm(1,1/2^.5) print using("#.############################",agm(1, 1/sqr(2))) function agm(agm,g) while agm an = (agm + g)/2 gn = sqr(agm*g) if abs(agm-g) <= abs(an-gn) then exit while agm = an g = gn wend end function  Output: txt 0.847213085 0.847213085 0.8472130847939791165772005376  ## Rust rust // Accepts two command line arguments // cargo run --name agm arg1 arg2 fn main () { let mut args = std::env::args(); let x = args.nth(1).expect("First argument not specified.").parse::().unwrap(); let y = args.next().expect("Second argument not specified.").parse::().unwrap(); let result = agm(x,y); println!("The arithmetic-geometric mean is {}", result); } fn agm (x: f32, y: f32) -> f32 { let e: f32 = 0.000001; let mut a = x; let mut g = y; let mut a1: f32; let mut g1: f32; if a * g < 0f32 { panic!("The arithmetric-geometric mean is undefined for numbers less than zero!"); } else { loop { a1 = (a + g) / 2.; g1 = (a * g).sqrt(); a = a1; g = g1; if (a - g).abs() < e { return a; } } } }  {{out}} Output of running with arguments 1, 0.70710678: txt The arithmetic-geometric mean is 1.456791  ## Scala scala def agm(a: Double, g: Double, eps: Double): Double = { if (math.abs(a - g) < eps) (a + g) / 2 else agm((a + g) / 2, math.sqrt(a * g), eps) } agm(1, math.sqrt(2)/2, 1e-15)  ## Scheme scheme (define agm (case-lambda ((a0 g0) ; call again with default value for tolerance (agm a0 g0 1e-8)) ((a0 g0 tolerance) ; called with three arguments (do ((a a0 (* (+ a g) 1/2)) (g g0 (sqrt (* a g)))) ((< (abs (- a g)) tolerance) a))))) (display (agm 1 (/ 1 (sqrt 2)))) (newline)  {{out}} txt 0.8472130848351929  ## Seed7 seed7$ include "seed7_05.s7i";
include "float.s7i";
include "math.s7i";

const func float: agm (in var float: a, in var float: g) is func
result
var float: agm is 0.0;
local
const float: iota is 1.0E-7;
var float: a1 is 0.0;
var float: g1 is 0.0;
begin
if a * g < 0.0 then
raise RANGE_ERROR;
else
while abs(a - g) > iota do
a1 := (a + g) / 2.0;
g1 := sqrt(a * g);
a := a1;
g := g1;
end while;
agm := a;
end if;
end func;

const proc: main is func
begin
writeln(agm(1.0, 2.0) digits 6);
writeln(agm(1.0, 1.0 / sqrt(2.0)) digits 6);
end func;


{{out}}

txt

1.456791
0.847213



## SequenceL

sequencel>import A and G, and the result will be returned in AGM. The performance is quite acceptable. Note that the subroutine clobbers  A and G, so you should save them if you want to use them again.

Better precision than this is not easily obtainable on the ZX81, unfortunately.

basic
10 LET A=1
20 LET G=1/SQR 2
30 GOSUB 100
40 PRINT AGM
50 STOP
100 LET A0=A
110 LET A=(A+G)/2
120 LET G=SQR (A0*G)
130 IF ABS(A-G)>.00000001 THEN GOTO 100
140 LET AGM=A
150 RETURN


{{out}}

txt
0.84721309


## SQL

{{works with|oracle|11.2 and higher}}
The solution uses recursive WITH clause (aka recursive CTE, recursive query, recursive factored subquery). Some, perhaps many, but not all SQL dialects support recursive WITH clause. The solution below was written and tested in Oracle SQL - Oracle has supported recursive WITH clause since version 11.2.

sql
with
rec (rn, a, g, diff) as (
select  1, 1, 1/sqrt(2), 1 - 1/sqrt(2)
from  dual
union all
select  rn + 1, (a + g)/2, sqrt(a * g), (a + g)/2 - sqrt(a * g)
from  rec
where diff > 1e-38
)
select *
from   rec
where  diff <= 1e-38
;


{{out}}

txt

RN                                         A                                          G                                       DIFF
-- ----------------------------------------- ------------------------------------------ ------------------------------------------
6 0.847213084793979086606499123482191636480 0.8472130847939790866064991234821916364792 0.0000000000000000000000000000000000000008


## Stata

stata
mata

real scalar agm(real scalar a, real scalar b) {
real scalar c
do {
c=0.5*(a+b)
b=sqrt(a*b)
a=c
} while (a-b>1e-15*a)
return(0.5*(a+b))
}

agm(1,1/sqrt(2))
end


{{out}}

txt
.8472130848


## Swift

Swift
import Darwin

enum AGRError : Error {
case undefined
}

func agm(_ a: Double, _ g: Double, _ iota: Double = 1e-8) throws -> Double {
var a = a
var g = g
var a1: Double = 0
var g1: Double = 0

guard a * g >= 0 else {
throw AGRError.undefined
}

while abs(a - g) > iota {
a1 = (a + g) / 2
g1 = sqrt(a * g)
a = a1
g = g1
}

return a
}

do {
try print(agm(1, 1 / sqrt(2)))
} catch {
print("agr is undefined when a * g < 0")
}


{{out}}

txt
0.847213084835193


## Tcl

The tricky thing about this implementation is that despite the finite precision available to IEEE doubles (which Tcl uses in its implementation of floating point arithmetic, in common with many other languages) the sequence of values does not ''quite'' converge to a single value; it gets to within a ULP and then errors prevent it from getting closer. This means that an additional termination condition is required: once a value does not change (hence the old_b variable) we have got as close as we can. Note also that we are using exact equality with floating point; this is reasonable because this is a rapidly converging sequence (it only takes 4 iterations in this case).

tcl
proc agm {a b} {
set old_b [expr {$b<0?inf:-inf}] while {$a != $b &&$b != $old_b} { set old_b$b
lassign [list [expr {0.5*($a+$b)}] [expr {sqrt($a*$b)}]] a b
}
return $a } puts [agm 1 [expr 1/sqrt(2)]]  Output: txt 0.8472130847939792  =={{header|TI-83 BASIC}}== ti83b 1→A:1/sqrt(2)→G While abs(A-G)>e-15 (A+G)/2→B sqrt(AG)→G:B→A End A  {{out}} txt .8472130848  ## UNIX Shell {{works with|ksh93}} ksh is one of the few unix shells that can do floating point arithmetic (bash does not). bash function agm { float a=$1 g=$2 eps=${3:-1e-11} tmp
while (( abs(a-g) > eps )); do
print "debug: a=$a\tg=$g"
tmp=$(( (a+g)/2.0 )) g=$(( sqrt(a*g) ))
a=$tmp done echo$a
}

agm $((1/sqrt(2))) 1  {{output}} txt debug: a=0.7071067812 g=1 debug: a=0.8535533906 g=0.8408964153 debug: a=0.8472249029 g=0.8472012668 debug: a=0.8472130848 g=0.8472130847 debug: a=0.8472130848 g=0.8472130848 debug: a=0.8472130848 g=0.8472130848 0.8472130848  You can get a more approximate convergence by changing the while condition to compare the numbers as strings: change bash while (( abs(a-g) > eps ))  to bash while [[$a != \$g ]]


## VBA

vb
Private Function agm(a As Double, g As Double, Optional tolerance As Double = 0.000000000000001) As Double
Do While Abs(a - g) > tolerance
tmp = a
a = (a + g) / 2
g = Sqr(tmp * g)
Debug.Print a
Loop
agm = a
End Function
Public Sub main()
Debug.Print agm(1, 1 / Sqr(2))
End Sub

{{out}}

txt
0,853553390593274
0,847224902923494
0,847213084835193
0,847213084793979
0,847213084793979


## VBScript

{{trans|BBC BASIC}}

vb

Function agm(a,g)
Do Until a = tmp_a
tmp_a = a
a = (a + g)/2
g = Sqr(tmp_a * g)
Loop
agm = a
End Function

WScript.Echo agm(1,1/Sqr(2))



{{Out}}

txt
0.847213084793979


## Visual Basic .NET

{{trans|C#}}

vbnet
Imports System
Imports System.Numerics

Module Module1
Function BIP(ByVal leadDig As Char, ByVal numDigs As Integer) As BigInteger
Return BigInteger.Parse(leadDig & New String("0", numDigs))
End Function

Function IntSqRoot(ByVal v As BigInteger) As BigInteger
Dim digs As Integer = Math.Max(0, v.ToString().Length / 2)
Dim term As BigInteger : IntSqRoot = BIP("3", digs)
While True
term = v / IntSqRoot
If Math.Abs(CDbl((term - IntSqRoot))) < 2 Then Exit While
IntSqRoot = (IntSqRoot + term) / 2
End While
End Function

Function CalcByAGM(ByVal digits As Integer) As BigInteger
Dim digs As Integer = digits + CInt((Math.Log(digits) / 2)), c As BigInteger,
d2 As Integer = digs * 2,
a As BigInteger = BIP("1", digs) ' initial value = 1
CalcByAGM = IntSqRoot(BIP("5", d2 - 1)) ' initial value = square root of 0.5
While True
c = a : a = ((a + CalcByAGM) / 2) : CalcByAGM = IntSqRoot(c * CalcByAGM)
If Math.Abs(CDbl((a - CalcByAGM))) <= 1 Then Exit While
End While
End Function

Sub Main(ByVal args As String())
Dim digits As Integer = 25000
If args.Length > 0 Then
Integer.TryParse(args(0), digits)
If digits < 1 OrElse digits > 999999 Then digits = 25000
End If
Console.WriteLine("0.{0}", CalcByAGM(digits).ToString())
End Sub
End Module



{{out}}
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4850842


## XPL0

XPL0
include c:\cxpl\codesi;
real A, A1, G;
[Format(0, 16);
A:= 1.0;  G:= 1.0/sqrt(2.0);
repeat	A1:= (A+G)/2.0;
G:= sqrt(A*G);
A:= A1;
RlOut(0, A);  RlOut(0, G);  RlOut(0, A-G);  CrLf(0);
until	A=G;
]


Output:

txt

8.5355339059327400E-001 8.4089641525371500E-001 1.2656975339559100E-002
8.4722490292349400E-001 8.4720126674689100E-001 2.3636176602726000E-005
8.4721308483519300E-001 8.4721308475276500E-001 8.2427509262572600E-011
8.4721308479397900E-001 8.4721308479397900E-001 0.0000000000000000E+000



## zkl

{{trans|XPL0}}

zkl
a:=1.0; g:=1.0/(2.0).sqrt();
while(not a.closeTo(g,1.0e-15)){
a1:=(a+g)/2.0; g=(a*g).sqrt(); a=a1;
println(a,"  ",g," ",a-g);
}


{{out}}

txt

0.853553  0.840896 0.012657
0.847225  0.847201 2.36362e-05
0.847213  0.847213 8.24275e-11
0.847213  0.847213 1.11022e-16



Or, using tail recursion

zkl
fcn(a=1.0, g=1.0/(2.0).sqrt()){ println(a," ",g," ",a-g);
if(a.closeTo(g,1.0e-15)) return(a) else return(self.fcn((a+g)/2.0, (a*g).sqrt()));
}()


{{out}}

txt

1 0.707107 0.292893
0.853553 0.840896 0.012657
0.847225 0.847201 2.36362e-05
0.847213 0.847213 8.24275e-11
0.847213 0.847213 1.11022e-16



## ZX Spectrum Basic

{{trans|ERRE}}

zxbasic
10 LET a=1: LET g=1/SQR 2
20 LET ta=a
30 LET a=(a+g)/2
40 LET g=SQR (ta*g)
50 IF a

`