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{{task}}{{wikipedia|Arithmetic-geometric mean}}
;task:
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 , and is equal to the limit of the sequence: : : Since the limit of tends (rapidly) to zero with iterations, this is an efficient method.
Demonstrate the function by calculating: :
;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
Ada
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
package N_IO is new Ada.Text_IO.Float_IO(Num);
package Math is new Ada.Numerics.Generic_Elementary_Functions(Num);
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
BASIC
=
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}}===
## 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_add (out1, in1, in2); 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> { private readonly double _maximumRelativeDifference; 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()); if (System.Diagnostics.Debugger.IsAttached) Console.ReadKey(); } } }
{{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
```
Now for our task example:
```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)
%8 = load double, double* %4, align 8 ; load a
%9 = load double, double* %3, align 8 ; load g
%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:
%13 = load double, double* %4, align 8 ; load a
%14 = load double, double* %3, align 8 ; load g
%15 = fsub double %13, %14 ; a - g
%16 = call double @llvm.fabs.f64(double %15) ; fabs(a - g)
%17 = load double, double* %5, align 8 ; load iota
%18 = fcmp ogt double %16, %17 ; fabs(a - g) > iota
br i1 %18, label %loop_body, label %eom
loop_body:
%19 = load double, double* %4, align 8 ; load a
%20 = load double, double* %3, align 8 ; load g
%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
%23 = load double, double* %4, align 8 ; load a
%24 = load double, double* %3, align 8 ; load g
%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
%27 = load double, double* %6, align 8 ; load a1
store double %27, double* %4, align 8 ; store a1 in a
%28 = load double, double* %7, align 8 ; load g1
store double %28, double* %3, align 8 ; store g1 in g
br label %loop
eom:
%29 = load double, double* %4, align 8 ; load a
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
%5 = load double, double* %2, align 8 ; reload y
%6 = load double, double* %1, align 8 ; reload x
%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
to about :a :b
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 {
Read a0, b0
Push Sqrt(a0*b0), (a0+b0)/2
' last pushed first read
} 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
```
=={{header|MATLAB}} / {{header|Octave}}==
```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))
```
=={{header|Oberon-2}}==
{{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}}
{{libheader|GMP}}
Port of the C example:
```pascal
Program ArithmeticGeometricMean;
uses
gmp;
procedure agm (in1, in2: mpf_t; var out1, out2: mpf_t);
begin
mpf_add (out1, in1, in2);
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#}}
{{Libheader|System.Numerics}}
```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())
If System.Diagnostics.Debugger.IsAttached Then Console.ReadKey()
End Sub
End Module
```
{{out}}
0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229942122285625233410963097962665830871059699713635983384251176326814289060389706768601616650048281188721897713309411767462019944392962902167289194499507231677897346863947606671057980557852173140349398304200422119216039839553595098193641293716340646029599967970599434351602031842648756950242174863855405981954581601742417887854192758804162719012085587685648326834140431218400804035809204559494313877815120926522254574397124286820766340954733674599621792665535348625686118543308626287287287563010835563193570668714785639088982115108836352147696979612621832943228417868113768445170018146021913694027020945996683513596327880804274345481744587363220025153952936265806614198365616491626259607434723706616902353080017375312847852558430631907454274934152685790655269406003147591020332746719686124796325510554648902820855297439651249940096625528660675804487353892185701401167716976535014084952476848993257321337028984668939194665861873752966387562266045914777044204681089256584408380320409106190031537067341195941010074743310599055058205243260099516927924174782169767810616836977141107392733439215501430220070873673659622721492587761928510523803670268904639096219076636442355380859029452340651900133423451058383417121805142550039237011113254111446126289062541335505266436535958245521562933975182514706501346410470569793556813066063293733450387109770972948759171790158173202815782884871499313408154933423677970447127859376185950851466773645546792016159342239971429840707888822790326567515965284358177957272848083564899635044041407342261101833835469759626633304220849998523007427039302772434749797179732645525465430198316949684610986907439050680137661192529197709384412997070158894931666611619945922650113111839663525025305616464315872084545229887754751772727476567216489829182392388952072076428397108847059603569219929218319015481412807665926982944644571492396663299730758139049576224389624231752095073190184244624423709864272811495111808228260538624846176751801409831274972576519837564923569028002161749055314272081534395405955635763711272816570597373374429700390560401563886630722257003892301591123769601215800817790778633512408624310735715837659265045466527873378744448344063102447570396812554539822664303534164130356138016341655752655897529445211668734512201912274667331915712407637538211069681410769263900748331757433967523196603308649735713838741960989838322028826948821913028193669499544222406972761686213695116578388850121990961606554546115432531481642493326947970041594914763231129205935165189979433500459762882172926259180894055084314663937825483351395501906533708720620640240770560758487964998436515927282645344286366154191425857771067561850172780332871751951893050318055052454260223355229007714181287986543511879180063562795936247682677864122494603381260826282540988953125276775346562432792145112295555160318184331336929617230417838551571255674049834166659269695800089537245730576945422753721602096871914703988784663672432627061911270717165908246400416799411204056571036408300024192943985530739946565396778104927010554103595133394321999250666762020783946955537605517964010097492188563113010178138885787938131720959480625392013009836502879176958279859052799477219417979970249430621584194688853281154977215799601944096234776861440850757392842988237593968232236705803341347746231128976258593243766317789749110772619097044895222045096307255155900938249040213648077920347672150485684460225544099928261631743126422857876289833806507220230103717531492635046310601885737725670066183812905806389545081270313113710437161358334880658339554312179013483988332164130576352447125115394720666703301013487165163241138288176398396295261211412632197959650986567867552507607604240959075175230219461045325643332496149012535333292237238689481278850201359663053760558493589283916304694038878549600274714871978014576595790495858022600660995249673643249668334617601066081567069751423818665036108388522097616550025160731149921612947757901997292486896382206038087602762816723701668191066335857751546503813342367223476420265585655884641601021054048985561871147358849763784064864267981865044863190774703822867114351511230036070865742988647714667473375011434581885279700605621172469217484718069486625119947289344427037830462070735493805287272062156063071882868580564521110696708028569906982576917722099867195996850779068144349493280497681154368046325993869307623507099951829512958112123570724538335482619075239515827309824818054966589790916886798407170779370595904577584091047341310960419411135775662072733779783320379730113767265853574771027978140972130961214239385473746276961504130795283737288205065871915225976508402779699176117539300672549249122984508236297556872271106584943553385049453263873648980460665597995436016950309279009245005785647723587619884898603441219534079536900299641197454906074160097885953766072290516077242859007090115663913836429904122082676962979786764903235649998199076599743987054864876909102491192709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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