Arithmetic-geometric mean
You are encouraged to solve this task according to the task description, using any language you may know.
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- Task
Write a function to compute the arithmetic-geometric mean of two numbers.
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
11l
<lang 11l>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)))</lang>
- Output:
0.847213
360 Assembly
For maximum compatibility, this program uses only the basic instruction set. <lang 360asm>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</lang>
- Output:
+00000000.84721
8th
<lang 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 </lang>
- Output:
agn(1, 1/sqrt(2)) = 0.8472130848
Ada
<lang 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;</lang>
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. <lang algol68> 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 </lang>
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
<lang APL> agd←{(⍺-⍵)<10*¯8:⍺⋄((⍺+⍵)÷2)∇(⍺×⍵)*÷2} 1 agd ÷2*÷2 </lang>
Output:
0.8472130848
AppleScript
By functional composition:
<lang AppleScript>-- 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</lang>
- Output:
0.847213084835
AutoHotkey
<lang AHK>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))</lang> Output:
0.847213084793979
AWK
<lang 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) } </lang> Output
0.8472130847939792
BASIC
Commodore BASIC
<lang commodorebasic>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</lang>
BBC BASIC
<lang bbcbasic> *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
</lang> Produces this output:
0.8472130847939792
IS-BASIC
<lang 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</lang>
bc
<lang 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)</lang>
- Output:
.84721308479397908659
C
Basic
<lang c>#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;
} </lang>
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
<lang cpp>/*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; }</lang>
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#
<lang csharp>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));
}
}
}
}</lang> Output:
0.847213084835193
Note that the last 5 digits are spurious, as maximumRelativeDifference was only specified to be 1e-5. Using 1e-11 instead will give the result 0.847213084793979, which is as far as double can take it.
Using Decimal Type
<lang csharp>using System; class Program {
static Decimal DecSqRoot(Decimal v) {
Decimal r = (Decimal)Math.Sqrt((double)v), t = 0, d = 0, ld = 1;
while (ld != d) { t = v / r; r = (r + t) / 2;
ld = d; d = t - r; } return t; }
static Decimal CalcAGM(Decimal a, Decimal b) {
Decimal c, d = 0, ld = 1; while (ld != d) { ld = d; c = a;
d = (a = (a + b) / 2) - (b = DecSqRoot(c * b)); } return b; }
static void Main(string[] args) {
Console.WriteLine(CalcAGM(1M, DecSqRoot(0.5M)));
if (System.Diagnostics.Debugger.IsAttached) Console.ReadKey();
}
}</lang>
- Output:
0.8472130847939790866064991235
C# with 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. <lang csharp>using static System.Math; using static System.Console; using BI = System.Numerics.BigInteger;
class Program {
static BI BIP(char leadDig, int numDigs) { // makes big constant
return BI.Parse(leadDig + new string('0', numDigs)); }
static BI IntSqRoot(BI v, BI res) { // res is the initial guess
BI term = 0, d = 0, dl = 1; while (dl != d) { term = v / res; res = (res + term) >> 1;
dl = d; d = term - res; } return term; }
static BI CalcByAGM(int digs) { // note: a and b are hard-coded for this RC task
BI a = BIP('1', digs), // value of 1, extended to required number of digits
b = BI.Parse(string.Format("{0:0.00000000000000000}", Sqrt(0.5)).Substring(2) +
new string('0', digs - 17)), // initial guess for square root of 0.5
c, // temporary variable to swap a and b
diff = 0, ldiff = 1; // difference of a and b, last difference
b = IntSqRoot(BIP('5', (digs << 1) - 1), b); // value of square root of 0.5
while (ldiff != diff) { ldiff = diff; c = a; a = (a + b) >> 1;
diff = a - (b = IntSqRoot(c * b, a)); } 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; }
WriteLine("0.{0}", CalcByAGM(digits));
if (System.Diagnostics.Debugger.IsAttached) ReadKey(); }
}</lang>
- Output:
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48
C++
<lang 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; } </lang>
Enter X and Y: 1.0 2.0 The Arithmetic-Geometric Mean of 1.0 and 2.0 is 1.45679103104690677028543177584651857614517211914062
Clojure
<lang 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)) </lang>
- Output:
8.47213084793979086606499123482191636481445910326942185060579372659734e-1
COBOL
<lang 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).</lang>
- Output:
0.8472130847939792
Common Lisp
<lang 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)))
</lang>
- Output:
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
<lang 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)));
}</lang>
- Output:
0.8472130847939790866
All the digits shown are exact.
Delphi
<lang Delphi> program geometric_mean;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
function Fabs(value: Double): Double; begin
Result := value; if Result < 0 then Result := -Result;
end;
function agm(a, g: Double):Double; var
iota, a1, g1: Double;
begin
iota := 1.0E-16;
if a * g < 0.0 then
begin
Writeln('arithmetic-geometric mean undefined when x*y<0');
exit(1);
end;
while Fabs(a - g) > iota do begin a1 := (a + g) / 2.0; g1 := sqrt(a * g); a := a1; g := g1; end; Exit(a);
end;
var
x, y: Double;
begin
Write('Enter two numbers:');
Readln(x, y);
writeln(format('The arithmetic-geometric mean is %.6f', [agm(x, y)]));
readln;
end.</lang>
- Output:
Enter two numbers:1 2 The arithmetic-geometric mean is 1,456791
EchoLisp
We use the (~= a b) operator which tests for |a - b| < ε = (math-precision). <lang scheme> (lib 'math)
(define (agm a g)
(if (~= a g) a
(agm (// (+ a g ) 2) (sqrt (* a g)))))
(math-precision)
→ 0.000001 ;; default
(agm 1 (/ 1 (sqrt 2)))
→ 0.8472130848351929
(math-precision 1.e-15)
→ 1e-15
(agm 1 (/ 1 (sqrt 2)))
→ 0.8472130847939792
</lang>
Elixir
<lang Elixir>defmodule ArithhGeom do
def mean(a,g,tol) when abs(a-g) <= tol, do: a def mean(a,g,tol) do mean((a+g)/2,:math.pow(a*g, 0.5),tol) end
end
IO.puts ArithhGeom.mean(1,1/:math.sqrt(2),0.0000000001)</lang>
- Output:
0.8472130848351929
Erlang
<lang Erlang>%% Arithmetic Geometric Mean of 1 and 1 / sqrt(2) %% Author: Abhay Jain
-module(agm_calculator). -export([find_agm/0]). -define(TOLERANCE, 0.0000000001).
find_agm() ->
A = 1,
B = 1 / (math:pow(2, 0.5)),
AGM = agm(A, B),
io:format("AGM = ~p", [AGM]).
agm (A, B) when abs(A-B) =< ?TOLERANCE ->
A;
agm (A, B) ->
A1 = (A+B) / 2, B1 = math:pow(A*B, 0.5), agm(A1, B1).</lang>
Output: <lang Erlang>AGM = 0.8472130848351929</lang>
ERRE
<lang> PROGRAM AGM
! ! for rosettacode.org !
!$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 </lang>
F#
<lang 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)</lang> Output
0.847213
Factor
<lang 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 .</lang>
- Output:
0.8472130847939792
Forth
<lang 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</lang>
Fortran
A Fortran 77 implementation <lang 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</lang>
FreeBASIC
<lang 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</lang>
- Output:
0.8472130847939792
Futhark
<lang 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)
</lang>
Go
<lang 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))
}</lang>
- Output:
0.8472130847939792
Groovy
Solution: <lang 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
}</lang>
Test: <lang 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</lang>
Output:
agm(1, 0.5**0.5) = agm(1, 0.7071067811865476) = 0.8472130847939792
Haskell
<lang 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 $ until eq 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 :: IO () main = do
let equal = (< 0.000000001) . relDiff print $ agm 1 (1 / sqrt 2) equal</lang>
- Output:
0.8472130847527654
Icon and Unicon
<lang>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</lang>
Output:
->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):
<lang j>mean=: +/ % #
(mean , */ %:~ #)^:_] 1,%%:2
0.8472130847939792 0.8472130847939791</lang>
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):
<lang j> ~.(mean , */ %:~ #)^:_] 1,%%:2 0.8472130847939792</lang>
Another variation would be to show intermediate values, in the limit process:
<lang j> (mean, */ %:~ #)^:a: 1,%%:2
1 0.7071067811865475
0.8535533905932737 0.8408964152537145 0.8472249029234942 0.8472012667468915 0.8472130848351929 0.8472130847527654 0.8472130847939792 0.8472130847939791</lang>
Arbitrary Precision
Another variation would be to use 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:
<lang 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&$: : (4 : 0) " 0
assert. 0<y m=. <.0.5+2^.y t=. (<:%>:) (x:!.0 y)%2x^m if. x<-:#":t do. t=. (1+x) round t end. ln2=. 2*+/1r3 (^%]) 1+2*i.>.0.5*(%3)^.0.5*0.1^x+>.10^.1>.m lnr=. 2*+/t (^%]) 1+2*i.>.0.5*(|t)^.0.5*0.1^x lnr + m * ln2
)
exp=: DP&$: : (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
)</lang>
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:
<lang J>fmt=:[: ;:inv DP&$: : (4 :0)&.>
x{.deb (x*2j1)":y
)
root=: ln@] exp@% [
epsilon=: 1r9^DP</lang>
Some example uses:
<lang J> fmt sqrt 2 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572
fmt *~sqrt 2
2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
fmt epsilon
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000418
fmt 2 root 2
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572</lang>
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:
<lang J>geomean=: */ root~ # geomean2=: [: sqrt */</lang>
A quick test to make sure these can be equivalent:
<lang J> fmt geomean 3 5 3.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673517
fmt geomean2 3 5
3.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673517</lang>
Now for our task example:
<lang 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</lang>
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
<lang 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)));
}
}</lang>
- Output:
0.8472130847939792
JavaScript
ES5
<lang 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));</lang>
ES6
<lang 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));
})();</lang>
- Output:
<lang JavaScript>0.8472130848351929</lang>
jq
Naive version that assumes tolerance is appropriately specified: <lang jq>def naive_agm(a; g; tolerance):
def abs: if . < 0 then -. else . end;
def _agm:
# state [an,gn]
if ((.[0] - .[1])|abs) > tolerance
then [add/2, ((.[0] * .[1])|sqrt)] | _agm
else .
end;
[a, g] | _agm | .[0] ;</lang>
This version avoids an infinite loop if the requested tolerance is too small: <lang 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)</lang>
- Output:
$ jq -n -f Arithmetic-geometric_mean.jq 0.8472130847939792
Julia
<lang 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)</lang> The ε for this calculation is given as a positive integer multiple of the machine ε for x.
- Output:
# 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
Klingphix
<lang Klingphix>include ..\Utilitys.tlhy
- agm [ over over + 2 / rot rot * sqrt ] [ over over tostr swap tostr # ] while drop ;
1 1 2 sqrt / agm
pstack
" " input</lang>
<lang Klingphix>include ..\Utilitys.tlhy
- agm %a %g %p !p !g !a
$p $a $g - abs > ( [$a] [.5 $a $g + * $a $g * sqrt $p agm] ) if ;
1 .5 sqrt 1e-15 agm
pstack
" " input</lang>
- Output:
(0.847213)
Kotlin
<lang 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<String>) {
println(agm(1.0, 1.0 / Math.sqrt(2.0)))
}</lang>
- Output:
0.8472130847939792
LFE
<lang 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)))
</lang>
Usage:
> (agm 1 (/ 1 (math:sqrt 2))) 0.8472130847939792
Liberty BASIC
<lang 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
</lang>
LiveCode
<lang 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</lang> Example <lang LiveCode>put agm(1, 1/sqrt(2)) -- ouput -- 0.847213</lang>
LLVM
<lang 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 }</lang>
- Output:
The arithmetic-geometric mean is 0.8472130847939791654
Logo
<lang 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 </lang>
Lua
<lang 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))))</lang>
Output:
0.847213084793979
M2000 Interpreter
<lang 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 </lang>
Maple
Maple provides this function under the name GaussAGM. To compute a floating point approximation, use evalf. <lang 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
</lang> Alternatively, if one or both arguments is already a float, Maple will compute a floating point approximation automatically. <lang Maple> > GaussAGM( 1.0, 1 / sqrt( 2 ) );
0.8472130847
</lang>
Mathematica/Wolfram Language
To any arbitrary precision, just increase PrecisionDigits <lang Mathematica>PrecisionDigits = 85; AGMean[a_, b_] := FixedPoint[{ Tr@#/2, Sqrt[Times@@#] }&, N[{a,b}, PrecisionDigits]]〚1〛</lang>
AGMean[1, 1/Sqrt[2]] 0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597232
MATLAB / Octave
<lang 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</lang>
octave:26> agm(1,1/sqrt(2)) ans = 0.84721
Maxima
<lang 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 */</lang>
МК-61/52
<lang>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</lang>
Modula-2
<lang 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.</lang>
- Output:
Enter two numbers: 1.0 2.0 1.456791031046900
Enter two numbers: 1.0 0.707 0.847154622368330
NetRexx
<lang 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
</lang> Output:
0.8472130847939792
NewLISP
<lang 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) ) </lang>
Nim
<lang 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))</lang>
Output:
8.4721308479397917e-01
See first 24 iterations:
<lang 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))</lang>
Oberon-2
<lang 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. </lang>
- Output:
8.4721308479397905E-1
Objeck
<lang 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));
}
} </lang>
Output:
0.847213085
OCaml
<lang 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)</lang> Output
0.8472130847939792
Oforth
<lang Oforth>: agm \ a b -- m
while( 2dup <> ) [ 2dup + 2 / -rot * sqrt ] drop ;</lang>
Usage : <lang Oforth>1 2 sqrt inv agm</lang>
- Output:
0.847213084793979
OOC
<lang ooc> import math // import for sqrt() function
amean: func (x: Double, y: Double) -> 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)))
} </lang> Output
0.8472130847939792
ooRexx
<lang 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</lang>
- Output:
0.8472130847939791968
PARI/GP
Built-in: <lang parigp>agm(1,1/sqrt(2))</lang>
Iteration: <lang parigp>agm2(x,y)=if(x==y,x,agm2((x+y)/2,sqrt(x*y))</lang>
Pascal
Port of the C example: <lang 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.</lang> Output is as long as the C example.
Perl
<lang 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 $a0;
}
print agm(1, 1/sqrt(2))."\n";</lang> Output:
0.847213084793979
Phix
function agm(atom a, atom g, atom tolerance=1.0e-15) while abs(a-g)>tolerance do {a,g} = {(a + g)/2,sqrt(a*g)} printf(1,"%0.15g\n",a) end while return a end function ?agm(1,1/sqrt(2)) -- (rounds to 10 d.p.)
- Output:
0.853553390593274 0.847224902923494 0.847213084835193 0.847213084793979 0.8472130848
Phixmonti
<lang Phixmonti>include ..\Utilitys.pmt
1.0e-15 var tolerance
def test
over over - abs tolerance >
enddef
def agm /# n1 n2 -- n3 #/
test while
over over + 2 / rot rot * sqrt
test endwhile
enddef
1 1 2 sqrt / agm tostr ?</lang>
PHP
<lang php> define('PRECISION', 13);
function agm($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 $an;
}
bcscale(PRECISION); echo agm(1, 1 / bcsqrt(2)); </lang>
- Output:
0.8472130848350
PicoLisp
<lang PicoLisp>(scl 80)
(de agm (A G)
(do 7
(prog1 (/ (+ A G) 2)
(setq G (sqrt A G) A @) ) ) )
(round
(agm 1.0 (*/ 1.0 1.0 (sqrt 2.0 1.0))) 70 )</lang>
Output:
-> "0.8472130847939790866064991234821916364814459103269421850605793726597340"
PL/I
<lang PL/I> arithmetic_geometric_mean: /* 31 August 2012 */
procedure options (main); declare (a, g, t) float (18);
a = 1; g = 1/sqrt(2.0q0);
put skip list ('The arithmetic-geometric mean of ' || a || ' and ' || g || ':');
do until (abs(a-g) < 1e-15*a);
t = (a + g)/2; g = sqrt(a*g);
a = t;
put skip data (a, g);
end;
put skip list ('The result is:', a);
end arithmetic_geometric_mean; </lang> Results:
The arithmetic-geometric mean of 1.00000000000000000E+0000 and 7.07106781186547524E-0001: A= 8.53553390593273762E-0001 G= 8.40896415253714543E-0001; A= 8.47224902923494153E-0001 G= 8.47201266746891460E-0001; A= 8.47213084835192807E-0001 G= 8.47213084752765367E-0001; A= 8.47213084793979087E-0001 G= 8.47213084793979087E-0001; The result is: 8.47213084793979087E-0001
Potion
Input values should be floating point <lang potion>sqrt = (x) :
xi = 1
7 times :
xi = (xi + x / xi) / 2
.
xi
.
agm = (x, y) :
7 times :
a = (x + y) / 2
g = sqrt(x * y)
x = a
y = g
.
x
.</lang>
PowerShell
<lang PowerShell> function agm ([Double]$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 = "$g"
}
} agm 1 (1/[Math]::Sqrt(2)) </lang> Output:
a g - - 0.847213084793979 0.847213084793979
Prolog
<lang Prolog> agm(A,G,A) :- abs(A-G) < 1.0e-15, !. agm(A,G,Res) :- A1 is (A+G)/2.0, G1 is sqrt(A*G),!, agm(A1,G1,Res).
?- agm(1,1/sqrt(2),Res). Res = 0.8472130847939792. </lang>
PureBasic
<lang purebasic>Procedure.d AGM(a.d, g.d, ErrLim.d=1e-15)
Protected.d ta=a+1, tg While ta <> a ta=a: tg=g a=(ta+tg)*0.5 g=Sqr(ta*tg) Wend ProcedureReturn a
EndProcedure
If OpenConsole()
PrintN(StrD(AGM(1, 1/Sqr(2)), 16)) Input() CloseConsole()
EndIf</lang>
0.8472130847939792
Python
The calculation generates two new values from two existing values which is the classic example for the use of assignment to a list of values in the one statement, so ensuring an gn are only calculated from an-1 gn-1.
Basic Version
<lang python>from math import sqrt
def agm(a0, g0, tolerance=1e-10):
""" Calculating the arithmetic-geometric mean of two numbers a0, g0.
tolerance the tolerance for the converged
value of the arithmetic-geometric mean
(default value = 1e-10)
"""
an, gn = (a0 + g0) / 2.0, sqrt(a0 * g0)
while abs(an - gn) > tolerance:
an, gn = (an + gn) / 2.0, sqrt(an * gn)
return an
print agm(1, 1 / sqrt(2))</lang>
- Output:
0.847213084835
Multi-Precision Version
<lang python>from decimal import Decimal, getcontext
def agm(a, g, tolerance=Decimal("1e-65")):
while True:
a, g = (a + g) / 2, (a * g).sqrt()
if abs(a - g) < tolerance:
return a
getcontext().prec = 70 print agm(Decimal(1), 1 / Decimal(2).sqrt())</lang>
- Output:
0.847213084793979086606499123482191636481445910326942185060579372659734
All the digits shown are correct.
R
<lang r>arithmeticMean <- function(a, b) { (a + b)/2 } geometricMean <- function(a, b) { sqrt(a * b) }
arithmeticGeometricMean <- function(a, b) {
rel_error <- abs(a - b) / pmax(a, b)
if (all(rel_error < .Machine$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))</lang>
- Output:
agm rel_error 1 0.8472130847939792 1.310441309927519e-16
This function also works on vectors a and b (following the spirit of R): <lang 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))</lang>
- Output:
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: <lang 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))) </lang> Output:
0.8472130847939792
This alternative version uses arbitrary precision floats: <lang racket>
- lang racket
(require math/bigfloat) (bf-precision 200) (bfagm 1.bf (bf/ (bfsqrt 2.bf))) </lang> Output:
(bf #e0.84721308479397908660649912348219163648144591032694218506057918)
Raku
(formerly Perl 6) <lang 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;</lang>
- Output:
0.84721308479397917
It's also possible to write it recursively: <lang perl6>sub agm( $a, $g ) {
$a ≅ $g ?? $a !! agm(|@$_)
given ($a + $g)/2, sqrt $a * $g;
}
say agm 1, 1/sqrt 2;</lang>
Raven
<lang 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</lang>
- Output:
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
Relation
<lang Relation> function agm(x,y) set a = x set g = y while abs(a - g) > 0.00000000001 set an = (a + g)/2 set gn = sqrt(a * g) set a = an set g = gn set i = i + 1 end while set result = g end function
set x = 1 set y = 1/sqrt(2) echo (x + y)/2 echo sqrt(x+y) echo agm(x,y) </lang>
0.853553391 0.840896415 0.847213085
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. <lang 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= 120 /*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 0 /*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; oy= y /*save the old value of X and 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</lang>
- output when using the default input:
1st # = 1 2nd # = 0.707106781186547524400844362104849039284835937688474036588339868995366239231053519425193767163820786367506923115456148513 AGM = 0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229942122285625233410963
Ring
<lang 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
</lang>
Ruby
Flt Version
The thing to note about this implementation is that it uses the Flt library for high-precision math. This lets you adapt context (including precision and epsilon) to a ridiculous-in-real-life degree. <lang 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)</lang>
- Output:
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 <lang 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)</lang>
- Output:
0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723201915677745718E0 0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597231986723114767413E0
Run BASIC
<lang 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</lang>Output:
0.847213085 0.847213085 0.8472130847939791165772005376
Rust
<lang 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::<f32>().unwrap();
let y = args.next().expect("Second argument not specified.").parse::<f32>().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; }
}
}
}</lang>
- Output:
Output of running with arguments 1, 0.70710678:
The arithmetic-geometric mean is 1.456791
Scala
<lang 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)
</lang>
Scheme
<lang 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) </lang>
- Output:
0.8472130848351929
Seed7
<lang 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;</lang>
- Output:
1.456791 0.847213
SequenceL
<lang sequencel>import <Utilities/Math.sl>;
agm(a, g) :=
let
iota := 1.0e-15;
arithmeticMean := 0.5 * (a + g);
geometricMean := sqrt(a * g);
in
a when abs(a-g) < iota
else
agm(arithmeticMean, geometricMean);
main := agm(1.0, 1.0 / sqrt(2));</lang>
- Output:
0.847213
Sidef
<lang ruby>func agm(a, g) {
loop {
var (a1, g1) = ((a+g)/2, sqrt(a*g))
[a1,g1] == [a,g] && return a
(a, g) = (a1, g1)
}
}
say agm(1, 1/sqrt(2))</lang>
- Output:
0.8472130847939790866064991234821916364814
Sinclair ZX81 BASIC
Works with 1k of RAM.
The specification calls for a function. Sadly that is not available to us, so this program uses a subroutine: pass the arguments in the global variables 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. <lang 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</lang>
- Output:
0.84721309
Smalltalk
That is simply a copy/paste of the already existing agm method in the Number class: <lang smalltalk>agm:y
"return the arithmetic-geometric mean agm(x, y)
of the receiver (x) and the argument, y.
See https://en.wikipedia.org/wiki/Arithmetic-geometric_mean"
|ai an gi gn epsilon delta|
ai := (self + y) / 2. gi := (self * y) sqrt. epsilon := self ulp.
[
an := (ai + gi) / 2.
gn := (ai * gi) sqrt.
delta := (an - ai) abs.
ai := an.
gi := gn.
] doUntil:[ delta < epsilon ].
^ ai</lang>
<lang smalltalk>Transcript showCR: (24 agm:6). Transcript showCR: ( (1/2) agm:(1/6) ). Transcript showCR: (1 agm:(1 / 2 sqrt)).</lang>
- Output:
13.4581714817256 0.310602797207483 0.847213084793979
SQL
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. <lang 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
- </lang>
- Output:
RN A G DIFF -- ----------------------------------------- ------------------------------------------ ------------------------------------------ 6 0.847213084793979086606499123482191636480 0.8472130847939790866064991234821916364792 0.0000000000000000000000000000000000000008
Standard ML
<lang sml> fun agm(a, g) = let
fun agm'(a, g, eps) =
if Real.abs(a-g) < eps then
a
else
agm'((a+g)/2.0, Math.sqrt(a*g), eps)
in agm'(a, g, 1e~15)
end;
</lang>
- Output:
agm(1.0, 1.0/Math.sqrt(2.0)) => 0.847213084794
Stata
<lang 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</lang>
- Output:
.8472130848
Swift
<lang 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") }</lang>
- Output:
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).
<lang 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)]]</lang> Output:
0.8472130847939792
TI-83 BASIC
<lang ti83b>1→A:1/sqrt(2)→G While abs(A-G)>e-15 (A+G)/2→B sqrt(AG)→G:B→A End A</lang>
- Output:
.8472130848
UNIX Shell
ksh is one of the few unix shells that can do floating point arithmetic (bash does not). <lang 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</lang>
- Output:
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 <lang bash>while (( abs(a-g) > eps ))</lang> to <lang bash>while $a != $g </lang>
VBA
<lang 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</lang>
- Output:
0,853553390593274 0,847224902923494 0,847213084835193 0,847213084793979 0,847213084793979
VBScript
<lang 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)) </lang>
- Output:
0.847213084793979
Visual Basic .NET
Double, Decimal Versions
<lang vbnet>Imports System.Math Imports System.Console
Module Module1
Function CalcAGM(ByVal a As Double, ByVal b As Double) As Double
Dim c As Double, d As Double = 0, ld As Double = 1
While ld <> d : c = a : a = (a + b) / 2 : b = Sqrt(c * b)
ld = d : d = a - b : End While : Return b
End Function
Function DecSqRoot(ByVal v As Decimal) As Decimal
Dim r As Decimal = CDec(Sqrt(CDbl(v))), t As Decimal = 0, d As Decimal = 0, ld As Decimal = 1
While ld <> d : t = v / r : r = (r + t) / 2
ld = d : d = t - r : End While : Return t
End Function
Function CalcAGM(ByVal a As Decimal, ByVal b As Decimal) As Decimal
Dim c As Decimal, d As Decimal = 0, ld As Decimal = 1
While ld <> d : c = a : a = (a + b) / 2 : b = DecSqRoot(c * b)
ld = d : d = a - b : End While : Return b
End Function
Sub Main(ByVal args As String())
WriteLine("Double result: {0}", CalcAGM(1.0, DecSqRoot(0.5)))
WriteLine("Decimal result: {0}", CalcAGM(1D, DecSqRoot(0.5D)))
If System.Diagnostics.Debugger.IsAttached Then ReadKey()
End Sub
End Module</lang>
- Output:
Double result: 0.847213084793979 Decimal result: 0.8472130847939790866064991235
System.Numerics
<lang vbnet>Imports System.Math Imports System.Console Imports BI = System.Numerics.BigInteger
Module Module1
Function BIP(ByVal leadDig As Char, ByVal numDigs As Integer) As BI
BIP = BI.Parse(leadDig & New String("0"c, numDigs))
End Function
Function IntSqRoot(ByVal v As BI, ByVal res As BI) As BI ' res is the initial guess of the square root
Dim d As BI = 0, dl As BI = 1
While dl <> d : IntSqRoot = v / res : res = (res + IntSqRoot) / 2
dl = d : d = IntSqRoot - res : End While
End Function
Function CalcByAGM(ByVal digits As Integer) As BI
Dim a As BI = BIP("1"c, digits), ' value is 1, extended to required number of digits
c as BI, ' a temporary variable for swapping a and b
diff As BI = 0, ldiff As BI = 1 ' difference of a and b, last difference
CalcByAGM = BI.Parse(String.Format("{0:0.00000000000000000}", ' initial value of square root of 0.5
Sqrt(0.5)).Substring(2) & New String("0"c, digits - 17))
CalcByAGM = IntSqRoot(BIP("5"c, (digits << 1) - 1), CalcByAGM) ' value is now the square root of 0.5
While ldiff <> diff : c = a : a = (a + CalcByAGM) >> 1 : CalcByAGM = IntSqRoot(c * CalcByAGM, a)
ldiff = diff : diff = a - CalcByAGM : 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
WriteLine("0.{0}", CalcByAGM(digits))
If System.Diagnostics.Debugger.IsAttached Then ReadKey()
End Sub
End Module</lang>
- Output:
0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229942122285625233410963097962665830871059699713635983384251176326814289060389706768601616650048281188721897713309411767462019944392962902167289194499507231677897346863947606671057980557852173140349398304200422119216039839553595098193641293716340646029599967970599434351602031842648756950242174863855405981954581601742417887854192758804162719012085587685648326834140431218400804035809204559494313877815120926522254574397124286820766340954733674599621792665535348625686118543308626287287287563010835563193570668714785639088982115108836352147696979612621832943228417868113768445170018146021913694027020945996683513596327880804274345481744587363220025153952936265806614198365616491626259607434723706616902353080017375312847852558430631907454274934152685790655269406003147591020332746719686124796325510554648902820855297439651249940096625528660675804487353892185701401167716976535014084952476848993257321337028984668939194665861873752966387562266045914777044204681089256584408380320409106190031537067341195941010074743310599055058205243260099516927924174782169767810616836977141107392733439215501430220070873673659622721492587761928510523803670268904639096219076636442355380859029452340651900133423451058383417121805142550039237011113254111446126289062541335505266436535958245521562933975182514706501346410470569793556813066063293733450387109770972948759171790158173202815782884871499313408154933423677970447127859376185950851466773645546792016159342239971429840707888822790326567515965284358177957272848083564899635044041407342261101833835469759626633304220849998523007427039302772434749797179732645525465430198316949684610986907439050680137661192529197709384412997070158894931666611619945922650113111839663525025305616464315872084545229887754751772727476567216489829182392388952072076428397108847059603569219929218319015481412807665926982944644571492396663299730758139049576224389624231752095073190184244624423709864272811495111808228260538624846176751801409831274972576519837564923569028002161749055314272081534395405955635763711272816570597373374429700390560401563886630722257003892301591123769601215800817790778633512408624310735715837659265045466527873378744448344063102447570396812554539822664303534164130356138016341655752655897529445211668734512201912274667331915712407637538211069681410769263900748331757433967523196603308649735713838741960989838322028826948821913028193669499544222406972761686213695116578388850121990961606554546115432531481642493326947970041594914763231129205935165189979433500459762882172926259180894055084314663937825483351395501906533708720620640240770560758487964998436515927282645344286366154191425857771067561850172780332871751951893050318055052454260223355229007714181287986543511879180063562795936247682677864122494603381260826282540988953125276775346562432792145112295555160318184331336929617230417838551571255674049834166659269695800089537245730576945422753721602096871914703988784663672432627061911270717165908246400416799411204056571036408300024192943985530739946565396778104927010554103595133394321999250666762020783946955537605517964010097492188563113010178138885787938131720959480625392013009836502879176958279859052799477219417979970249430621584194688853281154977215799601944096234776861440850757392842988237593968232236705803341347746231128976258593243766317789749110772619097044895222045096307255155900938249040213648077920347672150485684460225544099928261631743126422857876289833806507220230103717531492635046310601885737725670066183812905806389545081270313113710437161358334880658339554312179013483988332164130576352447125115394720666703301013487165163241138288176398396295261211412632197959650986567867552507607604240959075175230219461045325643332496149012535333292237238689481278850201359663053760558493589283916304694038878549600274714871978014576595790495858022600660995249673643249668334617601066081567069751423818665036108388522097616550025160731149921612947757901997292486896382206038087602762816723701668191066335857751546503813342367223476420265585655884641601021054048985561871147358849763784064864267981865044863190774703822867114351511230036070865742988647714667473375011434581885279700605621172469217484718069486625119947289344427037830462070735493805287272062156063071882868580564521110696708028569906982576917722099867195996850779068144349493280497681154368046325993869307623507099951829512958112123570724538335482619075239515827309824818054966589790916886798407170779370595904577584091047341310960419411135775662072733779783320379730113767265853574771027978140972130961214239385473746276961504130795283737288205065871915225976508402779699176117539300672549249122984508236297556872271106584943553385049453263873648980460665597995436016950309279009245005785647723587619884898603441219534079536900299641197454906074160097885953766072290516077242859007090115663913836429904122082676962979786764903235649998199076599743987054864876909102491192709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48
Wren
<lang ecmascript>var eps = 1e-14
var agm = Fn.new { |a, g|
while ((a-g).abs > a.abs * eps) {
var t = a
a = (a+g)/2
g = (t*g).sqrt
}
return a
}
System.print(agm.call(1, 1/2.sqrt))</lang>
- Output:
0.84721308479398
XPL0
<lang 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; ]</lang>
Output:
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
<lang 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);
}</lang>
- Output:
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 <lang 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()));
}()</lang>
- Output:
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
<lang 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<ta THEN GO TO 20 60 PRINT a </lang>
- Output:
0.84721309
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