# Feigenbaum constant calculation

Feigenbaum constant calculation
You are encouraged to solve this task according to the task description, using any language you may know.

Calculate the Feigenbaum constant.

See

## 11l

Translation of: Python

<lang 11l>V max_it = 13 V max_it_j = 10 V a1 = 1.0 V a2 = 0.0 V d1 = 3.2 V a = 0.0

print(‘ i d’) L(i) 2..max_it

```  a = a1 + (a1 - a2) / d1
L(j) 1..max_it_j
V x = 0.0
V y = 0.0
L(k) 1..(1 << i)
y = 1.0 - 2.0 * y * x
x = a - x * x
a = a - x / y
V d = (a1 - a2) / (a - a1)
print(‘#2    #.8’.format(i, d))
d1 = d
a2 = a1
a1 = a</lang>
```
Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

Translation of: Ring

procedure Main is

```  procedure feigenbaum is
subtype i_range is Integer range 2 .. 13;
subtype j_range is Integer range 1 .. 10;
```
```     -- the number of digits in type Real is reduced to 15 to produce the
-- results reported by C, C++, C# and Ring. Increasing the number of
-- digits in type Real produces the results reported by D.
```
```     type Real is digits 15;
package Real_Io is new Float_IO (Real);
use Real_Io;
```
```     a, x, y, d : Real;
a1         : Real := 1.0;
a2         : Real := 0.0;
d1         : Real := 3.2;
begin
Put_Line (" i       d");
for i in i_range loop
a := a1 + (a1 - a2) / d1;
for j in j_range loop
x := 0.0;
y := 0.0;
for k in 1 .. 2**i loop
y := 1.0 - 2.0 * x * y;
x := a - x * x;
end loop;
a := a - x / y;
end loop;
d := (a1 - a2) / (a - a1);
Put (Item => i, Width => 2);
Put (Item => d, Fore => 5, Aft => 8, Exp => 0);
New_Line;
d1 := d;
a2 := a1;
a1 := a;
end loop;
end feigenbaum;
```

begin

```  feigenbaum;
```

end Main; </lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
Translation of: Ring

<lang algol68># Calculate the Feigenbaum constant #

print( ( "Feigenbaum constant calculation:", newline ) ); INT max it = 13; INT max it j = 10; REAL a1 := 1.0; REAL a2 := 0.0; REAL d1 := 3.2; print( ( "i ", "d", newline ) ); FOR i FROM 2 TO max it DO

```    REAL a := a1 + (a1 - a2) / d1;
FOR j TO max it j DO
REAL x := 0;
REAL y := 0;
FOR k TO 2 ^ i DO
y := 1 - 2 * y * x;
x := a - x * x
OD;
a := a - x / y
OD;
REAL d = (a1 - a2) / (a - a1);
IF i < 10 THEN
print( ( whole( i, 0 ), "  ", fixed( d, -10, 8 ), newline ) )
ELSE
print( ( whole( i, 0 ), " ",  fixed( d, -10, 8 ), newline ) )
FI;
d1 := d;
a2 := a1;
a1 := a
```

OD</lang>

Output:
```Feigenbaum constant calculation:
i  d
2  3.21851142
3  4.38567760
4  4.60094928
5  4.65513050
6  4.66611195
7  4.66854858
8  4.66906066
9  4.66917155
10 4.66919515
11 4.66920026
12 4.66920098
13 4.66920537
```

## AWK

<lang AWK>

1. syntax: GAWK -f FEIGENBAUM_CONSTANT_CALCULATION.AWK

BEGIN {

```   a1 = 1
a2 = 0
d1 = 3.2
max_i = 13
max_j = 10
print(" i d")
for (i=2; i<=max_i; i++) {
a = a1 + (a1 - a2) / d1
for (j=1; j<=max_j; j++) {
x = y = 0
for (k=1; k<=2^i; k++) {
y = 1 - 2 * y * x
x = a - x * x
}
a -= x / y
}
d = (a1 - a2) / (a - a1)
printf("%2d %.8f\n",i,d)
d1 = d
a2 = a1
a1 = a
}
exit(0)
```

} </lang>

Output:
``` i d
2 3.21851142
3 4.38567760
4 4.60094928
5 4.65513050
6 4.66611195
7 4.66854858
8 4.66906066
9 4.66917155
10 4.66919515
11 4.66920026
12 4.66920098
13 4.66920537
```

## C

Translation of: Ring

<lang c>#include <stdio.h>

void feigenbaum() {

```   int i, j, k, max_it = 13, max_it_j = 10;
double a, x, y, d, a1 = 1.0, a2 = 0.0, d1 = 3.2;
printf(" i       d\n");
for (i = 2; i <= max_it; ++i) {
a = a1 + (a1 - a2) / d1;
for (j = 1; j <= max_it_j; ++j) {
x = 0.0;
y = 0.0;
for (k = 1; k <= 1 << i; ++k) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
d = (a1 - a2) / (a - a1);
printf("%2d    %.8f\n", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
```

}

int main() {

```   feigenbaum();
return 0;
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

## C#

Translation of: Kotlin

<lang csharp>using System;

namespace FeigenbaumConstant {

```   class Program {
static void Main(string[] args) {
var maxIt = 13;
var maxItJ = 10;
var a1 = 1.0;
var a2 = 0.0;
var d1 = 3.2;
Console.WriteLine(" i       d");
for (int i = 2; i <= maxIt; i++) {
var a = a1 + (a1 - a2) / d1;
for (int j = 1; j <= maxItJ; j++) {
var x = 0.0;
var y = 0.0;
for (int k = 1; k <= 1<<i; k++) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
var d = (a1 - a2) / (a - a1);
Console.WriteLine("{0,2:d}    {1:f8}", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
}
}
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## C++

Translation of: C

<lang cpp>#include <iostream>

int main() {

```   const int max_it = 13;
const int max_it_j = 10;
double a1 = 1.0, a2 = 0.0, d1 = 3.2;
```
```   std::cout << " i       d\n";
for (int i = 2; i <= max_it; ++i) {
double a = a1 + (a1 - a2) / d1;
for (int j = 1; j <= max_it_j; ++j) {
double x = 0.0;
double y = 0.0;
for (int k = 1; k <= 1 << i; ++k) {
y = 1.0 - 2.0*y*x;
x = a - x * x;
}
a -= x / y;
}
double d = (a1 - a2) / (a - a1);
printf("%2d    %.8f\n", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
```
```   return 0;
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## D

<lang d>import std.stdio;

void main() {

```   int max_it = 13;
int max_it_j = 10;
double a1 = 1.0;
double a2 = 0.0;
double d1 = 3.2;
double a;
```
```   writeln(" i       d");
for (int i=2; i<=max_it; i++) {
a = a1 + (a1 - a2) / d1;
for (int j=1; j<=max_it_j; j++) {
double x = 0.0;
double y = 0.0;
for (int k=1; k <= 1<<i; k++) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
double d = (a1 - a2) / (a - a1);
writefln("%2d    %.8f", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920028
12    4.66920099
13    4.66920555```

## F#

Translation of: C#

<lang fsharp>open System

[<EntryPoint>] let main _ =

```   let maxIt = 13
let maxItJ = 10
let mutable a1 = 1.0
let mutable a2 = 0.0
let mutable d1 = 3.2
Console.WriteLine(" i       d")
for i in 2 .. maxIt do
let mutable a = a1 + (a1 - a2) / d1
for j in 1 .. maxItJ do
let mutable x = 0.0
let mutable y = 0.0
for _ in 1 .. (1 <<< i) do
y <- 1.0 - 2.0 * y * x
x <- a - x * x
a <- a - x / y
let d = (a1 - a2) / (a - a1)
Console.WriteLine("{0,2:d}    {1:f8}", i, d)
d1 <- d
a2 <- a1
a1 <- a
0 // return an integer exit code</lang>
```
Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Factor

Translation of: Raku

<lang factor>USING: formatting io locals math math.ranges sequences ;

[let

```     1 :> a1!
0 :> a2!
3.2 :> d!
```
```   " i d" print
```
```   2 13 [a,b] [| exp |
a1 a2 - d /f a1 + :> a!
10 [
0 :> x!
0 :> y!
exp 2^ [
1 2 x y * * - y!
a x sq - x!
] times
a x y /f - a!
] times
a1 a2 - a a1 - /f d!
a1 a2! a a1!
exp d "%2d %.8f\n" printf
] each
```

]</lang>

Output:
``` i d
2 3.21851142
3 4.38567760
4 4.60094928
5 4.65513050
6 4.66611195
7 4.66854858
8 4.66906066
9 4.66917155
10 4.66919515
11 4.66920026
12 4.66920098
13 4.66920537
```

## Fortran

<lang fortran> program feigenbaum

```     implicit none
```
```     integer i, j, k
real ( KIND = 16 ) x, y, a, b, a1, a2, d1
```
```     print '(a4,a13)', 'i', 'd'
```
```     a1 = 1.0;
a2 = 0.0;
d1 = 3.2;
```
```     do i=2,20
a = a1 + (a1 - a2) / d1;
do j=1,10
x = 0
y = 0
do k=1,2**i
y = 1 - 2 * y * x;
x = a - x**2;
end do
a = a - x / y;
end do
```
```        d1 = (a1 - a2) / (a - a1);
a2 = a1;
a1 = a;
print '(i4,f13.10)', i, d1
end do
end</lang>
```
Output:
```   i            d
2 3.2185114220
3 4.3856775986
4 4.6009492765
5 4.6551304954
6 4.6661119478
7 4.6685485814
8 4.6690606606
9 4.6691715554
10 4.6691951560
11 4.6692002291
12 4.6692013133
13 4.6692015458
14 4.6692015955
15 4.6692016062
16 4.6692016085
17 4.6692016090
18 4.6692016091
19 4.6692016091
20 4.6692016091```

## FreeBASIC

<lang freebasic>' version 25-0-2019 ' compile with: fbc -s console

Dim As UInteger i, j, k, maxit = 13, maxitj = 13 Dim As Double x, y, a, a1 = 1, a2, d, d1 = 3.2

Print "Feigenbaum constant calculation:" Print Print " i d" Print "==================="

For i = 2 To maxIt

```   a = a1 + (a1 - a2) / d1
For j = 1 To maxItJ
x = 0 : y = 0
For k = 1 To 2 ^ i
y = 1 - 2 * y * x
x = a - x * x
Next
a = a - x / y
Next
d = (a1 - a2) / (a - a1)
Print Using "###    ##.#########"; i; d
d1 = d
a2 = a1
a1 = a
```

Next

' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>

Output:
```Feigenbaum constant calculation:

i     d
===================
2     3.218511422
3     4.385677599
4     4.600949277
5     4.655130495
6     4.666111948
7     4.668548581
8     4.669060660
9     4.669171555
10     4.669195148
11     4.669200285
12     4.669201301
13     4.669198656```

## Fōrmulæ

In this page you can see the solution of this task.

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.

The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.

## Go

Translation of: Ring

<lang go>package main

import "fmt"

func feigenbaum() {

```   maxIt, maxItJ := 13, 10
a1, a2, d1 := 1.0, 0.0, 3.2
fmt.Println(" i       d")
for i := 2; i <= maxIt; i++ {
a := a1 + (a1-a2)/d1
for j := 1; j <= maxItJ; j++ {
x, y := 0.0, 0.0
for k := 1; k <= 1<<uint(i); k++ {
y = 1.0 - 2.0*y*x
x = a - x*x
}
a -= x / y
}
d := (a1 - a2) / (a - a1)
fmt.Printf("%2d    %.8f\n", i, d)
d1, a2, a1 = d, a1, a
}
```

}

func main() {

```   feigenbaum()
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

## Groovy

Translation of: Java

<lang groovy>class Feigenbaum {

```   static void main(String[] args) {
int max_it = 13
int max_it_j = 10
double a1 = 1.0
double a2 = 0.0
double d1 = 3.2
double a
```
```       println(" i       d")
for (int i = 2; i <= max_it; i++) {
a = a1 + (a1 - a2) / d1
for (int j = 0; j < max_it_j; j++) {
double x = 0.0
double y = 0.0
for (int k = 0; k < 1 << i; k++) {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y
}
double d = (a1 - a2) / (a - a1)
printf("%2d    %.8f\n", i, d)
d1 = d
a2 = a1
a1 = a
}
}
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

<lang haskell>import Data.List (mapAccumL)

feigenbaumApprox :: Int -> [Double] feigenbaumApprox mx = snd \$ mitch mx 10

``` where
mitch :: Int -> Int -> ((Double, Double, Double), [Double])
mitch mx mxj =
mapAccumL
(\(a1, a2, d1) i ->
let a =
iterate
(\a ->
let (x, y) =
iterate
(\(x, y) -> (a - (x * x), 1.0 - ((2.0 * x) * y)))
(0.0, 0.0) !!
(2 ^ i)
in a - (x / y))
(a1 + (a1 - a2) / d1) !!
mxj
d = (a1 - a2) / (a - a1)
in ((a, a1, d), d))
(1.0, 0.0, 3.2)
[2 .. (1 + mx)]
```

-- TEST ------------------------------------------------------------------ main :: IO () main =

``` (putStrLn . unlines) \$
zipWith
(\i s -> justifyRight 2 ' ' (show i) ++ '\t' : s)
[1 ..]
(show <\$> feigenbaumApprox 13)
where
justifyRight n c s = drop (length s) (replicate n c ++ s)</lang>
```
Output:
``` 1    3.2185114220380866
2    4.3856775985683365
3    4.600949276538056
4    4.6551304953919646
5    4.666111947822846
6    4.668548581451485
7    4.66906066077106
8    4.669171554514976
9    4.669195154039278
10    4.669200256503637
11    4.669200975097843
12    4.669205372040318
13    4.669207514010413```

## J

Translated from the beautiful Fōrmulæ version. Rather than a verb, the conjunction pre-assigns m and n .

```Feigenbaum =: conjunction define  NB. use:  n Feigenbaum m
irange=: <. + i.@:>:@:|@:-  NB. inclusive range
a=. 0 1
delta=. , 3.2
for_i. 3 irange n do.
tmp=. ({: + ({:delta) *inv ({: - _2&{)) a
for. i. m do.
'b bp'=. 0
for. i. 2 ^ <: i do.
'b bp'=. (tmp - *: b) , 1 _2 p. b * bp
end.
tmp=. tmp - b % bp
end.
a=. a , tmp
delta=. delta , %/@:(-/"1) (- 2 3 ,: 1 2) { a
end.
2 14j6 14j6 ": (#\ i. # delta) ,. (}. a) ,. delta
)

8  Feigenbaum  13
1      1.000000      3.200000
2      1.310703      3.218511
3      1.381547      4.385678
4      1.396945      4.600949
5      1.400253      4.655130
6      1.400962      4.666112
7      1.401114      4.668549
8      1.401146      4.669061
9      1.401153      4.669172
10      1.401155      4.669195
11      1.401155      4.669200
12      1.401155      4.669201
```

## Java

Translation of: Kotlin

<lang java>public class Feigenbaum {

```   public static void main(String[] args) {
int max_it = 13;
int max_it_j = 10;
double a1 = 1.0;
double a2 = 0.0;
double d1 = 3.2;
double a;
```
```       System.out.println(" i       d");
for (int i = 2; i <= max_it; i++) {
a = a1 + (a1 - a2) / d1;
for (int j = 0; j < max_it_j; j++) {
double x = 0.0;
double y = 0.0;
for (int k = 0; k < 1 << i; k++) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
double d = (a1 - a2) / (a - a1);
System.out.printf("%2d    %.8f\n", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
}
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## jq

<lang jq>def feigenbaum_delta(imax; jmax):

``` def lpad: tostring | (" " *  (4 - length)) + .;
def pp(i;x): "\(i|lpad)   \(x)";
```
``` "Feigenbaum's delta constant incremental calculation:",
pp("i"; "δ"),
pp(1; "3.20"),
( foreach range(2; 1+imax) as \$i (
{a1: 1.0, a2: 0.0, d1: 3.2};
```
```        .a = .a1 + (.a1 - .a2) / .d1
| reduce range(1; 1+jmax) as \$j (.;
.x = 0 | .y = 0
| reduce range(1; 1+pow(2;\$i)) as \$k (.;
.y = (1 - 2 * .x * .y)
| .x = .a - (.x * .x) )
| .a -= (.x / .y) )
| .d = (.a1 - .a2) / (.a - .a1)
```

| .emit = pp(\$i; .d)

```        | .d1 = .d | .a2 = .a1 | .a1 = .a;
```

.emit ) ) ;

feigenbaum_delta(13; 10) </lang>

Output:

<lang sh> Feigenbaum's delta constant incremental calculation:

```  i   δ
1   3.20
2   3.2185114220380866
3   4.3856775985683365
4   4.600949276538056
5   4.6551304953919646
6   4.666111947822846
7   4.668548581451485
8   4.66906066077106
9   4.669171554514976
10   4.669195154039278
11   4.669200256503637
12   4.669200975097843
13   4.669205372040318
```

</lang>

## Julia

<lang julia># http://en.wikipedia.org/wiki/Feigenbaum_constant

function feigenbaum_delta(imax=23, jmax=20)

```   a1, a2, d1 = BigFloat(1.0), BigFloat(0.0), BigFloat(3.2)
println("Feigenbaum's delta constant incremental calculation:\ni   δ\n1   3.20")
for i in 2:imax
a = a1 + (a1 - a2) / d1
for j in 1:jmax
x, y = 0, 0
for k in 1:2^i
y = 1 - 2 * x * y
x = a - x * x
end
a -= x / y
end
d = (a1 - a2) / (a - a1)
d1, a2 = d, a1
a1 = a
end
```

end

feigenbaum_delta()

</lang>

Output:
```Feigenbaum's delta constant incremental calculation:
i   δ
1   3.20
2   3.218511422038087912270504530742813256028820377971082199141994437483271226037533
3   4.385677598568339085744948568775522346103216356576497808699630752612705940390646
4   4.600949276538075357811694698623834985023552496633543372295593454454329771521727
5   4.655130495391980136486254995856898819475460497385226078363311588165123307017281
6   4.66611194782857138833121369671177648071905897173694216397236891198998639455025
7   4.668548581446840948044543680148146265543287896654348757317309551400403337843036
8   4.66906066064826823913259982263027263779968209542149740052288679867743088942764
9   4.669171555379511388886004609897567088240676573170789783804375113804695091803033
10  4.669195156030017174021108801191492093392147908605756405516325961597435372704323
11  4.669200229086856497938353781004067217408888048906823830162962242800074595934665
12  4.669201313294204171164754941185571183728248888986548913352217226469150028661929
13  4.669201545780906707506058109930429736431564330452605295006142805341042630340361
14  4.669201595537493910292470639289646040074547412490596040512777985387237785978782
15  4.669201606198152157723831097078594524421336516011873717994000712976201143278191
16  4.669201608480804423294067945898622842792868381815074127672747764898152898198069
17  4.669201608969744700482485321938373343907385540992447405883605282416375303280911
18  4.669201609074452566227981520370886753946099646679618270214759101315481224820708
19  4.669201609096878794705135037864783677622666525741836726064298799595215295927305
20  4.66920160910168168118696016084580172992808889324407617097679098039831535247408
21  4.669201609102710327837210208629111857781724142614997392167298168695631199065625
22  4.669201609102930630539778141205517641783439121041016813735799961205502985593042
23  4.66920160910297781286849594159066394676896043144121209732784416240857379387701
```

## Kotlin

Translation of: Ring

<lang scala>// Version 1.2.40

fun feigenbaum() {

```   val maxIt = 13
val maxItJ = 10
var a1 = 1.0
var a2 = 0.0
var d1 = 3.2
println(" i       d")
for (i in 2..maxIt) {
var a = a1 + (a1 - a2) / d1
for (j in 1..maxItJ) {
var x = 0.0
var y = 0.0
for (k in 1..(1 shl i)) {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y
}
val d = (a1 - a2) / (a - a1)
println("%2d    %.8f".format(i,d))
d1 = d
a2 = a1
a1 = a
}
```

}

fun main(args: Array<String>) {

```   feigenbaum()
```

}</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

## Lambdatalk

Following the Python code in a recursive mode. <lang scheme> {feigenbaum 11} // on my computer stackoverflow for values greater than 11 -> [3.2185114220380866,4.3856775985683365,4.600949276538056,4.6551304953919646,4.666111947822846, 4.668548581451485,4.66906066077106,4.669171554514976,4.669195154039278,4.669200256503637]

with:

{def feigenbaum

```{lambda {:maxi}
{f3 :maxi 10 1 0 3.2 0 {A.new} 2}}}
```

{def f3

```{lambda {:maxi :maxj :a1 :a2 :d1 :a3 :s :i}
{if {< :i {+ :maxi 1}}
then {let { {:maxi :maxi} {:maxj :maxj} {:a1 :a1} {:a2 :a2}
{:a3 {f2 {+ :a1 {/ {- :a1 :a2} :d1}} :i :maxj 1} }
{:s :s} {:i :i}
} {f3 :maxi :maxj :a3 :a1 {/ {- :a1 :a2} {- :a3 :a1}} :a3
{A.addlast! {/ {- :a1 :a2} {- :a3 :a1}} :s} {+ :i 1}} }
else :s}}}
```

{def f2

```{lambda {:a :i :maxj :j}
{if {< :j {+ :maxj 1}}
then {f2 {f1 :a :i 0 0 1} :i :maxj {+ :j 1}}
else :a}}}
```

{def f1

```{lambda {:a :i :y :x :k}
{if {< :k {+ {pow 2 :i} 1}}
then {f1 :a :i {- 1 {* 2 :y :x}} {- :a {* :x :x}} {+ :k 1}}
else {- :a {/ :x :y}} }}}
```

</lang>

## Lua

<lang lua>function leftShift(n,p)

```   local r = n
while p>0 do
r = r * 2
p = p - 1
end
return r
```

end

-- main

local MAX_IT = 13 local MAX_IT_J = 10 local a1 = 1.0 local a2 = 0.0 local d1 = 3.2

print(" i d") for i=2,MAX_IT do

```   local a = a1 + (a1 - a2) / d1
for j=1,MAX_IT_J do
local x = 0.0
local y = 0.0
for k=1,leftShift(1,i) do
y = 1.0 - 2.0 * y * x
x = a - x * x
end
a = a - x / y
end
d = (a1 - a2) / (a - a1)
print(string.format("%2d    %.8f", i, d))
d1 = d
a2 = a1
a1 = a
```

end</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Modula-2

<lang modula2>MODULE Feigenbaum; FROM FormatString IMPORT FormatString; FROM LongStr IMPORT RealToStr; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;

VAR

```   buf : ARRAY[0..63] OF CHAR;
i,j,k,max_it,max_it_j : INTEGER;
a,x,y,d,a1,a2,d1 : LONGREAL;
```

BEGIN

```   max_it := 13;
max_it_j := 10;
```
```   a1 := 1.0;
a2 := 0.0;
d1 := 3.2;
```
```   WriteString(" i       d");
WriteLn;
FOR i:=2 TO max_it DO
a := a1 + (a1 - a2) / d1;
FOR j:=1 TO max_it_j DO
x := 0.0;
y := 0.0;
FOR k:=1 TO INT(1 SHL i) DO
y := 1.0 - 2.0 * y * x;
x := a - x * x
END;
a := a - x / y
END;
d := (a1 - a2) / (a - a1);
FormatString("%2i    ", buf, i);
WriteString(buf);
RealToStr(d, buf);
WriteString(buf);
WriteLn;
d1 := d;
a2 := a1;
a1 := a
END;
```
```   ReadChar
```

END Feigenbaum.</lang>

## Nim

Translation of: Kotlin

<lang Nim>import strformat

iterator feigenbaum(): tuple[n: int; δ: float] =

``` ## Yield successive approximations of Feigenbaum constant.
```
``` const
MaxI = 13
MaxJ = 10
var
a1 = 1.0
a2 = 0.0
δ = 3.2
```
``` for i in 2..MaxI:
var a = a1 + (a1 - a2) / δ
for j in 1..MaxJ:
var x, y = 0.0
for _ in 1..(1 shl i):
y = 1 - 2 * y * x
x = a - x * x
a -= x / y
```
```   δ = (a1 - a2) / (a - a1)
a2 = a1
a1 = a
yield (i, δ)
```

echo " i δ" for n, δ in feigenbaum():

``` echo fmt"{n:2d}    {δ:.8f}"</lang>
```
Output:
``` i         δ
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Perl

<lang perl>use strict; use warnings; use Math::AnyNum 'sqr';

my \$a1 = 1.0; my \$a2 = 0.0; my \$d1 = 3.2;

print " i δ\n";

for my \$i (2..13) {

```   my \$a = \$a1 + (\$a1 - \$a2)/\$d1;
for (1..10) {
my \$x = 0;
my \$y = 0;
for (1 .. 2**\$i) {
\$y = 1 - 2 * \$y * \$x;
\$x = \$a - sqr(\$x);
}
\$a -= \$x/\$y;
}
```
```   \$d1 = (\$a1 - \$a2) / (\$a - \$a1);
(\$a2, \$a1) = (\$a1, \$a);
printf "%2d %17.14f\n", \$i, \$d1;
```

}</lang>

Output:
``` 2  3.21851142203809
3  4.38567759856834
4  4.60094927653808
5  4.65513049539198
6  4.66611194782857
7  4.66854858144684
8  4.66906066064827
9  4.66917155537951
10  4.66919515603002
11  4.66920022908686
12  4.66920131329420
13  4.66920154578091```

## Phix

Translation of: Ring

<lang Phix>constant maxIt = 13,

```       maxItJ = 10
```

atom a1 = 1.0,

```    a2 = 0.0,
d1 = 3.2
```

puts(1," i d\n") for i=2 to maxIt do

```    atom a = a1 + (a1 - a2) / d1
for j=1 to maxItJ do
atom x = 0, y = 0
for k=1 to power(2,i) do
y = 1 - 2*y*x
x = a - x*x
end for
a = a - x/y
end for
atom d = (a1-a2)/(a-a1)
printf(1,"%2d %.8f\n",{i,d})
d1 = d
a2 = a1
a1 = a
```

end for</lang>

Output:
``` i d
2 3.21851142
3 4.38567760
4 4.60094928
5 4.65513050
6 4.66611195
7 4.66854858
8 4.66906066
9 4.66917155
10 4.66919515
11 4.66920026
12 4.66920098
13 4.66920537
```

## Python

Translation of: D

<lang python>max_it = 13 max_it_j = 10 a1 = 1.0 a2 = 0.0 d1 = 3.2 a = 0.0

print " i d" for i in range(2, max_it + 1):

```   a = a1 + (a1 - a2) / d1
for j in range(1, max_it_j + 1):
x = 0.0
y = 0.0
for k in range(1, (1 << i) + 1):
y = 1.0 - 2.0 * y * x
x = a - x * x
a = a - x / y
d = (a1 - a2) / (a - a1)
print("{0:2d}    {1:.8f}".format(i, d))
d1 = d
a2 = a1
a1 = a</lang>
```
Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Racket

Translation of: C

<lang racket>#lang racket (define (feigenbaum #:max-it (max-it 13) #:max-it-j (max-it-j 10))

``` (displayln " i       d" (current-error-port))
(define-values (_a _a1 d)
(for/fold ((a 1) (a1 0) (d 3.2))
((i (in-range 2 (add1 max-it))))
(let* ((a′ (for/fold ((a (+ a (/ (- a a1) d))))
((j (in-range max-it-j)))
(let-values (([x y] (for/fold ((x 0) (y 0))
((k (expt 2 i)))
(values (- a (* x x))
(- 1 (* 2 y x))))))
(- a (/ x y)))))
(d′ (/ (- a a1) (- a′ a))))
(eprintf "~a   ~a\n" (~a i #:width 2) (real->decimal-string d′ 8))
(values a′ a d′))))
d)
```

(module+ main

``` (feigenbaum))</lang>
```
Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10   4.66919515
11   4.66920026
12   4.66920098
13   4.66920537
4.669205372040318```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2018.04.01
Translation of: Ring

<lang perl6>my \$a1 = 1; my \$a2 = 0; my \$d = 3.2;

say ' i d';

for 2 .. 13 -> \$exp {

```   my \$a = \$a1 + (\$a1 - \$a2) / \$d;
do {
my \$x = 0;
my \$y = 0;
for ^2 ** \$exp {
\$y = 1 - 2 * \$y * \$x;
\$x = \$a - \$x²;
}
\$a -= \$x / \$y;
} xx 10;
\$d = (\$a1 - \$a2) / (\$a - \$a1);
(\$a2, \$a1) = (\$a1, \$a);
printf "%2d %.8f\n", \$exp, \$d;
```

}</lang>

Output:
``` i d
2 3.21851142
3 4.38567760
4 4.60094928
5 4.65513050
6 4.66611195
7 4.66854858
8 4.66906066
9 4.66917155
10 4.66919515
11 4.66920026
12 4.66920098
13 4.66920537```

## REXX

Translation of: Sidef

<lang rexx>/*REXX pgm calculates the (Mitchell) Feigenbaum bifurcation velocity, #digs can be given*/ parse arg digs maxi maxj . /*obtain optional argument from the CL.*/ if digs== | digs=="," then digs= 30 /*Not specified? Then use the default.*/ if maxi== | maxi=="," then maxi= 20 /* " " " " " " */ if maxJ== | maxJ=="," then maxJ= 10 /* " " " " " " */

1. = 4.669201609102990671853203820466201617258185577475768632745651343004134330211314737138,
```                     || 68974402394801381716    /*◄──Feigenbaum's constant, true value.*/
```

numeric digits digs /*use the specified # of decimal digits*/

```   a1=  1
a2=  0
d1=  3.2
```

say 'Using ' maxJ " iterations for maxJ, with " digs ' decimal digits:' say say copies(' ', 9) center("correct", 11) copies(' ', digs+1) say center('i', 9, "─") center('digits' , 11, "─") center('d', digs+1, "─")

```   do i=2  for maxi-1
a= a1  +  (a1 - a2) / d1
do maxJ
x= 0;   y= 0
do 2**i;       y= 1  -  2 * x * y
x= a  -  x*x
end   /*2**i*/
a= a  -  x / y
end   /*maxj*/
d= (a1 - a2)  /  (a - a1)                    /*compute the delta (D) of the function*/
t= max(0, compare(d, #)  - 2)                /*# true digs so far, ignore dec. point*/
say center(i, 9)     center(t, 11)     d     /*display values for  I & D ──►terminal*/
parse value  d  a1  a    with    d1  a2  a1  /*assign 3 variables with 3 new values.*/
end   /*i*/
/*stick a fork in it,  we're all done. */
```

say left(, 9 + 1 + 11 + 1 + t )"↑" /*show position of greatest accuracy. */ say ' true value= ' # / 1 /*true value of Feigenbaum's constant. */</lang>

output   when using the default inputs:
```Using  10  iterations for  maxJ,  with  30  decimal digits:

correct
────i──── ──digits─── ───────────────d───────────────
2          0      3.21851142203808791227050453077
3          1      4.3856775985683390857449485682
4          2      4.60094927653807535781169469969
5          2      4.65513049539198013648625498649
6          3      4.66611194782857138833121364654
7          3      4.66854858144684094804454708811
8          4      4.66906066064826823913257549468
9          4      4.6691715553795113888859465442
10          4      4.66919515603001717402161720542
11          6      4.66920022908685649793393149233
12          7      4.66920131329420417113719511412
13          7      4.66920154578090670783369507315
14          7      4.66920159553749390966169074155
15          9      4.66920160619815215840788706632
16          9      4.66920160848080435144581223484
17          9      4.66920160896974538458267849027
18         10      4.66920160907444981238909862845
19         10      4.66920160909687888294310165196
20         12      4.66920160910169069039564432665
↑
true value=  4.66920160910299067185320382047
```

## Ring

<lang ring># Project : Feigenbaum constant calculation

decimals(8) see "Feigenbaum constant calculation:" + nl maxIt = 13 maxItJ = 10 a1 = 1.0 a2 = 0.0 d1 = 3.2 see "i " + "d" + nl for i = 2 to maxIt

```    a = a1 + (a1 - a2) / d1
for j = 1 to maxItJ
x = 0
y = 0
for k = 1 to pow(2,i)
y = 1 - 2 * y * x
x = a - x * x
next
a = a - x / y
next
d = (a1 - a2) / (a - a1)
if i < 10
see "" + i + "    " + d + nl
else
see "" + i + "  " + d + nl
ok
d1 = d
a2 = a1
a1 = a
```

next</lang> Output:

```Feigenbaum constant calculation:
i  d
2  3.21851142
3  4.38567760
4  4.60094928
5  4.65513050
6  4.66611195
7  4.66854858
8  4.66906066
9  4.66917155
10 4.66919515
11 4.66920026
12 4.66920098
13 4.66920537```

## Ruby

Translation of: C#

<lang ruby>def main

```   maxIt = 13
maxItJ = 10
a1 = 1.0
a2 = 0.0
d1 = 3.2
puts " i       d"
for i in 2 .. maxIt
a = a1 + (a1 - a2) / d1
for j in 1 .. maxItJ
x = 0.0
y = 0.0
for k in 1 .. 1 << i
y = 1.0 - 2.0 * y * x
x = a - x * x
end
a = a - x / y
end
d = (a1 - a2) / (a - a1)
print "%2d    %.8f\n" % [i, d]
d1 = d
a2 = a1
a1 = a
end
```

end

main()</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Scala

### Imperative, ugly

<lang Scala>object Feigenbaum1 extends App {

``` val (max_it, max_it_j) = (13, 10)
var (a1, a2, d1, a) = (1.0, 0.0, 3.2, 0.0)
```
``` println(" i       d")
var i: Int = 2
while (i <= max_it) {
a = a1 + (a1 - a2) / d1
for (_ <- 0 until max_it_j) {
var (x, y) = (0.0, 0.0)
for (_ <- 0 until 1 << i) {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y
}
val d: Double = (a1 - a2) / (a - a1)
printf("%2d    %.8f\n", i, d)
d1 = d
a2 = a1
a1 = a
i += 1
}
```

}</lang>

### Functional Style, Tail recursive

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

<lang Scala>object Feigenbaum2 extends App {

``` private val (max_it, max_it_j) = (13, 10)
```
``` private def result = {
```
```   @scala.annotation.tailrec
def outer(i: Int, d1: Double, a2: Double, a1: Double, acc: Seq[Double]): Seq[Double] = {
@scala.annotation.tailrec
def center(j: Int, a: Double): Double = {
@scala.annotation.tailrec
def inner(k: Int, end: Int, x: Double, y: Double): (Double, Double) =
if (k < end) inner(k + 1, end, a - x * x, 1.0 - 2.0 * y * x) else (x, y)
```
```       val (x, y) = inner(0, 1 << i, 0.0, 0.0)
if (j < max_it_j) {
center(j + 1, a - (x / y))
} else a
}
```
```     if (i <= max_it) {
val a = center(0, a1 + (a1 - a2) / d1)
val d: Double = (a1 - a2) / (a - a1)
```
```       outer(i + 1, d, a1, a, acc :+ d)
} else acc
}
```
```   outer(2, 3.2, 0, 1.0, Seq[Double]()).zipWithIndex
}
```
``` println(" i     ≈ δ")
result.foreach { case (δ, i) => println(f"\${i + 2}%2d  \$δ%.8f") }
```

}</lang>

## Sidef

Translation of: Raku

<lang ruby>var a1 = 1 var a2 = 0 var δ = 3.2.float

say " i\tδ"

for i in (2..15) {

```   var a0 = ((a1 - a2)/δ + a1)
10.times {
var (x, y) = (0, 0)
2**i -> times {
y = (1 - 2*x*y)
x = (a0 - x²)
}
a0 -= x/y
}
δ = ((a1 - a2) / (a0 - a1))
(a2, a1) = (a1, a0)
printf("%2d %.8f\n", i, δ)
```

}</lang>

Output:
``` i	δ
2 3.21851142
3 4.38567760
4 4.60094928
5 4.65513050
6 4.66611195
7 4.66854858
8 4.66906066
9 4.66917156
10 4.66919516
11 4.66920023
12 4.66920131
13 4.66920155
14 4.66920160
15 4.66920161
```

## Swift

Translation of: C

<lang swift>import Foundation

func feigenbaum(iterations: Int = 13) {

``` var a = 0.0
var a1 = 1.0
var a2 = 0.0
var d = 0.0
var d1 = 3.2
```
``` print(" i       d")
```
``` for i in 2...iterations {
a = a1 + (a1 - a2) / d1
```
```   for _ in 1...10 {
var x = 0.0
var y = 0.0
```
```     for _ in 1...1<<i {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
```
```     a -= x / y
}
```
```   d = (a1 - a2) / (a - a1)
d1 = d
(a1, a2) = (a, a1)
```
```   print(String(format: "%2d    %.8f", i, d))
}
```

}

feigenbaum()</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Visual Basic .NET

Translation of: C#

<lang vbnet>Module Module1

```   Sub Main()
Dim maxIt = 13
Dim maxItJ = 10
Dim a1 = 1.0
Dim a2 = 0.0
Dim d1 = 3.2
Console.WriteLine(" i       d")
For i = 2 To maxIt
Dim a = a1 + (a1 - a2) / d1
For j = 1 To maxItJ
Dim x = 0.0
Dim y = 0.0
For k = 1 To 1 << i
y = 1.0 - 2.0 * y * x
x = a - x * x
Next
a -= x / y
Next
Dim d = (a1 - a2) / (a - a1)
Console.WriteLine("{0,2:d}    {1:f8}", i, d)
d1 = d
a2 = a1
a1 = a
Next
End Sub
```

End Module</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537```

## Wren

Translation of: Ring
Library: Wren-fmt

<lang ecmascript>import "/fmt" for Fmt

var feigenbaum = Fn.new {

```   var maxIt = 13
var maxItJ = 10
var a1 = 1
var a2 = 0
var d1 = 3.2
System.print(" i       d")
for (i in 2..maxIt) {
var a = a1 + (a1 - a2)/d1
for (j in 1..maxItJ) {
var x = 0
var y = 0
for (k in 1..(1<<i)) {
y = 1 - 2*y*x
x = a - x*x
}
a = a - x/y
}
var d = (a1 - a2)/(a - a1)
System.print("%(Fmt.d(2, i))    %(Fmt.f(0, d, 8))")
d1 = d
a2 = a1
a1 = a
}
```

}

feigenbaum.call()</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

## zkl

Translation of: Kotlin

<lang zkl>fcn feigenbaum{

```  maxIt,maxItJ,a1,a2,d1,a,d := 13, 10, 1.0, 0.0, 3.2, 0, 0;
println(" i       d");
foreach i in ([2..maxIt]){
a=a1 + (a1 - a2)/d1;
foreach j in ([1..maxItJ]){
x,y := 0.0, 0.0;
```

foreach k in ([1..(1).shiftLeft(i)]){ y,x = 1.0 - 2.0*y*x, a - x*x; } a-=x/y

```     }
d=(a1 - a2)/(a - a1);
println("%2d    %.8f".fmt(i,d));
d1,a2,a1 = d,a1,a;
}
```

}();</lang>

Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```