Continued fraction/Arithmetic/Construct from rational number

From Rosetta Code
Task
Continued fraction/Arithmetic/Construct from rational number
You are encouraged to solve this task according to the task description, using any language you may know.
To understand this task in context please see Continued fraction arithmetic

The purpose of this task is to write a function , or , which will output a continued fraction assuming:

is the numerator
is the denominator

The function should output its results one digit at a time each time it is called, in a manner sometimes described as lazy evaluation.

To achieve this it must determine: the integer part; and remainder part, of divided by . It then sets to and to the determined remainder part. It then outputs the determined integer part. It does this until is zero.

Demonstrate the function by outputing the continued fraction for:

1/2
3
23/8
13/11
22/7
-151/77

should approach try ever closer rational approximations until boredom gets the better of you:

14142,10000
141421,100000
1414214,1000000
14142136,10000000

Try :

31,10
314,100
3142,1000
31428,10000
314285,100000
3142857,1000000
31428571,10000000
314285714,100000000

Observe how this rational number behaves differently to and convince yourself that, in the same way as may be represented as when an extra decimal place is required, may be represented as when an extra term is required.

C[edit]

C does not implement Lazy evaluation and it is this particular feature which is the real challenge of this particular example. It can however be simulated. The following example uses pointers. It seems that the same data is being passed but since the function accepts pointers, the variables are being changed. One other way to simulate laziness would be to use global variables. Then although it would seem that the same values are being passed even as constants, the job is actually getting done. In my view, that would be plain cheating.

 
/*Abhishek Ghosh, 8th November 2013, Rotterdam*/
 
#include<stdio.h>
 
typedef struct{
int num,den;
}fraction;
 
fraction examples[] = {{1,2}, {3,1}, {23,8}, {13,11}, {22,7}, {-151,77}};
fraction sqrt2[] = {{14142,10000}, {141421,100000}, {1414214,1000000}, {14142136,10000000}};
fraction pi[] = {{31,10}, {314,100}, {3142,1000}, {31428,10000}, {314285,100000}, {3142857,1000000}, {31428571,10000000}, {314285714,100000000}};
 
int r2cf(int *numerator,int *denominator)
{
int quotient=0,temp;
 
if(denominator != 0)
{
quotient = *numerator / *denominator;
 
temp = *numerator;
 
*numerator = *denominator;
 
*denominator = temp % *denominator;
}
 
return quotient;
}
 
int main()
{
int i;
 
printf("Running the examples :");
 
for(i=0;i<sizeof(examples)/sizeof(fraction);i++)
{
printf("\nFor N = %d, D = %d :",examples[i].num,examples[i].den);
 
while(examples[i].den != 0){
printf(" %d ",r2cf(&examples[i].num,&examples[i].den));
}
}
 
printf("\n\nRunning for %c2 :",251); /* 251 is the ASCII code for the square root symbol */
 
for(i=0;i<sizeof(sqrt2)/sizeof(fraction);i++)
{
printf("\nFor N = %d, D = %d :",sqrt2[i].num,sqrt2[i].den);
 
while(sqrt2[i].den != 0){
printf(" %d ",r2cf(&sqrt2[i].num,&sqrt2[i].den));
}
}
 
printf("\n\nRunning for %c :",227); /* 227 is the ASCII code for Pi's symbol */
 
for(i=0;i<sizeof(pi)/sizeof(fraction);i++)
{
printf("\nFor N = %d, D = %d :",pi[i].num,pi[i].den);
 
while(pi[i].den != 0){
printf(" %d ",r2cf(&pi[i].num,&pi[i].den));
}
}
 
 
 
return 0;
}
 
 

And the run gives :

Running the examples :
For N = 1, D = 2 : 0  2
For N = 3, D = 1 : 3
For N = 23, D = 8 : 2  1  7
For N = 13, D = 11 : 1  5  2
For N = 22, D = 7 : 3  7
For N = -151, D = 77 : -1  -1  -24  -1  -2

Running for √2 :
For N = 14142, D = 10000 : 1  2  2  2  2  2  1  1  29
For N = 141421, D = 100000 : 1  2  2  2  2  2  2  3  1  1  3  1  7  2
For N = 1414214, D = 1000000 : 1  2  2  2  2  2  2  2  3  6  1  2  1  12
For N = 14142136, D = 10000000 : 1  2  2  2  2  2  2  2  2  2  6  1  2  4  1  1  2

Running for π :
For N = 31, D = 10 : 3  10
For N = 314, D = 100 : 3  7  7
For N = 3142, D = 1000 : 3  7  23  1  2
For N = 31428, D = 10000 : 3  7  357
For N = 314285, D = 100000 : 3  7  2857
For N = 3142857, D = 1000000 : 3  7  142857
For N = 31428571, D = 10000000 : 3  7  476190  3
For N = 314285714, D = 100000000 : 3  7  7142857

C++[edit]

#include <iostream>
/* Interface for all Continued Fractions
Nigel Galloway, February 9th., 2013.
*/

class ContinuedFraction {
public:
virtual const int nextTerm(){};
virtual const bool moreTerms(){};
};
/* Create a continued fraction from a rational number
Nigel Galloway, February 9th., 2013.
*/

class r2cf : public ContinuedFraction {
private: int n1, n2;
public:
r2cf(const int numerator, const int denominator): n1(numerator), n2(denominator){}
const int nextTerm() {
const int thisTerm = n1/n2;
const int t2 = n2; n2 = n1 - thisTerm * n2; n1 = t2;
return thisTerm;
}
const bool moreTerms() {return fabs(n2) > 0;}
};
/* Generate a continued fraction for sqrt of 2
Nigel Galloway, February 9th., 2013.
*/

class SQRT2 : public ContinuedFraction {
private: bool first=true;
public:
const int nextTerm() {if (first) {first = false; return 1;} else return 2;}
const bool moreTerms() {return true;}
};

Testing[edit]

1/2 3 23/8 13/11 22/7 -151/77[edit]

int main() {
for(r2cf n(1,2); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(3,1); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(23,8); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(13,11); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(22,7); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf cf(-151,77); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
std::cout << std::endl;
return 0;
}
Output:
0 2
3
2 1 7
1 5 2
3 7
-1 -1 -24 -1 -2

2 {\displaystyle {\sqrt {2}}}

int main() {
int i = 0;
for(SQRT2 n; i++ < 20; std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(14142,10000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(14142136,10000000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
return 0;
}
Output:
1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 2 2 2 2 2 1 1 29
1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2

Real approximations of a rational number[edit]

int main() {
for(r2cf n(31,10); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(314,100); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(3142,1000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(31428,10000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(314285,100000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(3142857,1000000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(31428571,10000000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf n(314285714,100000000); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
return 0;
}
Output:
3 10
3 7 7
3 7 23 1 2
3 7 357
3 7 2857
3 7 142857
3 7 476190 3
3 7 7142857

C#[edit]

using System;
using System.Collections.Generic;
 
class Program
{
static IEnumerable<int> r2cf(int n1, int n2)
{
while (Math.Abs(n2) > 0)
{
int t1 = n1 / n2;
int t2 = n2;
n2 = n1 - t1 * n2;
n1 = t2;
yield return t1;
}
}
 
static void spit(IEnumerable<int> f)
{
foreach (int n in f) Console.Write(" {0}", n);
Console.WriteLine();
}
 
static void Main(string[] args)
{
spit(r2cf(1, 2));
spit(r2cf(3, 1));
spit(r2cf(23, 8));
spit(r2cf(13, 11));
spit(r2cf(22, 7));
spit(r2cf(-151, 77));
for (int scale = 10; scale <= 10000000; scale *= 10)
{
spit(r2cf((int)(Math.Sqrt(2) * scale), scale));
}
spit(r2cf(31, 10));
spit(r2cf(314, 100));
spit(r2cf(3142, 1000));
spit(r2cf(31428, 10000));
spit(r2cf(314285, 100000));
spit(r2cf(3142857, 1000000));
spit(r2cf(31428571, 10000000));
spit(r2cf(314285714, 100000000));
}
}
 

Output

 0 2
 3
 2 1 7
 1 5 2
 3 7
 -1 -1 -24 -1 -2
 1 2 2
 1 2 2 3 1 1 2
 1 2 2 2 2 5 3
 1 2 2 2 2 2 1 1 29
 1 2 2 2 2 2 2 3 1 1 3 1 7 2
 1 2 2 2 2 2 2 2 1 1 4 1 1 1 1 1 2 1 6
 1 2 2 2 2 2 2 2 2 2 1 594
 3 10
 3 7 7
 3 7 23 1 2
 3 7 357
 3 7 2857
 3 7 142857
 3 7 476190 3
 3 7 7142857

Clojure[edit]

(defn r2cf [n d]
(if-not (= d 0) (cons (quot n d) (lazy-seq (r2cf d (rem n d))))))
 
; Example usage
(def demo '((1 2)
(3 1)
(23 8)
(13 11)
(22 7)
(-151 77)
(14142 10000)
(141421 100000)
(1414214 1000000)
(14142136 10000000)
(31 10)
(314 100)
(3142 1000)
(31428 10000)
(314285 100000)
(3142857 1000000)
(31428571 10000000)
(314285714 100000000)
(3141592653589793 1000000000000000)))
 
(doseq [inputs demo
 :let [outputs (r2cf (first inputs) (last inputs))]]
(println inputs ";" outputs))
Output:
(1 2) ; (0 2)
(3 1) ; (3)
(23 8) ; (2 1 7)
(13 11) ; (1 5 2)
(22 7) ; (3 7)
(-151 77) ; (-1 -1 -24 -1 -2)
(14142 10000) ; (1 2 2 2 2 2 1 1 29)
(141421 100000) ; (1 2 2 2 2 2 2 3 1 1 3 1 7 2)
(1414214 1000000) ; (1 2 2 2 2 2 2 2 3 6 1 2 1 12)
(14142136 10000000) ; (1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2)
(31 10) ; (3 10)
(314 100) ; (3 7 7)
(3142 1000) ; (3 7 23 1 2)
(31428 10000) ; (3 7 357)
(314285 100000) ; (3 7 2857)
(3142857 1000000) ; (3 7 142857)
(31428571 10000000) ; (3 7 476190 3)
(314285714 100000000) ; (3 7 7142857)
(3141592653589793 1000000000000000) ; (3 7 15 1 292 1 1 1 2 1 3 1 14 4 2 3 1 12 5 1 5 20 1 11 1 1 1 2)

Common Lisp[edit]

(defun r2cf (n1 n2)
(lambda ()
(unless (zerop n2)
(multiple-value-bind (t1 r)
(floor n1 n2)
(setf n1 n2 n2 r)
t1))))
 
;; Example usage
 
(defun demo-generator (numbers)
(let* ((n1 (car numbers))
(n2 (cadr numbers))
(gen (r2cf n1 n2)))
(format t "~S  ; ~S~%"
`(r2cf ,n1 ,n2)
(loop
:for r = (funcall gen)
:until (null r)
:collect r))))
 
(mapcar #'demo-generator
'((1 2)
(3 1)
(23 8)
(13 11)
(22 7)
(-151 77)
(14142 10000)
(141421 100000)
(1414214 1000000)
(14142136 10000000)
(31 10)
(314 100)
(3142 1000)
(31428 10000)
(314285 100000)
(3142857 1000000)
(31428571 10000000)
(314285714 100000000)
(3141592653589793 1000000000000000)))

Output:

(R2CF 3 1)  ; (3)
(R2CF 23 8)  ; (2 1 7)
(R2CF 13 11)  ; (1 5 2)
(R2CF 22 7)  ; (3 7)
(R2CF -151 77)  ; (-2 25 1 2)
(R2CF 14142 10000)  ; (1 2 2 2 2 2 1 1 29)
(R2CF 141421 100000)  ; (1 2 2 2 2 2 2 3 1 1 3 1 7 2)
(R2CF 1414214 1000000)  ; (1 2 2 2 2 2 2 2 3 6 1 2 1 12)
(R2CF 14142136 10000000)  ; (1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2)
(R2CF 31 10)  ; (3 10)
(R2CF 314 100)  ; (3 7 7)
(R2CF 3142 1000)  ; (3 7 23 1 2)
(R2CF 31428 10000)  ; (3 7 357)
(R2CF 314285 100000)  ; (3 7 2857)
(R2CF 3142857 1000000)  ; (3 7 142857)
(R2CF 31428571 10000000)  ; (3 7 476190 3)
(R2CF 314285714 100000000)  ; (3 7 7142857)
(R2CF 3141592653589793 1000000000000000)  ; (3 7 15 1 292 1 1 1 2 1 3 1 14 4 2 3 1 12 5 1 5 20 1 11 1 1 1 2)

F#[edit]

let rec r2cf n d =
if d = LanguagePrimitives.GenericZero then []
else let q = n / d in q :: (r2cf d (n - q * d))
 
[<EntryPoint>]
let main argv =
printfn "%A" (r2cf 1 2)
printfn "%A" (r2cf 3 1)
printfn "%A" (r2cf 23 8)
printfn "%A" (r2cf 13 11)
printfn "%A" (r2cf 22 7)
printfn "%A" (r2cf -151 77)
printfn "%A" (r2cf 141 100)
printfn "%A" (r2cf 1414 1000)
printfn "%A" (r2cf 14142 10000)
printfn "%A" (r2cf 141421 100000)
printfn "%A" (r2cf 1414214 1000000)
printfn "%A" (r2cf 14142136 10000000)
0

Output

[0; 2]
[3]
[2; 1; 7]
[1; 5; 2]
[3; 7]
[-1; -1; -24; -1; -2]
[1; 2; 2; 3; 1; 1; 2]
[1; 2; 2; 2; 2; 5; 3]
[1; 2; 2; 2; 2; 2; 1; 1; 29]
[1; 2; 2; 2; 2; 2; 2; 3; 1; 1; 3; 1; 7; 2]
[1; 2; 2; 2; 2; 2; 2; 2; 3; 6; 1; 2; 1; 12]
[1; 2; 2; 2; 2; 2; 2; 2; 2; 2; 6; 1; 2; 4; 1; 1; 2]

Haskell[edit]

Translation of: Python

This more general version generates a continued fraction from any real number (with rationals as a special case):

import Data.Ratio ((%))
 
real2cf :: (RealFrac a, Integral b) => a -> [b]
real2cf x =
let (i, f) = properFraction x
in i :
if f == 0
then []
else real2cf (1 / f)
 
main :: IO ()
main =
mapM_
print
[ real2cf (13 % 11)
, take 20 $ real2cf (sqrt 2)
]
Output:
[1,5,2]
[1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]

J[edit]

Note that the continued fractions shown in this task differ from those in the Continued fraction task as b here is implicitly always 1.

Tacit version 1[edit]

This version is a modification of an explicit version shown in http://www.jsoftware.com/jwiki/Essays/Continued%20Fractions to comply with the task specifications.

cf=: _1 1 ,@}. (, <.)@%@-/ ::]^:a:@(, <.)@(%&x:/)

Examples[edit]

   cf each 1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77
┌───┬─┬─────┬─────┬───┬─────────────────────────────────┬─────────┐
0 232 1 71 5 23 71 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2_2 25 1 2
└───┴─┴─────┴─────┴───┴─────────────────────────────────┴─────────┘
cf each 14142 10000;141421 100000;1414214 1000000;14142136 10000000
┌──────────────────┬───────────────────────────┬────────────────────────────┬─────────────────────────────────┐
1 2 2 2 2 2 1 1 291 2 2 2 2 2 2 3 1 1 3 1 7 21 2 2 2 2 2 2 2 3 6 1 2 1 121 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2
└──────────────────┴───────────────────────────┴────────────────────────────┴─────────────────────────────────┘
cf each 31 10;314 100;3142 1000;31428 10000;314285 100000;3142857 1000000;31428571 10000000;314285714 100000000
┌────┬─────┬──────────┬───────┬────────┬──────────┬────────────┬───────────┐
3 103 7 73 7 23 1 23 7 3573 7 28573 7 1428573 7 476190 33 7 7142857
└────┴─────┴──────────┴───────┴────────┴──────────┴────────────┴───────────┘

This tacit version first produces the answer with a trailing ∞ (represented by _ in J) which is then removed by the last operation (_1 1 ,@}. ...). A continued fraction can be evaluated using the verb ((+%)/) and both representations produce equal results,

   3 7 =&((+ %)/) 3 7 _
1

Incidentally, J and Tcl report a different representation for -151/77 versus the representation of some other implementations; however, both representations produce equal results.

   _2 25 1 2 =&((+ %)/) _1 _1 _24 _1 _2
1

Tacit version 2[edit]

Translation of python

r2cf=:1 1{."1@}.({:,(0,{:)#:{.)^:(*@{:)^:a:

Example use:

   ((":@{.,'/',":@{:),':  ',":@r2cf)@>1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77;14142 10000;141421 100000;1414214 1000000;14142136 10000000;31 10;314 100;3142 1000;31428 10000;314285 100000;3142857 1000000;31428571 10000000;314285714 100000000 
1/2: 0 2
3/1: 3
23/8: 2 1 7
13/11: 1 5 2
22/7: 3 7
14142136/10000000: 1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2
_151/77: _2 25 1 2
14142/10000: 1 2 2 2 2 2 1 1 29
141421/100000: 1 2 2 2 2 2 2 3 1 1 3 1 7 2
1414214/1000000: 1 2 2 2 2 2 2 2 3 6 1 2 1 12
14142136/10000000: 1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2
31/10: 3 10
314/100: 3 7 7
3142/1000: 3 7 23 1 2
31428/10000: 3 7 357
314285/100000: 3 7 2857
3142857/1000000: 3 7 142857
31428571/10000000: 3 7 476190 3
314285714/100000000: 3 7 7142857

Explicit versions[edit]

version 1[edit]

Implemented as a class, r2cf preserves state in a separate locale. I've used some contrivances to jam the examples onto one line.

 
coclass'cf'
create =: dyad def 'EMPTY [ N =: x , y'
destroy =: codestroy
r2cf =: monad define
if. 0 (= {:) N do. _ return. end.
RV =. <[email protected]:(%/) N
N =: ({. , |/)@:|. N
RV
)
 
cocurrent'base'
CF =: conew'cf'
 
Until =: conjunction def 'u^:([email protected]:v)^:_'
 
(,. }[email protected]:}:@:((,r2cf__CF)Until(_-:{:))@:(8[create__CF/)&.>)1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77
Note 'Output'
┌─────────────────┬─────────────────────────────────┐
│1 2 │0 2 │
├─────────────────┼─────────────────────────────────┤
│3 1 │3 │
├─────────────────┼─────────────────────────────────┤
│23 8 │2 1 7 │
├─────────────────┼─────────────────────────────────┤
│13 11 │1 5 2 │
├─────────────────┼─────────────────────────────────┤
│22 7 │3 7 │
├─────────────────┼─────────────────────────────────┤
│14142136 10000000│1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2│
├─────────────────┼─────────────────────────────────┤
│_151 77 │_2 25 1 2 │
└─────────────────┴─────────────────────────────────┘
)

version 2[edit]

 
f =: 3 : 0
a =. {.y
b =. {:y
out=. <. a%b
while. b > 1 do.
'a b' =. b; b|a
out=. out , <. a%b
end.
)
f each 1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77
┌───┬─┬─────┬─────┬───┬───────────────────────────────────┬─────────┐
0 232 1 71 5 23 71 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2 __2 25 1 2
└───┴─┴─────┴─────┴───┴───────────────────────────────────┴─────────┘

version 3[edit]

translation of python:

r2cf=:3 :0
'n1 n2'=. y
r=.''
while.n2 do.
'n1 t1 n2'=. n2,(0,n2)#:n1
r=.r,t1
end.
)

Example:

   r2cf each 1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77
┌───┬─┬─────┬─────┬───┬─────────────────────────────────┬─────────┐
0 232 1 71 5 23 71 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2_2 25 1 2
└───┴─┴─────┴─────┴───┴─────────────────────────────────┴─────────┘

Julia[edit]

# It's most appropriate to define a Julia iterable object for this task
# Julia doesn't have Python's yield, the closest to it is produce/consume calls with Julia tasks
# but for various reasons they don't work out for this task
# This solution works with two integers, a Julia rational or a real
 
type R2cf
n1::Union{Int,Float64} # numerator or real
n2::Int # denominator or 1 if real
t1::Int # generated coefficient
f::Float64 # aux. field for working with real inputs
end
 
# constructors for all possible input types
R2cf(n::Union{Int,Float64})=R2cf(n,1,0,0.1)
R2cf(n1::Int,n2::Int)=R2cf(n1,n2,0,0.1)
R2cf(r::Rational{Int})=R2cf(num(r),den(r),0,0.1)
 
# methods to make our object iterable
Base.start(::R2cf)=nothing
 
# generates the next coefficient
function Base.next(cf::R2cf,s)
if typeof(cf.n1)==Int
cf.n1, (cf.t1, cf.n2) = cf.n2, divrem(cf.n1, cf.n2)
else
cf.t1,cf.f=divrem(cf.n1,1)
if cf.f!=0.0 cf.n1=1/cf.f end
end
return (cf.t1,nothing)
end
 
# returns true if we've prepared the continued fraction
function Base.done(cf::R2cf,s)
if typeof(cf.n1)==Int
return cf.n2==0
else
return cf.f==0.0
end
end
 
# tell Julia that this object always returns ints (all coeffs are integers)
Base.eltype(::Type{R2cf})=Int
 
# overload the default collect function so that we can collect the first maxiter coeffs of infinite continued fractions
# array slicing doesn't work as Julia crashes before the slicing due to our infinitely long array
function Base.collect(itr::R2cf,maxiter::Int)
r=Array{Int,1}() # all results are ints
i=1
for v=itr
push!(r,v)
i+=1
if i==maxiter break end
end
return r
end
 
# test cases according to task description with outputs in comments
println(collect(R2cf(1,2))) # => [0, 2]
println(collect(R2cf(3,1))) # => [3]
println(collect(R2cf(23,8))) # => [2, 1, 7]
println(collect(R2cf(13,11))) # => [1, 5, 2]
println(collect(R2cf(22,7))) # => [3, 7]
println(collect(R2cf(14142,10000))) # => [1, 2, 2, 2, 2, 2, 1, 1, 29]
println(collect(R2cf(141421,100000))) # => [1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 7, 2]
println(collect(R2cf(1414214,1000000))) # => [1, 2, 2, 2, 2, 2, 2, 2, 3, 6, 1, 2, 1, 12]
println(collect(R2cf(14142136,10000000))) # => [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1, 2, 4, 1, 1, 2]
 
println(collect(R2cf(13//11))) # => [1, 5, 2]
println(collect(R2cf(2 ^ 0.5), 20)) # => [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

Kotlin[edit]

// version 1.1.2
// compile with -Xcoroutines=enable flag from command line
 
import kotlin.coroutines.experimental.buildSequence
 
fun r2cf(frac: Pair<Int, Int>) =
buildSequence {
var num = frac.first
var den = frac.second
while (Math.abs(den) != 0) {
val div = num / den
val rem = num % den
num = den
den = rem
yield(div)
}
}
 
fun iterate(seq: Sequence<Int>) {
for (i in seq) print("$i ")
println()
}
 
fun main(args: Array<String>) {
val fracs = arrayOf(1 to 2, 3 to 1, 23 to 8, 13 to 11, 22 to 7, -151 to 77)
for (frac in fracs) {
print("${"%4d".format(frac.first)} / ${"%-2d".format(frac.second)} = ")
iterate(r2cf(frac))
}
val root2 = arrayOf(14142 to 10000, 141421 to 100000,
1414214 to 1000000, 14142136 to 10000000)
println("\nSqrt(2) ->")
for (frac in root2) {
print("${"%8d".format(frac.first)} / ${"%-8d".format(frac.second)} = ")
iterate(r2cf(frac))
}
val pi = arrayOf(31 to 10, 314 to 100, 3142 to 1000, 31428 to 10000,
314285 to 100000, 3142857 to 1000000,
31428571 to 10000000, 314285714 to 100000000)
println("\nPi ->")
for (frac in pi) {
print("${"%9d".format(frac.first)} / ${"%-9d".format(frac.second)} = ")
iterate(r2cf(frac))
}
}
Output:
   1 / 2  = 0 2 
   3 / 1  = 3 
  23 / 8  = 2 1 7 
  13 / 11 = 1 5 2 
  22 / 7  = 3 7 
-151 / 77 = -1 -1 -24 -1 -2 

Sqrt(2) ->
   14142 / 10000    = 1 2 2 2 2 2 1 1 29 
  141421 / 100000   = 1 2 2 2 2 2 2 3 1 1 3 1 7 2 
 1414214 / 1000000  = 1 2 2 2 2 2 2 2 3 6 1 2 1 12 
14142136 / 10000000 = 1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2 

Pi ->
       31 / 10        = 3 10 
      314 / 100       = 3 7 7 
     3142 / 1000      = 3 7 23 1 2 
    31428 / 10000     = 3 7 357 
   314285 / 100000    = 3 7 2857 
  3142857 / 1000000   = 3 7 142857 
 31428571 / 10000000  = 3 7 476190 3 
314285714 / 100000000 = 3 7 7142857 

Mathematica / Wolfram Language[edit]

Mathematica has a build-in function ContinuedFraction.

ContinuedFraction[1/2]
ContinuedFraction[3]
ContinuedFraction[23/8]
ContinuedFraction[13/11]
ContinuedFraction[22/7]
ContinuedFraction[-151/77]
ContinuedFraction[14142/10000]
ContinuedFraction[141421/100000]
ContinuedFraction[1414214/1000000]
ContinuedFraction[14142136/10000000]
Output:
{0, 2}
{3}
{2, 1, 7}
{1, 5, 2}
{3, 7}
{-1, -1, -24, -1, -2}
{1, 2, 2, 2, 2, 2, 1, 1, 29}
{1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 7, 2}
{1, 2, 2, 2, 2, 2, 2, 2, 3, 6, 1, 2, 1, 12}
{1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1, 2, 4, 1, 1, 2}

PARI/GP[edit]

apply(contfrac,[1/2,3,23/8,13/11,22/7,-151/77])
Output:
[[0, 2], [3], [2, 1, 7], [1, 5, 2], [3, 7], [-2, 25, 1, 2]]

Perl[edit]

To do output one digit at a time, we first turn off buffering to be pedantic, then use a closure that yields one term per call.

$|=1;
 
sub rc2f {
my($num, $den) = @_;
return sub { return unless $den;
my $q = int($num/$den);
($num, $den) = ($den, $num - $q*$den);
$q; };
}
 
sub rcshow {
my $sub = shift;
print "[";
my $n = $sub->();
print "$n" if defined $n;
print "; $n" while defined($n = $sub->());
print "]\n";
}
 
rcshow(rc2f(@$_))
for ([1,2],[3,1],[23,8],[13,11],[22,7],[-151,77]);
print "\n";
rcshow(rc2f(@$_))
for ([14142,10000],[141421,100000],[1414214,1000000],[14142136,10000000]);
print "\n";
rcshow(rc2f(314285714,100000000));
Output:
[0; 2]
[3]
[2; 1; 7]
[1; 5; 2]
[3; 7]
[-1; -1; -24; -1; -2]

[1; 2; 2; 2; 2; 2; 1; 1; 29]
[1; 2; 2; 2; 2; 2; 2; 3; 1; 1; 3; 1; 7; 2]
[1; 2; 2; 2; 2; 2; 2; 2; 3; 6; 1; 2; 1; 12]
[1; 2; 2; 2; 2; 2; 2; 2; 2; 2; 6; 1; 2; 4; 1; 1; 2]

[3; 7; 7142857]

Perl 6[edit]

Straightforward implementation:

sub r2cf(Rat $x is copy) {
gather loop {
$x -= take $x.floor;
last unless $x > 0;
$x = 1 / $x;
}
}
 
say r2cf(.Rat) for <1/2 3 23/8 13/11 22/7 1.41 1.4142136>;
Output:
(0 2)
(3)
(2 1 7)
(1 5 2)
(3 7)
(1 2 2 3 1 1 2)
(1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2)

As a silly one-liner:

sub r2cf(Rat $x is copy) { gather $x [R/]= 1 while ($x -= take $x.floor) > 0 }

Python[edit]

Translation of: Ruby
def r2cf(n1,n2):
while n2:
n1, (t1, n2) = n2, divmod(n1, n2)
yield t1
 
print(list(r2cf(1,2))) # => [0, 2]
print(list(r2cf(3,1))) # => [3]
print(list(r2cf(23,8))) # => [2, 1, 7]
print(list(r2cf(13,11))) # => [1, 5, 2]
print(list(r2cf(22,7))) # => [3, 7]
print(list(r2cf(14142,10000))) # => [1, 2, 2, 2, 2, 2, 1, 1, 29]
print(list(r2cf(141421,100000))) # => [1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 7, 2]
print(list(r2cf(1414214,1000000))) # => [1, 2, 2, 2, 2, 2, 2, 2, 3, 6, 1, 2, 1, 12]
print(list(r2cf(14142136,10000000))) # => [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1, 2, 4, 1, 1, 2]

This version generates it from any real number (with rationals as a special case):

def real2cf(x):
while True:
t1, f = divmod(x, 1)
yield int(t1)
if not f:
break
x = 1/f
 
from fractions import Fraction
from itertools import islice
 
print(list(real2cf(Fraction(13, 11)))) # => [1, 5, 2]
print(list(islice(real2cf(2 ** 0.5), 20))) # => [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

Racket[edit]

 
#lang racket
 
(define ((r2cf n d))
(or (zero? d)
(let-values ([(q r) (quotient/remainder n d)])
(set! n d)
(set! d r)
q)))
 
(define (r->cf n d)
(for/list ([i (in-producer (r2cf n d) #t)]) i))
 
(define (real->cf x places)
(define d (expt 10 places))
(define n (exact-floor (* x d)))
(r->cf n d))
 
(map r->cf
'(1 3 23 13 22 -151)
'(2 1 8 11 7 77))
(real->cf (sqrt 2) 10)
(real->cf pi 10)
 
Output:
'((0 2) (3) (2 1 7) (1 5 2) (3 7) (-1 -1 -24 -1 -2))
'(1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 1 3 8 9 1 20 1 2)
'(3 7 15 1 292 1 1 6 2 13 3 1 12 3)

REXX[edit]

Programming notes:

  •   Increasing   numeric digits   to a higher value will generate more terms.
  •   Two subroutines,   sqrt   and   pi,   were included here to demonstrate terms for   √ 2   and   pi.
  •   The subroutine   $maxfact   was included and is only needed if the number used for   r2cf   is a decimal fraction.
  •   Checks were included to verify that the arguments being passed to   r2cf   are indeed numeric and also not zero.
  •   This REXX version also handles negative numbers.
/*REXX program converts a  decimal  or  rational fraction  to a  continued fraction.    */
numeric digits 230 /*determines how many terms to be gened*/
say ' 1/2 ──► CF: ' r2cf( '1/2' )
say ' 3 ──► CF: ' r2cf( 3 )
say ' 23/8 ──► CF: ' r2cf( '23/8' )
say ' 13/11 ──► CF: ' r2cf( '13/11' )
say ' 22/7 ──► CF: ' r2cf( '22/7 ' )
say ' ___'
say '───────── attempts at √ 2.'
say '14142/1e4 ──► CF: ' r2cf( '14142/1e4 ' )
say '141421/1e5 ──► CF: ' r2cf( '141421/1e5 ' )
say '1414214/1e6 ──► CF: ' r2cf( '1414214/1e6 ' )
say '14142136/1e7 ──► CF: ' r2cf( '14142136/1e7 ' )
say '141421356/1e8 ──► CF: ' r2cf( '141421356/1e8 ' )
say '1414213562/1e9 ──► CF: ' r2cf( '1414213562/1e9 ' )
say '14142135624/1e10 ──► CF: ' r2cf( '14142135624/1e10 ' )
say '141421356237/1e11 ──► CF: ' r2cf( '141421356237/1e11 ' )
say '1414213562373/1e12 ──► CF: ' r2cf( '1414213562373/1e12 ' )
say '√2 ──► CF: ' r2cf( sqrt(2) )
say
say '───────── an attempt at pi'
say 'pi ──► CF: ' r2cf( pi() )
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
$maxFact: procedure; parse arg x 1 _x,y; y=10**(digits()-1); b=0; h=1; a=1; g=0
do while a<=y & g<=y; n=trunc(_x); _=a; a=n*a+b; b=_; _=g; g=n*g+h; h=_
if n=_x | a/g=x then do; if a>y | g>y then iterate; b=a; h=g; leave; end
_x=1/(_x-n); end; return b'/'h
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi: return 3.1415926535897932384626433832795028841971693993751058209749445923078164062862,
|| 089986280348253421170679821480865132823066470938446095505822317253594081284,
|| 811174502841027019385211055596446229489549303819644288109756659334461284756,
|| 48233786783165271 /* ··· should ≥ NUMERIC DIGITS */
/*──────────────────────────────────────────────────────────────────────────────────────*/
r2cf: procedure; parse arg g 1 s 2; $=; if s=='-' then g=substr(g, 2)
else s=
if pos(., g)\==0 then do; if \datatype(g, 'N') then call serr 'not numeric:' g
g=$maxfact(g)
end
if pos('/', g)==0 then g=g"/"1
parse var g n '/' d
if \datatype(n, 'W') then call serr "a numerator isn't an integer:" n
if \datatype(d, 'W') then call serr "a denominator isn't an integer:" d
if d=0 then call serr 'a denominator is zero'
n=abs(n) /*ensure numerator is positive. */
do while d\==0; _=d /*where the rubber meets the road*/
$=$ s || (n%d) /*append another number to list. */
d=n // d; n=_ /* % is int div, // is modulus.*/
end /*while*/
return strip($)
/*──────────────────────────────────────────────────────────────────────────────────────*/
serr: say; say '***error***'; say; say arg(1); say; exit 13
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); h=d+6; numeric form
m.=9; 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*/
numeric digits d; return g/1
output   when using the default (internal) inputs:
              1/2  ──► CF:  0 2
               3   ──► CF:  3
             23/8  ──► CF:  2 1 7
             13/11 ──► CF:  1 5 2
             22/7  ──► CF:  3 7
                       ___
───────── attempts at √ 2.
14142/1e4          ──► CF:  1 2 2 2 2 2 1 1 29
141421/1e5         ──► CF:  1 2 2 2 2 2 2 3 1 1 3 1 7 2
1414214/1e6        ──► CF:  1 2 2 2 2 2 2 2 3 6 1 2 1 12
14142136/1e7       ──► CF:  1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2
141421356/1e8      ──► CF:  1 2 2 2 2 2 2 2 2 2 2 3 4 1 1 2 6 8
1414213562/1e9     ──► CF:  1 2 2 2 2 2 2 2 2 2 2 2 1 1 14 1 238 1 3
14142135624/1e10   ──► CF:  1 2 2 2 2 2 2 2 2 2 2 2 2 2 5 4 1 8 4 2 1 4
141421356237/1e11  ──► CF:  1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 4 1 2 1 63 2 1 1 1 4 2
1414213562373/1e12 ──► CF:  1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 11 2 3 2 1 1 1 25 1 2 3
√2                 ──► CF:  1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

───────── an attempt at pi
pi                 ──► CF:  3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2 1 84 2 1 1 15 3 13 1 4 2 6 6 99 1 2 2 6 3 5 1 1 6 8 1 7 1 2 3 7 1 2 1 1 12 1 1 1 3 1 1 8 1 1 2 1 6 1 1 5 2 2 3 1 2 4 4 16 1 161 45 1 22 1 2 2 1 4 1 2 24 1 2 1 3 1 2 1 1 10 2 5 4 1 2 2 8 1 5 2 2 26 1 4 1 1 8 2 42 2 1 7 3 3 1 1 7 2 4 9 7 2 3 1 57 1 18 1 9 19 1 2 18 1 3 7 30 1 1 1 3 3 3 1 2 8 1 1 2 1 15 1 2 13 1 2 1 4 1 12 1 1 3 3 28 1 10 3 2 20 1 1 1 1 4 1 1 1 5 3 2 1 6 1 4 1 120 2 1 1 3 1 23 1 15 1 3 7 1 16 1 2 1 21 2 1 1 2 9 1 6 4

Ruby[edit]

# Generate a continued fraction from a rational number
 
def r2cf(n1,n2)
while n2 > 0
n1, (t1, n2) = n2, n1.divmod(n2)
yield t1
end
end

Testing[edit]

Test 1:

[[1,2], [3,1], [23,8], [13,11], [22,7], [-151,77]].each do |n1,n2|
print "%10s : " % "#{n1} / #{n2}"
r2cf(n1,n2) {|n| print "#{n} "}
puts
end
Output:
     1 / 2 : 0 2 
     3 / 1 : 3 
    23 / 8 : 2 1 7 
   13 / 11 : 1 5 2 
    22 / 7 : 3 7 
 -151 / 77 : -2 25 1 2 

Test 2:

(5..8).each do |digit|
n2 = 10 ** (digit-1)
n1 = (Math.sqrt(2) * n2).round
print "%-8s / %-8s : " % [n1, n2]
r2cf(n1,n2) {|n| print "#{n} "}
puts
end
Output:
14142    / 10000    : 1 2 2 2 2 2 1 1 29 
141421   / 100000   : 1 2 2 2 2 2 2 3 1 1 3 1 7 2 
1414214  / 1000000  : 1 2 2 2 2 2 2 2 3 6 1 2 1 12 
14142136 / 10000000 : 1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2 

Test 3:

a =[ [31,10],
[314,100],
[3142,1000],
[31428,10000],
[314285,100000],
[3142857,1000000],
[31428571,10000000],
[314285714,100000000]
]
a.each do |n1,n2|
print "%-9s / %-9s : " % [n1, n2]
r2cf(n1,n2) {|n| print "#{n} "}
puts
end
Output:
31        / 10        : 3 10 
314       / 100       : 3 7 7 
3142      / 1000      : 3 7 23 1 2 
31428     / 10000     : 3 7 357 
314285    / 100000    : 3 7 2857 
3142857   / 1000000   : 3 7 142857 
31428571  / 10000000  : 3 7 476190 3 
314285714 / 100000000 : 3 7 7142857 

Rust[edit]

 
struct R2cf {
n1: i64,
n2: i64
}
 
// This iterator generates the continued fraction representation from the
// specified rational number.
impl Iterator for R2cf {
type Item = i64;
 
fn next(&mut self) -> Option<i64> {
if self.n2 == 0 {
None
}
else {
let t1 = self.n1 / self.n2;
let t2 = self.n2;
self.n2 = self.n1 - t1 * t2;
self.n1 = t2;
Some(t1)
}
}
}
 
fn r2cf(n1: i64, n2: i64) -> R2cf {
R2cf { n1: n1, n2: n2 }
}
 
macro_rules! printcf {
($x:expr, $y:expr) => (println!("{:?}", r2cf($x, $y).collect::<Vec<_>>()));
}
 
fn main() {
printcf!(1, 2);
printcf!(3, 1);
printcf!(23, 8);
printcf!(13, 11);
printcf!(22, 7);
printcf!(-152, 77);
 
printcf!(14_142, 10_000);
printcf!(141_421, 100_000);
printcf!(1_414_214, 1_000_000);
printcf!(14_142_136, 10_000_000);
 
printcf!(31, 10);
printcf!(314, 100);
printcf!(3142, 1000);
printcf!(31_428, 10_000);
printcf!(314_285, 100_000);
printcf!(3_142_857, 1_000_000);
printcf!(31_428_571, 10_000_000);
printcf!(314_285_714, 100_000_000);
}
 
Output:
[0, 2]
[3]
[2, 1, 7]
[1, 5, 2]
[3, 7]
[-1, -1, -37, -2]
[1, 2, 2, 2, 2, 2, 1, 1, 29]
[1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 7, 2]
[1, 2, 2, 2, 2, 2, 2, 2, 3, 6, 1, 2, 1, 12]
[1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1, 2, 4, 1, 1, 2]
[3, 10]
[3, 7, 7]
[3, 7, 23, 1, 2]
[3, 7, 357]
[3, 7, 2857]
[3, 7, 142857]
[3, 7, 476190, 3]
[3, 7, 7142857]

Sidef[edit]

Translation of: Perl
func r2cf(num, den) {
func() {
den || return nil
var q = num//den
(num, den) = (den, num - q*den)
return q
}
}
 
func showcf(f) {
print "["
var n = f()
print "#{n}" if defined(n)
print "; #{n}" while defined(n = f())
print "]\n"
}
 
[
[1/2, 3/1, 23/8, 13/11, 22/7, -151/77],
[14142/10000, 141421/100000, 1414214/1000000, 14142136/10000000],
[314285714/100000000],
].each { |seq|
seq.each { |r| showcf(r2cf(r.nude)) }
print "\n"
}
Output:
[0; 2]
[3]
[2; 1; 7]
[1; 5; 2]
[3; 7]
[-1; -1; -24; -1; -2]

[1; 2; 2; 2; 2; 2; 1; 1; 29]
[1; 2; 2; 2; 2; 2; 2; 3; 1; 1; 3; 1; 7; 2]
[1; 2; 2; 2; 2; 2; 2; 2; 3; 6; 1; 2; 1; 12]
[1; 2; 2; 2; 2; 2; 2; 2; 2; 2; 6; 1; 2; 4; 1; 1; 2]

[3; 7; 7142857]

Tcl[edit]

Works with: Tcl version 8.6
Translation of: Ruby

Direct translation[edit]

package require Tcl 8.6
 
proc r2cf {n1 {n2 1}} {
# Convert a decimal fraction (e.g., 1.23) into a form we can handle
if {$n1 != int($n1) && [regexp {\.(\d+)} $n1 -> suffix]} {
set pow [string length $suffix]
set n1 [expr {int($n1 * 10**$pow)}]
set n2 [expr {$n2 * 10**$pow}]
}
# Construct the continued fraction as a coroutine that yields the digits in sequence
coroutine cf\#[incr ::cfcounter] apply {{n1 n2} {
yield [info coroutine]
while {$n2 > 0} {
yield [expr {$n1 / $n2}]
set n2 [expr {$n1 % [set n1 $n2]}]
}
return -code break
}} $n1 $n2
}

Demonstrating:

proc printcf {name cf} {
puts -nonewline "$name -> "
while 1 {
puts -nonewline "[$cf],"
}
puts "\b "
}
 
foreach {n1 n2} {
1 2
3 1
23 8
13 11
22 7
-151 77
14142 10000
141421 100000
1414214 1000000
14142136 10000000
31 10
314 100
3142 1000
31428 10000
314285 100000
3142857 1000000
31428571 10000000
314285714 100000000
3141592653589793 1000000000000000
} {
printcf "\[$n1;$n2\]" [r2cf $n1 $n2]
}
Output:
[1;2] -> 0,2 
[3;1] -> 3 
[23;8] -> 2,1,7 
[13;11] -> 1,5,2 
[22;7] -> 3,7 
[-151;77] -> -2,25,1,2 
[14142;10000] -> 1,2,2,2,2,2,1,1,29 
[141421;100000] -> 1,2,2,2,2,2,2,3,1,1,3,1,7,2 
[1414214;1000000] -> 1,2,2,2,2,2,2,2,3,6,1,2,1,12 
[14142136;10000000] -> 1,2,2,2,2,2,2,2,2,2,6,1,2,4,1,1,2 
[31;10] -> 3,10 
[314;100] -> 3,7,7 
[3142;1000] -> 3,7,23,1,2 
[31428;10000] -> 3,7,357 
[314285;100000] -> 3,7,2857 
[3142857;1000000] -> 3,7,142857 
[31428571;10000000] -> 3,7,476190,3 
[314285714;100000000] -> 3,7,7142857 
[3141592653589793;1000000000000000] -> 3,7,15,1,292,1,1,1,2,1,3,1,14,4,2,3,1,12,5,1,5,20,1,11,1,1,1,2 

Objectified version[edit]

package require Tcl 8.6
 
# General generator class based on coroutines
oo::class create Generator {
constructor {} {
coroutine [namespace current]::coro my Apply
}
destructor {
catch {rename [namespace current]::coro {}}
}
method Apply {} {
yield
# Call the method (defined in subclasses) that actually produces values
my Produce
my destroy
return -code break
}
forward generate coro
method unknown args {
if {![llength $args]} {
tailcall coro
}
next {*}$args
}
 
# Various ways to get the sequence from the generator
method collect {} {
set result {}
while 1 {
lappend result [my generate]
}
return $result
}
method take {n {suffix ""}} {
set result {}
for {set i 0} {$i < $n} {incr i} {
lappend result [my generate]
}
while {$suffix ne ""} {
my generate
lappend result $suffix
break
}
return $result
}
}
 
oo::class create R2CF {
superclass Generator
variable a b
# The constructor converts other kinds of fraction (e.g., 1.23, 22/7) into a
# form we can handle.
constructor {n1 {n2 1}} {
next; # Delegate to superclass for coroutine management
if {[regexp {(.*)/(.*)} $n1 -> a b]} {
# Nothing more to do; assume we can ignore second argument here
} elseif {$n1 != int($n1) && [regexp {\.(\d+)} $n1 -> suffix]} {
set pow [string length $suffix]
set a [expr {int($n1 * 10**$pow)}]
set b [expr {$n2 * 10**$pow}]
} else {
set a $n1
set b $n2
}
}
# How to actually produce the values of the sequence
method Produce {} {
while {$b > 0} {
yield [expr {$a / $b}]
set b [expr {$a % [set a $b]}]
}
}
}
 
proc printcf {name cf {take ""}} {
if {$take ne ""} {
set terms [$cf take $take \u2026]
} else {
set terms [$cf collect]
}
puts [format "%-15s-> %s" $name [join $terms ,]]
}
 
foreach {n1 n2} {
1 2
3 1
23 8
13 11
22 7
-151 77
14142 10000
141421 100000
1414214 1000000
14142136 10000000
31 10
314 100
3142 1000
31428 10000
314285 100000
3142857 1000000
31428571 10000000
314285714 100000000
3141592653589793 1000000000000000
} {
printcf "\[$n1;$n2\]" [R2CF new $n1 $n2]
}
# Demonstrate parsing of input in forms other than a direct pair of decimals
printcf "1.5" [R2CF new 1.5]
printcf "23/7" [R2CF new 23/7]
Output:
[1;2]          -> 0,2
[3;1]          -> 3
[23;8]         -> 2,1,7
[13;11]        -> 1,5,2
[22;7]         -> 3,7
[-151;77]      -> -2,25,1,2
[14142;10000]  -> 1,2,2,2,2,2,1,1,29
[141421;100000]-> 1,2,2,2,2,2,2,3,1,1,3,1,7,2
[1414214;1000000]-> 1,2,2,2,2,2,2,2,3,6,1,2,1,12
[14142136;10000000]-> 1,2,2,2,2,2,2,2,2,2,6,1,2,4,1,1,2
[31;10]        -> 3,10
[314;100]      -> 3,7,7
[3142;1000]    -> 3,7,23,1,2
[31428;10000]  -> 3,7,357
[314285;100000]-> 3,7,2857
[3142857;1000000]-> 3,7,142857
[31428571;10000000]-> 3,7,476190,3
[314285714;100000000]-> 3,7,7142857
[3141592653589793;1000000000000000]-> 3,7,15,1,292,1,1,1,2,1,3,1,14,4,2,3,1,12,5,1,5,20,1,11,1,1,1,2
1.5            -> 1,2
23/7           -> 3,3,2

XPL0[edit]

include c:\cxpl\codes;
real Val;
 
proc R2CF(N1, N2, Lev); \Output continued fraction for N1/N2
int N1, N2, Lev;
int Quot, Rem;
[if Lev=0 then Val:= 0.0;
Quot:= N1/N2;
Rem:= rem(0);
IntOut(0, Quot);
if Rem then [ChOut(0, if Lev then ^, else ^;); R2CF(N2, Rem, Lev+1)];
Val:= Val + float(Quot); \generate value from continued fraction
if Lev then Val:= 1.0/Val;
];
 
int I, Data;
[Data:= [1,2, 3,1, 23,8, 13,11, 22,7, 0];
Format(0, 15);
I:= 0;
while Data(I) do
[IntOut(0, Data(I)); ChOut(0, ^/); IntOut(0, Data(I+1)); ChOut(0, 9\tab\);
ChOut(0, ^[); R2CF(Data(I), Data(I+1), 0); ChOut(0, ^]); ChOut(0, 9\tab\);
RlOut(0, Val); CrLf(0);
I:= I+2];
]
Output:
1/2     [0;2]    5.000000000000000E-001
3/1     [3]      3.000000000000000E+000
23/8    [2;1,7]  2.875000000000000E+000
13/11   [1;5,2]  1.181818181818180E+000
22/7    [3;7]    3.142857142857140E+000

zkl[edit]

Two iterators; one light weight, one heavy weight.

Light weight, explicit state:

fcn r2cf(nom,dnom){ // -->Walker (iterator)
Walker.tweak(fcn(state){
nom,dnom:=state;
if(dnom==0) return(Void.Stop);
n,d:=nom.divr(dnom);
state.clear(dnom,d);
n
}.fp(List(nom,dnom))) // partial application (light weight closure)
}

Heavy weight, implicit state:

fcn r2cf2(nom,dnom){ // -->Generator (heavy weight Walker)
Utils.Generator(fcn(nom,dnom){
while(dnom){
n,d:=nom.divr(dnom); nom,dnom=dnom,d;
vm.yield(n);
}
Void.Stop;
},nom,dnom)
}

Both of the above return an iterator so they function the same:

foreach nom,dnom in (T(T(1,2), T(3,1), T(23,8), T(13,11), T(22,7), 
T(14142,10000), T(141421,100000), T(1414214,1000000),
T(14142136,10000000))){
r2cf(nom,dnom).walk(25).println(); // print up to 25 numbers
}
Output:
L(0,2)
L(3)
L(2,1,7)
L(1,5,2)
L(3,7)
L(1,2,2,2,2,2,1,1,29)
L(1,2,2,2,2,2,2,3,1,1,3,1,7,2)
L(1,2,2,2,2,2,2,2,3,6,1,2,1,12)
L(1,2,2,2,2,2,2,2,2,2,6,1,2,4,1,1,2)