Continued fraction/Arithmetic/Construct from rational number

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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

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.

<lang C> /*Abhishek Ghosh, 8th November 2013, Rotterdam*/

  1. 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; }

</lang> 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++

<lang cpp>#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;} };</lang>

Testing

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

<lang cpp>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; }</lang>

Output:
0 2
3
2 1 7
1 5 2
3 7
-1 -1 -24 -1 -2

<lang cpp>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; }</lang>

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

<lang cpp>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;

}</lang>

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#

<lang csharp>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));
   }

} </lang> 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

<lang clojure>(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))</lang>
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

<lang lisp>(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)))</lang>

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)

D

Translation of: Kotlin

<lang D>import std.concurrency; import std.stdio;

struct Pair { 

   int first, second;

}

auto r2cf(Pair frac) { 

   return new Generator!int({
       auto num = frac.first;
       auto den = frac.second;
       while (den != 0) {
           auto div = num / den;
           auto rem = num % den;
           num = den;
           den = rem;
           div.yield();
       }
   });

}

void iterate(Generator!int seq) { 

   foreach(i; seq) {
       write(i, " ");
   }
   writeln();

}

void main() { 

   auto fracs = [
       Pair(   1,  2),
       Pair(   3,  1),
       Pair(  23,  8),
       Pair(  13, 11),
       Pair(  22,  7),
       Pair(-151, 77),
   ];
   foreach(frac; fracs) {
       writef("%4d / %-2d = ", frac.first, frac.second);
       frac.r2cf.iterate;
   }
   writeln;

   auto root2 = [
       Pair(    14_142,     10_000),
       Pair(   141_421,    100_000),
       Pair( 1_414_214,  1_000_000),
       Pair(14_142_136, 10_000_000),
   ];
   writeln("Sqrt(2) ->");
   foreach(frac; root2) {
       writef("%8d / %-8d = ", frac.first, frac.second);
       frac.r2cf.iterate;
   }
   writeln;

   auto pi = [
       Pair(         31,          10),
       Pair(        314,         100),
       Pair(      3_142,       1_000),
       Pair(     31_428,      10_000),
       Pair(    314_285,     100_000),
       Pair(  3_142_857,   1_000_000),
       Pair( 31_428_571,  10_000_000),
       Pair(314_285_714, 100_000_000),
   ];
   writeln("Pi ->");
   foreach(frac; pi) {
       writef("%9d / %-9d = ", frac.first, frac.second);
       frac.r2cf.iterate;
   }

}</lang>

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

F#

<lang fsharp>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</lang>

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

Translation of: Python

This more general version generates a continued fraction from any real number (with rationals as a special case): <lang haskell>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)
   ]</lang>
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

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

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

Examples

<lang j> cf each 1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77 ┌───┬─┬─────┬─────┬───┬─────────────────────────────────┬─────────┐ │0 2│3│2 1 7│1 5 2│3 7│1 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 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│ └──────────────────┴───────────────────────────┴────────────────────────────┴─────────────────────────────────┘ 

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

┌────┬─────┬──────────┬───────┬────────┬──────────┬────────────┬───────────┐ │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│ └────┴─────┴──────────┴───────┴────────┴──────────┴────────────┴───────────┘</lang> 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, <lang j> 3 7 =&((+ %)/) 3 7 _ 1</lang> Incidentally, J and Tcl report a different representation for -151/77 versus the representation of some other implementations; however, both representations produce equal results. <lang j> _2 25 1 2 =&((+ %)/) _1 _1 _24 _1 _2 1</lang>

Tacit version 2

Translation of python

<lang J>r2cf=:1 1{."1@}.({:,(0,{:)#:{.)^:(*@{:)^:a:</lang>

Example use:

<lang J> ((":@{.,'/',":@{:),': ',":@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 </lang>

Explicit versions

version 1

Implemented as a class, r2cf preserves state in a separate locale. I've used some contrivances to jam the examples onto one line. <lang J> coclass'cf' create =: dyad def 'EMPTY [ N =: x , y' destroy =: codestroy r2cf =: monad define 

if. 0 (= {:) N do. _ return. end.
RV =. <.@:(%/) N
N =: ({. , |/)@:|. N
RV

)

cocurrent'base' CF =: conew'cf'

Until =: conjunction def 'u^:(-.@:v)^:_'

(,. }.@:}:@:((,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 │ └─────────────────┴─────────────────────────────────┘ )</lang>

version 2

<lang J> 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 2│3│2 1 7│1 5 2│3 7│1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2 _│_2 25 1 2│ └───┴─┴─────┴─────┴───┴───────────────────────────────────┴─────────┘</lang>

version 3

translation of python:

<lang J>r2cf=:3 :0 

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

)</lang>

Example:

<lang J> r2cf each 1 2;3 1;23 8;13 11;22 7;14142136 10000000;_151 77 ┌───┬─┬─────┬─────┬───┬─────────────────────────────────┬─────────┐ │0 2│3│2 1 7│1 5 2│3 7│1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2│_2 25 1 2│ └───┴─┴─────┴─────┴───┴─────────────────────────────────┴─────────┘</lang>

Java

Translation of: Kotlin
Works with: Java version 9

<lang Java>import java.util.Iterator; import java.util.List; import java.util.Map;

public class ConstructFromRationalNumber { 

   private static class R2cf implements Iterator<Integer> {
       private int num;
       private int den;

       R2cf(int num, int den) {
           this.num = num;
           this.den = den;
       }

       @Override
       public boolean hasNext() {
           return den != 0;
       }

       @Override
       public Integer next() {
           int div = num / den;
           int rem = num % den;
           num = den;
           den = rem;
           return div;
       }
   }

   private static void iterate(R2cf generator) {
       generator.forEachRemaining(n -> System.out.printf("%d ", n));
       System.out.println();
   }

   public static void main(String[] args) {
       List<Map.Entry<Integer, Integer>> fracs = List.of(
               Map.entry(1, 2),
               Map.entry(3, 1),
               Map.entry(23, 8),
               Map.entry(13, 11),
               Map.entry(22, 7),
               Map.entry(-151, 77)
       );
       for (Map.Entry<Integer, Integer> frac : fracs) {
           System.out.printf("%4d / %-2d = ", frac.getKey(), frac.getValue());
           iterate(new R2cf(frac.getKey(), frac.getValue()));
       }

       System.out.println("\nSqrt(2) ->");
       List<Map.Entry<Integer, Integer>> root2 = List.of(
               Map.entry(    14_142,     10_000),
               Map.entry(   141_421,    100_000),
               Map.entry( 1_414_214,  1_000_000),
               Map.entry(14_142_136, 10_000_000)
       );
       for (Map.Entry<Integer, Integer> frac : root2) {
           System.out.printf("%8d / %-8d = ", frac.getKey(), frac.getValue());
           iterate(new R2cf(frac.getKey(), frac.getValue()));
       }

       System.out.println("\nPi ->");
       List<Map.Entry<Integer, Integer>> pi = List.of(
               Map.entry(         31,        10),
               Map.entry(        314,       100),
               Map.entry(      3_142,      1_000),
               Map.entry(     31_428,     10_000),
               Map.entry(    314_285,    100_000),
               Map.entry(  3_142_857,   1_000_000),
               Map.entry( 31_428_571,  10_000_000),
               Map.entry(314_285_714, 100_000_000)
       );
       for (Map.Entry<Integer, Integer> frac : pi) {
           System.out.printf("%9d / %-9d = ", frac.getKey(), frac.getValue());
           iterate(new R2cf(frac.getKey(), frac.getValue()));
       }
   }

}</lang>

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

Julia

Works with: Julia version 0.6

<lang julia># It'st most appropriate to define a Julia iterable object for this task

  1. Julia doesn't have Python'st yield, the closest to it is produce/consume calls with Julia tasks
  2. but for various reasons they don't work out for this task
  3. This solution works with two integers, a Julia rational or a real

mutable struct ContinuedFraction{T<:Integer} 

   n1::T # numerator or real
   n2::T # denominator or 1 if real
   t1::T # generated coefficient

end

  1. Constructors for all possible input types

ContinuedFraction{T<:Integer}(n1::T, n2::T) = ContinuedFraction(n1, n2, 0) ContinuedFraction(n::Rational) = ContinuedFraction(numerator(n), denominator(n)) ContinuedFraction(n::AbstractFloat) = ContinuedFraction(Rational(n))

  1. Methods to make our object iterable

Base.start(::ContinuedFraction) = nothing

  1. Returns true if we've prepared the continued fraction

Base.done(cf::ContinuedFraction, st) = cf.n2 == 0

  1. Generates the next coefficient

function Base.next(cf::ContinuedFraction, st) 

   cf.n1, (cf.t1, cf.n2) = cf.n2, divrem(cf.n1, cf.n2)
   return cf.t1, nothing

end

  1. Tell Julia that this object always returns ints (all coeffs are integers)

Base.eltype{T}(::Type{ContinuedFraction{T}}) = T

  1. Overload the default collect function so that we can collect the first maxiter coeffs of infinite continued fractions
  2. array slicing doesn't work as Julia crashes before the slicing due to our infinitely long array

function Base.collect(itr::ContinuedFraction, maxiter::Integer = 100) 

   r = Array{eltype(itr)}(maxiter)
   i = 1
   for v in itr
       r[i] = v
       i += 1
       if i > maxiter break end
   end
   return r[1:i-1]

end

  1. Test cases according to task description with outputs in comments

println(collect(ContinuedFraction(1, 2))) # => [0, 2] println(collect(ContinuedFraction(3, 1))) # => [3] println(collect(ContinuedFraction(23, 8))) # => [2, 1, 7] println(collect(ContinuedFraction(13, 11))) # => [1, 5, 2] println(collect(ContinuedFraction(22, 7))) # => [3, 7] println(collect(ContinuedFraction(14142, 10000))) # => [1, 2, 2, 2, 2, 2, 1, 1, 29] println(collect(ContinuedFraction(141421, 100000))) # => [1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 7, 2] println(collect(ContinuedFraction(1414214, 1000000))) # => [1, 2, 2, 2, 2, 2, 2, 2, 3, 6, 1, 2, 1, 12] println(collect(ContinuedFraction(14142136, 10000000))) # => [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1, 2, 4, 1, 1, 2]

println(collect(ContinuedFraction(13 // 11))) # => [1, 5, 2] println(collect(ContinuedFraction(√2), 20)) # => [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]</lang>

Kotlin

<lang scala>// 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))
   }

}</lang>

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

Mathematica has a build-in function ContinuedFraction. <lang mathematica>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]</lang>

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

<lang parigp>apply(contfrac,[1/2,3,23/8,13/11,22/7,-151/77])</lang>

Output:
[[0, 2], [3], [2, 1, 7], [1, 5, 2], [3, 7], [-2, 25, 1, 2]]

Perl

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. <lang perl>$|=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));</lang>

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

Straightforward implementation: <lang perl6>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>;</lang>

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: <lang perl6>sub r2cf(Rat $x is copy) { gather $x [R/]= 1 while ($x -= take $x.floor) > 0 }</lang>

Python

Translation of: Ruby

<lang python>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]</lang> This version generates it from any real number (with rationals as a special case): <lang python>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]</lang>

Racket

<lang racket>

  1. 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) </lang>

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

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.

<lang rexx>/*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</lang>
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

<lang ruby># 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</lang>

Testing

Test 1: <lang ruby>[[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</lang>

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: <lang ruby>(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</lang>

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: <lang ruby>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</lang>

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

<lang rust> 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);

} </lang>

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

Translation of: Perl

<lang ruby>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"

}</lang>

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

Works with: Tcl version 8.6
Translation of: Ruby

Direct translation

<lang tcl>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

}</lang> Demonstrating: <lang tcl>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]

}</lang>

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

<lang tcl>package require Tcl 8.6

  1. 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]

}

  1. 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]</lang>

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

<lang XPL0>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];

]</lang>

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

Two iterators; one light weight, one heavy weight.

Light weight, explicit state: <lang zkl>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)

}</lang> Heavy weight, implicit state: <lang zkl>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)

}</lang> Both of the above return an iterator so they function the same: <lang zkl>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

}</lang>

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)
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