# Continued fraction/Arithmetic/Construct from rational number

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.

The purpose of this task is to write a function ${\displaystyle {\mathit {r2cf}}(\mathrm {int} }$ ${\displaystyle N_{1},\mathrm {int} }$ ${\displaystyle N_{2})}$, or ${\displaystyle {\mathit {r2cf}}(\mathrm {Fraction} }$ ${\displaystyle N)}$, which will output a continued fraction assuming:

${\displaystyle N_{1}}$ is the numerator
${\displaystyle N_{2}}$ 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 ${\displaystyle N_{1}}$ divided by ${\displaystyle N_{2}}$. It then sets ${\displaystyle N_{1}}$ to ${\displaystyle N_{2}}$ and ${\displaystyle N_{2}}$ to the determined remainder part. It then outputs the determined integer part. It does this until ${\displaystyle \mathrm {abs} (N_{2})}$ is zero.

Demonstrate the function by outputing the continued fraction for:

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

${\displaystyle {\sqrt {2}}}$ should approach ${\displaystyle [1;2,2,2,2,\ldots ]}$ 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 ${\displaystyle {\sqrt {2}}}$ and convince yourself that, in the same way as ${\displaystyle 3.7}$ may be represented as ${\displaystyle 3.70}$ when an extra decimal place is required, ${\displaystyle [3;7]}$ may be represented as ${\displaystyle [3;7,\infty ]}$ 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.

 #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++

#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

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

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


#### ${\displaystyle {\sqrt {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

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#

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

(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

(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)


## D

Translation of: Kotlin
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;    }}
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#

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]

## Go

(Note, the files making up this package are re-used as presented here for for the Continued_fraction/Arithmetic/G(matrix_NG,_Contined_Fraction_N)#Go and Continued_fraction/Arithmetic/G(matrix_NG,_Contined_Fraction_N1,_Contined_Fraction_N2)#Go tasks.)

File cf.go:

package cf import (	"fmt"	"strings") // ContinuedFraction is a regular continued fraction.type ContinuedFraction func() NextFn // NextFn is a function/closure that can return// a posibly infinite sequence of values.type NextFn func() (term int64, ok bool) // String implements fmt.Stringer.// It formats a maximum of 20 values, ending the// sequence with ", ..." if the sequence is longer.func (cf ContinuedFraction) String() string {	var buf strings.Builder	buf.WriteByte('[')	sep := "; "	const maxTerms = 20	next := cf()	for n := 0; ; n++ {		t, ok := next()		if !ok {			break		}		if n > 0 {			buf.WriteString(sep)			sep = ", "		}		if n >= maxTerms {			buf.WriteString("...")			break		}		fmt.Fprint(&buf, t)	}	buf.WriteByte(']')	return buf.String()} // Sqrt2 is the continued fraction for √2, [1; 2, 2, 2, ...].func Sqrt2() NextFn {	first := true	return func() (int64, bool) {		if first {			first = false			return 1, true		}		return 2, true	}} // Phi is the continued fraction for ϕ, [1; 1, 1, 1, ...].func Phi() NextFn {	return func() (int64, bool) { return 1, true }} // E is the continued fraction for e,// [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, ...].func E() NextFn {	var i int	return func() (int64, bool) {		i++		switch {		case i == 1:			return 2, true		case i%3 == 0:			return int64(i/3) * 2, true		default:			return 1, true		}	}}

File rat.go:

package cf import "fmt" // A Rat represents a quotient N/D.type Rat struct {	N, D int64} // String implements fmt.Stringer and returns a string// representation of r in the form "N/D" (even if D == 1).func (r Rat) String() string {	return fmt.Sprintf("%d/%d", r.N, r.D)} // As ContinuedFraction returns a contined fraction representation of r.func (r Rat) AsContinuedFraction() ContinuedFraction { return r.CFTerms }func (r Rat) CFTerms() NextFn {	return func() (int64, bool) {		if r.D == 0 {			return 0, false		}		q := r.N / r.D		r.N, r.D = r.D, r.N-q*r.D		return q, true	}} // Rosetta Code task explicitly asked for this function,// so here it is. We'll just use the types above instead.func r2cf(n1, n2 int64) ContinuedFraction { return Rat{n1, n2}.CFTerms }

File rat_test.go:

package cf import (	"fmt"	"math") func Example_ConstructFromRational() {	cases := [...]Rat{		{1, 2},		{3, 1},		{23, 8},		{13, 11},		{22, 7},		{-151, 77},	}	for _, r := range cases {		fmt.Printf("%7s = %s\n", r, r.AsContinuedFraction())	} 	for _, tc := range [...]struct {		name   string		approx float64		cf     ContinuedFraction		d1, d2 int64	}{		{"√2", math.Sqrt2, Sqrt2, 1e4, 1e8},		{"π", math.Pi, nil, 10, 1e10},		{"ϕ", math.Phi, Phi, 10, 1e5},		{"e", math.E, E, 1e5, 1e9},	} {		fmt.Printf("\nApproximating %s ≅ %v:\n", tc.name, tc.approx)		for d := tc.d1; d < tc.d2; d *= 10 {			n := int64(math.Round(tc.approx * float64(d)))			r := Rat{n, d}			fmt.Println(r, "=", r.AsContinuedFraction())		}		if tc.cf != nil {			wid := int(math.Log10(float64(tc.d2)))*2 + 2 // ick			fmt.Printf("%*s: %v\n", wid, "Actual", tc.cf)		}	} 	// Output:	// [… commented output used by go test omitted for	//    Rosetta Code listing; it is the same as below …]}
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]

Approximating √2 ≅ 1.4142135623730951:
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]
Actual: [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]

Approximating π ≅ 3.141592653589793:
31/10 = [3; 10]
314/100 = [3; 7, 7]
3142/1000 = [3; 7, 23, 1, 2]
31416/10000 = [3; 7, 16, 11]
314159/100000 = [3; 7, 15, 1, 25, 1, 7, 4]
3141593/1000000 = [3; 7, 16, 983, 4, 2]
31415927/10000000 = [3; 7, 15, 1, 354, 2, 6, 1, 4, 1, 2]
314159265/100000000 = [3; 7, 15, 1, 288, 1, 2, 1, 3, 1, 7, 4]
3141592654/1000000000 = [3; 7, 15, 1, 293, 11, 1, 1, 7, 2, 1, 3, 3, 2]

Approximating ϕ ≅ 1.618033988749895:
16/10 = [1; 1, 1, 2]
162/100 = [1; 1, 1, 1, 1, 1, 2, 2]
1618/1000 = [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5]
16180/10000 = [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5]
Actual: [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...]

Approximating e ≅ 2.718281828459045:
271828/100000 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 10, 1, 1, 2]
2718282/1000000 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 3, 141]
27182818/10000000 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 11, 1, 2, 10, 6, 2]
271828183/100000000 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 2, 1, 1, 17, 6, 1, 1, 1, ...]
Actual: [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, ...]


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 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. cf=: _1 1 ,@}. (, <.)@%@-/ ::]^:a:@(, <.)@(%&x:/) #### Examples  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│└────┴─────┴──────────┴───────┴────────┴──────────┴────────────┴───────────┘ 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 _21 ### Tacit version 2 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 1 231/10: 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 #### 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.  coclass'cf'create =: dyad def 'EMPTY [ N =: x , y'destroy =: codestroyr2cf =: 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 77Note '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  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│└───┴─┴─────┴─────┴───┴───────────────────────────────────┴─────────┘ #### version 3 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 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│└───┴─┴─────┴─────┴───┴─────────────────────────────────┴─────────┘ ## Java Translation of: Kotlin Works with: Java version 9 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())); } }} 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 # It'st most appropriate to define a Julia iterable object for this task# Julia doesn't have Python'st 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 mutable struct ContinuedFraction{T<:Integer} n1::T # numerator or real n2::T # denominator or 1 if real t1::T # generated coefficientend # Constructors for all possible input typesContinuedFraction{T<:Integer}(n1::T, n2::T) = ContinuedFraction(n1, n2, 0)ContinuedFraction(n::Rational) = ContinuedFraction(numerator(n), denominator(n))ContinuedFraction(n::AbstractFloat) = ContinuedFraction(Rational(n)) # Methods to make our object iterableBase.start(::ContinuedFraction) = nothing# Returns true if we've prepared the continued fractionBase.done(cf::ContinuedFraction, st) = cf.n2 == 0# Generates the next coefficientfunction Base.next(cf::ContinuedFraction, st) cf.n1, (cf.t1, cf.n2) = cf.n2, divrem(cf.n1, cf.n2) return cf.t1, nothingend # Tell Julia that this object always returns ints (all coeffs are integers)Base.eltype{T}(::Type{ContinuedFraction{T}}) = T # 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 arrayfunction 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 # Test cases according to task description with outputs in commentsprintln(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] ## Kotlin // 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

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}

## Modula-2

MODULE ConstructFromrationalNumber;FROM FormatString IMPORT FormatString;FROM Terminal IMPORT WriteString,WriteLn,ReadChar; TYPE R2cf = RECORD    num,den : INTEGER;END; PROCEDURE HasNext(self : R2cf) : BOOLEAN;BEGIN    RETURN self.den # 0;END HasNext; PROCEDURE Next(VAR self : R2cf) : INTEGER;VAR div,rem : INTEGER;BEGIN    div := self.num / self.den;    rem := self.num REM self.den;    self.num := self.den;    self.den := rem;    RETURN div;END Next; PROCEDURE Iterate(self : R2cf);VAR buf : ARRAY[0..64] OF CHAR;BEGIN    WHILE HasNext(self) DO        FormatString("%i ", buf, Next(self));        WriteString(buf);    END;    WriteLn;END Iterate; PROCEDURE Print(num,den : INTEGER);VAR frac : R2cf;VAR buf : ARRAY[0..64] OF CHAR;BEGIN    FormatString("%9i / %-9i = ", buf, num, den);    WriteString(buf);     frac.num := num;    frac.den := den;    Iterate(frac);END Print; VAR frac : R2cf;BEGIN    Print(1,2);    Print(3,1);    Print(23,8);    Print(13,11);    Print(22,7);    Print(-151,77);     WriteLn;    WriteString("Sqrt(2) ->");    WriteLn;    Print(14142,10000);    Print(141421,100000);    Print(1414214,1000000);    Print(14142136,10000000);     WriteLn;    WriteString("Pi ->");    WriteLn;    Print(31,10);    Print(314,100);    Print(3142,1000);    Print(31428,10000);    Print(314285,100000);    Print(3142857,1000000);    Print(31428571,10000000);    Print(314285714,100000000);     ReadChar;END ConstructFromrationalNumber.

## PARI/GP

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

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

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 }

## Phix

Translation of: C
function r2cf(sequence fraction)    integer {numerator, denominator} = fraction    integer quotient = 0    if denominator!=0 then        quotient = floor(numerator/denominator)        {numerator,denominator} = {denominator,mod(numerator,denominator)}    end if    return {quotient,{numerator,denominator}}end function constant DENOMINATOR = 2procedure test(string txt, sequence tests)sequence fractioninteger quotient    printf(1,"Running %s :",{txt})    for i=1 to length(tests) do        fraction = tests[i]        printf(1,"\nFor N = %d, D = %d :",fraction)        while fraction[DENOMINATOR]!=0 do            {quotient,fraction} = r2cf(fraction)            printf(1," %d ",quotient)        end while    end for    printf(1,"\n\n")end procedure constant examples = {{1,2}, {3,1}, {23,8}, {13,11}, {22,7}, {-151,77}},         sqrt2 = {{14142,10000}, {141421,100000}, {1414214,1000000}, {14142136,10000000}},         pi = {{31,10}, {314,100}, {3142,1000}, {31428,10000}, {314285,100000}, {3142857,1000000}, {31428571,10000000}, {314285714,100000000}} test("the examples",examples)test("for sqrt(2)",sqrt2)test("for pi",pi)
Output:
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 : -2  25  1  2

Running for sqrt(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 pi :
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


## Python

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 Fractionfrom 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

 #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

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   $maxfact was included and is only needed if the number used for r2cf is a decimal fraction. • Checks were included to verify that the arguments being passed to r2cf are indeed numeric and also not zero. • This REXX version also handles negative numbers. /*REXX program converts a decimal or rational fraction to a continued fraction. */numeric digits 230 /*determines how many terms to be gened*/say ' 1/2 ──► CF: ' r2cf( '1/2' )say ' 3 ──► CF: ' r2cf( 3 )say ' 23/8 ──► CF: ' r2cf( '23/8' )say ' 13/11 ──► CF: ' r2cf( '13/11' )say ' 22/7 ──► CF: ' r2cf( '22/7 ' )say ' ___'say '───────── attempts at √ 2.'say '14142/1e4 ──► CF: ' r2cf( '14142/1e4 ' )say '141421/1e5 ──► CF: ' r2cf( '141421/1e5 ' )say '1414214/1e6 ──► CF: ' r2cf( '1414214/1e6 ' )say '14142136/1e7 ──► CF: ' r2cf( '14142136/1e7 ' )say '141421356/1e8 ──► CF: ' r2cf( '141421356/1e8 ' )say '1414213562/1e9 ──► CF: ' r2cf( '1414213562/1e9 ' )say '14142135624/1e10 ──► CF: ' r2cf( '14142135624/1e10 ' )say '141421356237/1e11 ──► CF: ' r2cf( '141421356237/1e11 ' )say '1414213562373/1e12 ──► CF: ' r2cf( '1414213562373/1e12 ' )say '√2 ──► CF: ' r2cf( sqrt(2) )saysay '───────── an attempt at pi'say 'pi ──► CF: ' r2cf( pi() )exit /*stick a fork in it, we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/$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 # Generate a continued fraction from a rational number def r2cf(n1,n2) while n2 > 0 n1, (t1, n2) = n2, n1.divmod(n2) yield t1 endend ### Testing 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} "} putsend 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: ${\displaystyle {\sqrt {2}}}$ (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} "} putsend 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} "} putsend 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  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, $y).collect::<Vec<_>>()));} fn main() { printcf!(1, 2); printcf!(3, 1); printcf!(23, 8); printcf!(13, 11); printcf!(22, 7); printcf!(-152, 77); printcf!(14_142, 10_000); printcf!(141_421, 100_000); printcf!(1_414_214, 1_000_000); printcf!(14_142_136, 10_000_000); printcf!(31, 10); printcf!(314, 100); printcf!(3142, 1000); printcf!(31_428, 10_000); printcf!(314_285, 100_000); printcf!(3_142_857, 1_000_000); printcf!(31_428_571, 10_000_000); printcf!(314_285_714, 100_000_000);}  Output: [0, 2] [3] [2, 1, 7] [1, 5, 2] [3, 7] [-1, -1, -37, -2] [1, 2, 2, 2, 2, 2, 1, 1, 29] [1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 7, 2] [1, 2, 2, 2, 2, 2, 2, 2, 3, 6, 1, 2, 1, 12] [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1, 2, 4, 1, 1, 2] [3, 10] [3, 7, 7] [3, 7, 23, 1, 2] [3, 7, 357] [3, 7, 2857] [3, 7, 142857] [3, 7, 476190, 3] [3, 7, 7142857]  ## Sidef Translation of: Perl func r2cf(num, den) { func() { den || return nil var q = num//den (num, den) = (den, num - q*den) return q }} func showcf(f) { print "[" var n = f() print "#{n}" if defined(n) print "; #{n}" while defined(n = f()) print "]\n"} [ [1/2, 3/1, 23/8, 13/11, 22/7, -151/77], [14142/10000, 141421/100000, 1414214/1000000, 14142136/10000000], [314285714/100000000],].each { |seq| seq.each { |r| showcf(r2cf(r.nude)) } print "\n"} Output: [0; 2] [3] [2; 1; 7] [1; 5; 2] [3; 7] [-1; -1; -24; -1; -2] [1; 2; 2; 2; 2; 2; 1; 1; 29] [1; 2; 2; 2; 2; 2; 2; 3; 1; 1; 3; 1; 7; 2] [1; 2; 2; 2; 2; 2; 2; 2; 3; 6; 1; 2; 1; 12] [1; 2; 2; 2; 2; 2; 2; 2; 2; 2; 6; 1; 2; 4; 1; 1; 2] [3; 7; 7142857]  ## Tcl Works with: Tcl version 8.6 Translation of: Ruby ### Direct translation package require Tcl 8.6 proc r2cf {n1 {n2 1}} { # Convert a decimal fraction (e.g., 1.23) into a form we can handle if {$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

package require Tcl 8.6 # General generator class based on coroutinesoo::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 decimalsprintcf "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

include c:\cxpl\codes;real Val; proc R2CF(N1, N2, Lev);         \Output continued fraction for N1/N2int  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 fractionif 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

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)
`