Convert decimal number to rational: Difference between revisions

{{trans|Nim}}

Int numerator

Int denominator

print(rationalize(0.99))

print(rationalize(0.909))

{{out}}

Specification of a procedure Real_To_Rational, which is searching for the best approximation of a real number. The procedure is generic. I.e., you can instantiate it by your favorite "Real" type (Float, Long_Float, ...).

type Real is digits <>;

procedure Real_To_Rational(R: Real;

Bound: Positive;

Nominator: out Integer;

The implementation (just brute-force search for the best approximation with Denominator less or equal Bound):

Bound: Positive;

Nominator: out Integer;

Nominator := Integer(Real'Rounding(Real(Denominator) * R));

The main program, called "Convert_Decimal_To_Rational", reads reals from the standard input until 0.0. It outputs progressively better rational approximations of the reals, where "progressively better" means a larger Bound for the Denominator:

procedure Convert_Decimal_To_Rational is

end loop;

end Convert_Decimal_To_Rational;

Finally, the output (reading the input from a file):

=={{header|AppleScript}}==

-- approxRatio :: Real -> Real -> Ratio

set my text item delimiters to dlm

s

{{Out}}

<pre>0.9054054 -> 67/74

=={{header|AutoHotkey}}==

inputbox, string, Enter Number

stringsplit, string, string, .

MsgBox % String . " -> " . String1 . " " . Ans

reload

<pre>

0.9054054 -> 67/74

=={{header|Bracmat}}==

= integerPart decimalPart z

. @(!arg:?integerPart "." ?decimalPart)

)

)

Output:

<pre>0.9054054054 = 4527027027/5000000000 (approx. 67/74)

=={{header|C}}==

#include <stdlib.h>

#include <math.h>

return 0;

denom <= 1: 0/1

denom <= 16: 1/7

denom <= 65536: 104348/33215

denom <= 1048576: 3126535/995207

=={{header|C sharp|C#}}==

{{trans|C}}

using System.Text;

}

}

{{out}}

=={{header|D}}==

{{trans|Ada}}

alias Fraction = Tuple!(int,"nominator", uint,"denominator");

writeln();

}

{{out}}

<pre>0.750000000 1/1 3/4 3/4 3/4 3/4 3/4

Thanks Rudy Velthuis for Velthuis.BigRationals and Velthuis.BigDecimals library[https://github.com/rvelthuis/DelphiBigNumbers].

{{Trans|Go}}

program Convert_decimal_number_to_rational;

end;

Readln;

=={{header|EchoLisp}}==

The '''rationalize''' function uses a Stern-Brocot tree [http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree] to find the best rational approximation of an inexact (floating point) number, for a given precision. The '''inexact->exact''' function returns a rational approximation for the default precision 0.0001 .

(exact->inexact 67/74)

→ 0.9054054054054054

"precision:10^-13 PI = 5419351/1725033"

"precision:10^-14 PI = 58466453/18610450"

=={{header|Factor}}==

{{out}}

<pre>

=={{header|Forth}}==

\ Brute force search, optimized to search only within integer bounds surrounding target

\ Forth 200x compliant

2.71828e 1000 RealToRational swap . . 1264 465

0.9054054e 100 RealToRational swap . . 67 74

=={{header|Fortran}}==

Rather than engage in fancy schemes, here are two "brute force" methods. The first simply multiplies the value by a large power of ten, then casts out common factors in P/Q = x*1000000000/100000000. But if the best value for Q involves factors other than two and five, this won't work well. The second method is to jiggle either P or Q upwards depending on whether P/Q is smaller or larger than X, reporting improvements as it goes. Once beyond small numbers there are many small improvements to be found, so only those much better than the previous best are reported. Loosely speaking, the number of digits correct in good values of P/Q should be the sum of the number of digits in P and Q, and more still for happy fits, but a factor of eight suffices to suppress the rabble. Thus for Pi, the famous 22/7 and 355/113 appear as desired. Later pairs use lots more digits without a surprise hit, except for the short decimal sequence which comes out as 314159/100000 that reconstitutes the given decimal fraction exactly. Which is not a good approximation for Pi, and its pairs diverge from those of the more accurate value. In other words, one must assess the precision of the given value and not be distracted by the spurious precision offered by the larger P/Q pairs, so for 3·14159 with six digits, there is little point in going further than 355/113 - with their six digits. Contrariwise, if a P/Q with few digits matches many more digits of the given number, then a source rational number can be suspected. But if given just a few digits, such as 0·518 (or 0·519, when rounded), 13/25 could be just as likely a source number as 14/27 which is further away.

INTEGER MSG !Output unit number.

CONTAINS !One good routine.

CALL RATIONALISE(0.518518D0,"Two repeats, truncated.")

CALL RATIONALISE(0.518519D0,"Two repeats, rounded.")

Some examples. Each rational value is followed by X - P/Q. Notice that 0·(518) repeating, presented as 0·518518, is ''not'' correctly rounded.

=={{header|FreeBASIC}}==

'' (no error checking, limited to 64-bit signed math)

type number as longint

if n = "" then exit do

print n & ":", parserational(n)

=={{header|Fōrmulæ}}==

=={{header|Go}}==

Go has no native decimal representation so strings are used as input here. The program parses it into a Go rational number, which automatically reduces.

import (

}

}

Output:

<pre>

'''Test:'''

[

].each{

printf "%30.27f %s\n", it, it as Rational

'''Output:'''

The first map finds the simplest fractions within a given radius, because the floating-point representation is not exact. The second line shows that the numbers could be parsed into fractions at compile time if they are given the right type. The last converts the string representation of the given values directly to fractions.

[67 % 74,14 % 27,3 % 4]

Prelude> [0.9054054, 0.518518, 0.75] :: [Rational]

[4527027 % 5000000,259259 % 500000,3 % 4]

Prelude> map (fst . head . Numeric.readFloat) ["0.9054054", "0.518518", "0.75"] :: [Rational]

=={{header|J}}==

J's <code>x:</code> built-in will find a rational number which "best matches" a floating point number.

These numbers are ratios where the integer on the left of the <code>r</code> is the numerator and the integer on the right of the <code>r</code> is the denominator. (Note that this use is in analogy with floating point notion, though it is true that hexadecimal notation and some languages' typed numeric notations use letters within numbers. Using letters rather than other characters makes lexical analysis simpler to remember - both letters and numbers are almost always "word forming characters".)

Note that the concept of "best" has to do with the expected precision of the argument:

9r10 1r2

x: 0.9054 0.5185

67r74 14r27

x: 0.9054054054054054 0.5185185185185185

Note that J allows us to specify an epsilon, for the purpose of recognizing a best fit, but the allowed values must be rather small. In J version 6, the value 5e_11 was nearly the largest epsilon allowed:

(Note that epsilon will be scaled by magnitude of the largest number involved in a comparison when testing floating point representations of numbers for "equality". Note also that this J implementation uses 64 bit ieee floating point numbers.)

Here are some other alternatives for dealing with decimals and fractions:

0.9054054000 0.5185180000 0.7500000000

0j10": x:inv 67r74 42r81 3r4 NB. decimal representation (shown to 10 decimal places)

0.9054054054 0.5185185185 0.7500000000

x: x:inv 67r74 42r81 3r4 NB. invertible

=={{header|Java}}==

public class Test {

System.out.printf("%-12s : %s%n", d, new BigFraction(d, 0.00000002D, 10000));

}

<pre>0.75 : 3 / 4

0.518518 : 37031 / 71417

=={{header|JavaScript}}==

Deriving an approximation within a specified tolerance:

'use strict';

// MAIN ---

return main();

{{Out}}

<pre>[

(*) With the caveat that the C implementation of jq does not support unlimited-precision integer arithmetic.

# Input: any JSON number, not only a decimal

| (.e | if length > 0 then tonumber else 0 end) as $e

| .s | dtor(null; $e)

'''Examples'''

0.518518,

0.75,

1e308

{{out}}

<pre>

Julia has a native Rational type, and provides [http://docs.julialang.org/en/latest/manual/conversion-and-promotion/#case-study-rational-conversions a convenience conversion function] that implements a standard algorithm for approximating a floating-point number by a ratio of integers to within a given tolerance, which defaults to machine epsilon.

rationalize(0.518518)

4527027//5000000

Here is the core algorithm written in Julia 1.0, without handling various data types and corner cases:

p, q, pp, qq = copysign(1,x), 0, 0, 1

x, y = abs(x), 1.0

i = Int(cld(x-tt,y+t)) # find optimal semiconvergent: smallest i such that x-i*y < i*t+tt

return (i*p+pp) // (i*q+qq)

=={{header|Kotlin}}==

class Rational(val num: Long, val den: Long) {

for (decimal in decimals)

println("${decimal.toString().padEnd(9)} = ${decimalToRational(decimal)}")

{{out}}

=={{header|Liberty BASIC}}==

' Uses convention that one repeating sequence implies infinitely repeating sequence..

' Non-recurring fractions are limited to nd number of digits in nuerator & denominator

if length /2 =int( length /2) then if mid$( i$, 3, length /2) =mid$( i$, 3 +length /2, length /2) then check$ ="recurring"

end function

<pre>

0.5 is non-recurring

=={{header|Lua}}==

Brute force, and why not?

local n, d, dmax, eps = 1, 1, 1e7, 1e-15

while math.abs(n/d-v)>eps and d<dmax do d=d+1 n=math.floor(v*d) end

print(string.format("%15.13f --> %d / %d", v, n, d))

{{out}}

<pre>0.9054054000000 --> 4527027 / 5000000

=={{header|Maple}}==

> map( convert, [ 0.9054054, 0.518518, 0.75 ], 'rational', 'exact' );

4527027 259259

[-------, ------, 3/4]

5000000 500000

=={{header|Mathematica}} / {{header|Wolfram Language}}==

=={{header|MATLAB}} / {{header|Octave}}==

[a,b]=rat(.75)

[a,b]=rat(.518518)

[a,b]=rat(.9054054)

Output:

{{improve|МК-61/52|This example is not clear, please improve it or delete it!}}

* П1 БП 02 ИП4 10^x П0 ПA ИП1 ПB

ИПA ИПB / П9 КИП9 ИПA ИПB ПA ИП9 *

- ПB x=0 20 ИПA ИП0 ИПA / П0 ИП1

=={{header|NetRexx}}==

Now the nearly equivalent program.

/*NetRexx program to convert decimal numbers to fractions *************

* 16.08.2012 Walter Pachl derived from Rexx Version 2

end

If den=1 Then Return nom /* an integer */

Output is the same as for Rexx Version 2.

=={{header|Nim}}==

Similar to the Rust and Julia implementations.

import fenv

echo rationalize(0.99)

echo rationalize(0.909)

{{out}}

Nim also has a rationals library for this, though it does not allow you to set tolerances like the code above.

echo toRational(0.9054054054)

echo toRational(0.9)

echo toRational(0.99)

{{out}}

=={{header|PARI/GP}}==

Quick and dirty.

my(n=0);

while(x-floor(x*10^n)/10^n!=0.,n++);

floor(x*10^n)/10^n

To convert a number into a rational with a denominator not dividing a power of 10, use <code>contfrac</code> and the Gauss-Kuzmin distribution to distinguish (hopefully!) where to truncate.

=={{header|Perl}}==

Note: the following is considerably more complicated than what was specified, because the specification is not, well, specific. Three methods are provided with different interpretation of what "conversion" means: keeping the string representation the same, keeping machine representation the same, or find best approximation with denominator in a reasonable range. None of them takes integer overflow seriously (though the best_approx is not as badly subject to it), so not ready for real use.

my ($m, $n) = @_;

($m, $n) = ($n, $m % $n) while $n;

for (map { 10 ** $_ } 1 .. 10) {

printf "approx below %g: %d / %d\n", $_, best_approx($x, $_)

Output: <pre>3/8 = 0.375:

machine: 3/8

=={{header|Phix}}==

<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>

<span style="color: #008080;">function</span> <span style="color: #000000;">decrat</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span>

<span style="color: #0000FF;">?</span><span style="color: #000000;">decrat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"0.518518"</span><span style="color: #0000FF;">)</span>

<span style="color: #0000FF;">?</span><span style="color: #000000;">decrat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"0.75"</span><span style="color: #0000FF;">)</span>

{{out}}

<pre>

=={{header|PHP}}==

{{works with|PHP|5.3+}}

{

if ($val == (int) $val) {

echo asRational(1/4)."\n"; // "1/4"

echo asRational(1/3)."\n"; // "1/3"

=={{header|PL/I}}==

Convert_Decimal_To_Rational: procedure options (main); /* 14 January 2014, from Ada */

end;

end;

Output:

<pre>

=={{header|PureBasic}}==

Define t.i : If a < b : Swap a, b : EndIf

While a%b : t=a : a=b : b=t%a : Wend : ProcedureReturn b

DataSection

Data.d 0.9054054,0.518518,0.75,0.0

<pre>

0.9054054 -> 4527027/5000000

The first loop limits the size of the denominator, because the floating-point representation is not exact. The second converts the string representation of the given values directly to fractions.

>>> for d in (0.9054054, 0.518518, 0.75): print(d, Fraction.from_float(d).limit_denominator(100))

0.518518 259259/500000

0.75 3/4

Or, writing our own '''approxRatio''' function:

{{Works with|Python|3.7}}

from math import (floor, gcd)

# MAIN ---

if __name__ == '__main__':

{{Out}}

<pre>Approximate rationals from decimals (epsilon of 1/10000):

=={{header|R}}==

ratio<-function(decimal){

denominator=1

return(str)

}

{{Out}}

<pre>

Using the Quackery big number rational arithmetic library <code>bigrat.qky</code>.

[ dup echo$

say "." cr ] is task ( $ --> )

{{out}}

Racket has builtin exact and inexact representantions of numbers, 3/4 is a valid number syntactically, and one can change between the exact and inexact values with the functions showed in the example.

They have some amount of inaccuracy, but i guess it can be tolerated.

(inexact->exact 0.75) ; -> 3/4

(exact->inexact 67/74) ; -> 0.9054054054054054

=={{header|Raku}}==

(formerly Perl 6)

Decimals are natively represented as rationals in Raku, so if the task does not need to handle repeating decimals, it is trivially handled by the <tt>.nude</tt> method, which returns the numerator and denominator:

{{out}}

<pre>4527027/5000000

3/4</pre>

However, if we want to take repeating decimals into account, then we can get a bit fancier.

my ( $int, $dec ) = ( $n ~~ /^ (\d+) \. (\d+) $/ )».Str or die;

for @a -> [ $n, $d ] {

say "$n with $d repeating digits = ", decimal_to_fraction( $n, $d );

{{out}}

<pre>0.9054 with 3 repeating digits = 67/74

<br>REXX can support almost any number of decimal digits, but &nbsp; '''10''' &nbsp; was chosen for practicality for this task.

numeric digits 10 /*use ten decimal digits of precision. */

parse arg orig 1 n.1 "/" n.2; if n.2='' then n.2=1 /*get the fraction.*/

end /*while*/

if h==1 then return b /*don't return number ÷ by 1.*/

'''output''' &nbsp; when using various inputs (which are displayed as part of the output):

<br>(Multiple runs are shown, outputs are separated by a blank line.)

===version 2===

* 15.08.2012 Walter Pachl derived from above for readability

* It took me time to understand :-) I need descriptive variable names

if den=1 then return nom /* denominator 1: integer */

return nom'/'den /* otherwise a fraction */

Output:

<pre>

===version 3===

* 13.02.2014 Walter Pachl

* specify the number as xxx.yyy(pqr) pqr is the period

Parse Arg a,b

if b = 0 then return abs(a)

'''Output:'''

<pre>5.55555 = 111111/20000 ok

This converts the string representation of the given values directly to fractions.

0.9054054 4527027/5000000

0.518518 259259/500000

0.75 3/4

This loop finds the simplest fractions within a given radius, because the floating-point representation is not exact.

# =>0.9054054 67/74

# =>0.518518 14/27

# =>0.75 3/4

{{works with|Ruby|2.1.0+}}

A suffix for integer and float literals was introduced:

=={{header|Rust}}==

extern crate rand;

extern crate num;

}

}

First test the function with 1_000_000 random double floats :

<pre>

=={{header|Scala}}==

{{Out}}Best seen running in your browser [https://scastie.scala-lang.org/rrlFnuTURgirBiTsH3Kqrg Scastie (remote JVM)].

object Number2Fraction extends App {

for (d <- n)

println(f"$d%-12s : ${new BigFraction(d, 0.00000002D, 10000)}%s")

=={{header|Seed7}}==

defines the operator [http://seed7.sourceforge.net/libraries/bigrat.htm#%28attr_bigRational%29parse%28in_var_string%29 parse],

which accepts, besides fractions, also a decimal number with repeating decimals.

include "bigrat.s7i";

writeln(bigRational parse "31.415926536");

writeln(bigRational parse "0.000000000");

{{out}}

<pre>67/74

=={{header|Sidef}}==

By default, literal numbers are represented in rational form:

say 0.518518.as_frac #=> 259259/500000

Additionally, '''Num(str)''' can be used for parsing a decimal expansion into rational form:

say Num(str).as_frac

{{out}}

For rational approximations, the Number '''.rat_approx''' method can be used:

=={{header|Tcl}}==

{{works with|Tcl|8.4+}}

Here is a complete script with the implemented function and a small test suite (which is executed when this script is called directly from a shell) - originally on http://wiki.tcl.tk/752:

proc dbl2frac {dbl {eps 0.000001}} {

}

}

Running it shows one unexpected result, but on closer examination it is clear that 14/27 equals 42/81, so it should indeed be the right solution:

~ $ fractional.tcl

=={{header|Vala}}==

{{trans|C}}

public long d;

public long n;

print("%11ld/%ld\n", r.n, r.d);

}

{{out}}

=={{header|VBA}}==

{{trans|D}}

If r = 0 Then

Next i

End Sub

{{out}}

<pre>

{{libheader|Wren-rat}}

{{libheader|Wren-fmt}}

import "/fmt" for Fmt

var r = Rat.fromFloat(test)

System.print("%(Fmt.s(-9, test)) -> %(r)")

{{out}}

=={{header|zkl}}==

{{trans|D}}

if (r == 0.0) return(0,1);

if (r < 0.0){

return((r*best).round().toInt(),best);

}

0.142857143, 3.141592654, 2.718281828,

-0.423310825, 31.415926536);

{ print(" %d/%d".fmt(real2Rational(r,(10).pow(i)).xplode())) }

println();

{{out}}

<pre>