Convert decimal number to rational: Difference between revisions
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{{task}}
{{clarify task}}
The task is to write a program to transform a decimal number into a fraction in lowest terms.
Line 19 ⟶ 21:
<br><br>
=={{header|
{{trans|Nim}}
<syntaxhighlight lang="11l">T Rational
Int numerator
Int denominator
F (numerator, denominator)
.numerator = numerator
.denominator = denominator
F String()
I .denominator == 1
R String(.numerator)
E
R .numerator‘//’(.denominator)
F rationalize(x, tol = 1e-12)
V xx = x
V flagNeg = xx < 0.0
I flagNeg
xx = -xx
I xx < 1e-10
R Rational(0, 1)
I abs(xx - round(xx)) < tol
R Rational(Int(xx), 1)
V a = 0
V b = 1
V c = Int(ceil(xx))
V d = 1
V aux1 = 7FFF'FFFF I/ 2
L c < aux1 & d < aux1
V aux2 = (Float(a) + Float(c)) / (Float(b) + Float(d))
I abs(xx - aux2) < tol
L.break
I xx > aux2
a += c
b += d
E
c += a
d += b
V g = gcd(a + c, b + d)
I flagNeg
R Rational(-(a + c) I/ g, (b + d) I/ g)
E
R Rational((a + c) I/ g, (b + d) I/ g)
print(rationalize(0.9054054054))
print(rationalize(0.9054054054, 0.0001))
print(rationalize(0.5185185185))
print(rationalize(0.5185185185, 0.0001))
print(rationalize(0.75))
print(rationalize(0.1428571428, 0.001))
print(rationalize(35.000))
print(rationalize(35.001))
print(rationalize(0.9))
print(rationalize(0.99))
print(rationalize(0.909))
print(rationalize(0.909, 0.001))</syntaxhighlight>
{{out}}
<pre>
1910136854//2109703391
67//74
983902907//1897527035
14//27
3//4
1//7
35
35001//1000
9//10
99//100
909//1000
10//11
</pre>
=={{header|Ada}}==
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;
Denominator: out Positive);</
The implementation (just brute-force search for the best approximation with Denominator less or equal Bound):
<
Bound: Positive;
Nominator: out Integer;
Line 60 ⟶ 137:
Nominator := Integer(Real'Rounding(Real(Denominator) * R));
end Real_To_Rational;</
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
Line 89 ⟶ 166:
end loop;
end Convert_Decimal_To_Rational;
</syntaxhighlight>
Finally, the output (reading the input from a file):
Line 104 ⟶ 181:
0.000000000
</pre>
=={{header|AppleScript}}==
<syntaxhighlight lang="applescript">--------- RATIONAL APPROXIMATION TO DECIMAL NUMBER -------
-- approxRatio :: Real -> Real -> Ratio
on approxRatio(epsilon, n)
if {real, integer} contains (class of epsilon) and 0 < epsilon then
-- Given
set e to epsilon
else
-- Default
set e to 1 / 10000
end if
script gcde
on |λ|(e, x, y)
script _gcd
on |λ|(a, b)
if b < e then
a
else
|λ|(b, a mod b)
end if
end |λ|
end script
|λ|(abs(x), abs(y)) of _gcd
end |λ|
end script
set c to |λ|(e, 1, n) of gcde
Ratio((n div c), (1 div c))
end approxRatio
-- Ratio :: Int -> Int -> Ratio
on Ratio(n, d)
{type:"Ratio", n:n, d:d}
end Ratio
-- showRatio :: Ratio -> String
on showRatio(r)
(n of r as string) & "/" & (d of r as string)
end showRatio
--------------------------- TEST -------------------------
on run
script ratioString
-- Using a tolerance epsilon of 1/10000
on |λ|(x)
(x as string) & " -> " & showRatio(approxRatio(1.0E-4, x))
end |λ|
end script
unlines(map(ratioString, ¬
{0.9054054, 0.518518, 0.75}))
-- 0.9054054 -> 67/74
-- 0.518518 -> 14/27
-- 0.75 -> 3/4
end run
-------------------- GENERIC FUNCTIONS -------------------
-- abs :: Num -> Num
on abs(x)
if 0 > x then
-x
else
x
end if
end abs
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- unlines :: [String] -> String
on unlines(xs)
-- A single string formed by the intercalation
-- of a list of strings with the newline character.
set {dlm, my text item delimiters} to ¬
{my text item delimiters, linefeed}
set s to xs as text
set my text item delimiters to dlm
s
end unlines</syntaxhighlight>
{{Out}}
<pre>0.9054054 -> 67/74
0.518518 -> 14/27
0.75 -> 3/4</pre>
=={{header|AutoHotkey}}==
<syntaxhighlight lang="autohotkey"> Array := []
inputbox, string, Enter Number
stringsplit, string, string, .
Line 196 ⟶ 386:
MsgBox % String . " -> " . String1 . " " . Ans
reload
</syntaxhighlight>
<pre>
0.9054054 -> 67/74
Line 204 ⟶ 394:
=={{header|Bracmat}}==
<
= integerPart decimalPart z
. @(!arg:?integerPart "." ?decimalPart)
Line 275 ⟶ 465:
)
)
);</
Output:
<pre>0.9054054054 = 4527027027/5000000000 (approx. 67/74)
Line 293 ⟶ 483:
=={{header|C}}==
Since the intention of the task giver is entirely unclear, here's another version of best rational approximation of a floating point number. It's somewhat more rigorous than the Perl version below, but is still not quite complete.<
#include <stdlib.h>
#include <math.h>
Line 365 ⟶ 555:
return 0;
}</
denom <= 1: 0/1
denom <= 16: 1/7
Line 381 ⟶ 571:
denom <= 65536: 104348/33215
denom <= 1048576: 3126535/995207
denom <= 16777216: 47627751/15160384</
=={{header|C sharp|C#}}==
{{trans|C}}
<syntaxhighlight lang="csharp">using System;
using System.Text;
namespace RosettaDecimalToFraction
{
public class Fraction
{
public Int64 Numerator;
public Int64 Denominator;
public Fraction(double f, Int64 MaximumDenominator = 4096)
{
/* Translated from the C version. */
/* a: continued fraction coefficients. */
Int64 a;
var h = new Int64[3] { 0, 1, 0 };
var k = new Int64[3] { 1, 0, 0 };
Int64 x, d, n = 1;
int i, neg = 0;
if (MaximumDenominator <= 1)
{
Denominator = 1;
Numerator = (Int64)f;
return;
}
if (f < 0) { neg = 1; f = -f; }
while (f != Math.Floor(f)) { n <<= 1; f *= 2; }
d = (Int64)f;
/* continued fraction and check denominator each step */
for (i = 0; i < 64; i++)
{
a = (n != 0) ? d / n : 0;
if ((i != 0) && (a == 0)) break;
x = d; d = n; n = x % n;
x = a;
if (k[1] * a + k[0] >= MaximumDenominator)
{
x = (MaximumDenominator - k[0]) / k[1];
if (x * 2 >= a || k[1] >= MaximumDenominator)
i = 65;
else
break;
}
h[2] = x * h[1] + h[0]; h[0] = h[1]; h[1] = h[2];
k[2] = x * k[1] + k[0]; k[0] = k[1]; k[1] = k[2];
}
Denominator = k[1];
Numerator = neg != 0 ? -h[1] : h[1];
}
public override string ToString()
{
return string.Format("{0} / {1}", Numerator, Denominator);
}
}
class Program
{
static void Main(string[] args)
{
Console.OutputEncoding = UTF8Encoding.UTF8;
foreach (double d in new double[] { 0.9054054, 0.518518, 0.75, 0.4285714, 0.833333,
0.90909, 3.14159265358979, 2.7182818284590451 })
{
var f = new Fraction(d, d >= 2 ? 65536 : 4096);
Console.WriteLine("{0,20} → {1}", d, f);
}
}
}
}
</syntaxhighlight>
{{out}}
<pre> 0.9054054 → 67 / 74
0.518518 → 14 / 27
0.75 → 3 / 4
0.4285714 → 3 / 7
0.833333 → 5 / 6
0.90909 → 10 / 11
3.14159265358979 → 104348 / 33215
2.71828182845905 → 49171 / 18089</pre>
=={{header|C++}}==
<syntaxhighlight lang="c++">
#include <cmath>
#include <cstdint>
#include <iostream>
#include <limits>
#include <numeric>
#include <vector>
class Rational {
public:
Rational(const int64_t& numer, const uint64_t& denom) : numerator(numer), denominator(denom) { }
Rational negate() {
return Rational(-numerator, denominator);
}
std::string to_string() {
return std::to_string(numerator) + " / " + std::to_string(denominator);
}
private:
int64_t numerator;
uint64_t denominator;
};
/**
* Return a Rational such that its numerator / denominator = 'decimal', correct to dp decimal places,
* where dp is minimum of 'decimal_places' and the number of decimal places in 'decimal'.
*/
Rational decimal_to_rational(double decimal, const uint32_t& decimal_places) {
const double epsilon = 1.0 / std::pow(10, decimal_places);
const bool negative = ( decimal < 0.0 );
if ( negative ) {
decimal = -decimal;
}
if ( decimal < std::numeric_limits<double>::min() ) {
return Rational(0, 1);
}
if ( std::abs( decimal - std::round(decimal) ) < epsilon ) {
return Rational(std::round(decimal), 1);
}
uint64_t a = 0;
uint64_t b = 1;
uint64_t c = static_cast<uint64_t>(std::ceil(decimal));
uint64_t d = 1;
const uint64_t auxiliary_1 = std::numeric_limits<uint64_t>::max() / 2;
while ( c < auxiliary_1 && d < auxiliary_1 ) {
const double auxiliary_2 = static_cast<double>( a + c ) / ( b + d );
if ( std::abs(decimal - auxiliary_2) < epsilon ) {
break;
}
if ( decimal > auxiliary_2 ) {
a = a + c;
b = b + d;
} else {
c = a + c;
d = b + d;
}
}
const uint64_t divisor = std::gcd(( a + c ), ( b + d ));
Rational result(( a + c ) / divisor, ( b + d ) / divisor);
return ( negative ) ? result.negate() : result;
}
int main() {
for ( const double& decimal : { 3.1415926535, 0.518518, -0.75, 0.518518518518, -0.9054054054054, -0.0, 2.0 } ) {
std::cout << decimal_to_rational(decimal, 9).to_string() << std::endl;
}
}
</syntaxhighlight>
{{ out }}
<pre>
104348 / 33215
36975 / 71309
-3 / 4
14 / 27
-67 / 74
0 / 1
2 / 1
</pre>
=={{header|Clojure}}==
Line 420 ⟶ 789:
=={{header|D}}==
{{trans|Ada}}
<
alias Fraction = Tuple!(int,"nominator", uint,"denominator");
Line 459 ⟶ 828:
writeln();
}
}</
{{out}}
<pre>0.750000000 1/1 3/4 3/4 3/4 3/4 3/4
Line 469 ⟶ 838:
-0.423310825 0/1 -3/7 -11/26 -69/163 -1253/2960 -10093/23843
31.415926536 31/1 157/5 377/12 3550/113 208696/6643 2918194/92889</pre>
=={{header|Delphi}}==
{{libheader| Velthuis.BigRationals}}
{{libheader| Velthuis.BigDecimals}}
Thanks Rudy Velthuis for Velthuis.BigRationals and Velthuis.BigDecimals library[https://github.com/rvelthuis/DelphiBigNumbers].
{{Trans|Go}}
<syntaxhighlight lang="delphi">
program Convert_decimal_number_to_rational;
{$APPTYPE CONSOLE}
uses
Velthuis.BigRationals,
Velthuis.BigDecimals;
const
Tests: TArray<string> = ['0.9054054', '0.518518', '0.75'];
var
Rational: BigRational;
Decimal: BigDecimal;
begin
for var test in Tests do
begin
Decimal := test;
Rational := Decimal;
Writeln(test, ' = ', Rational.ToString);
end;
Readln;
end.</syntaxhighlight>
=={{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
Line 500 ⟶ 898:
"precision:10^-13 PI = 5419351/1725033"
"precision:10^-14 PI = 58466453/18610450"
</syntaxhighlight>
=={{header|Factor}}==
<syntaxhighlight lang="factor">USING: kernel math.floating-point prettyprint ;
0.9054054 0.518518 0.75 [ double>ratio . ] tri@</syntaxhighlight>
{{out}}
<pre>
4077583422059235/4503599627370496
2335197471584895/4503599627370496
3/4
</pre>
=={{header|Fermat}}==
Fermat does this automatically.
{{out}}
<pre>
>3.14
157 / 50; or 3.1400000000000000000000000000000000000000
>10.00000
10
>77.7777
777777 / 10000; or 77.7777000000000000000000000000000000000000
>-5.5
-11 / 2; or -5.5000000000000000000000000000000000000000
</pre>
=={{header|Forth}}==
<syntaxhighlight lang="forth">
\ Brute force search, optimized to search only within integer bounds surrounding target
\ Forth 200x compliant
: RealToRational ( float_target int_denominator_limit -- numerator denominator )
{: f: thereal denlimit | realscale numtor denom neg? f: besterror f: temperror :}
0 to numtor
0 to denom
9999999e to besterror \ very large error that will surely be improved upon
thereal F0< to neg? \ save sign for later
thereal FABS to thereal
thereal FTRUNC f>s 1+ to realscale \ realscale helps set integer bounds around target
denlimit 1+ 1 ?DO \ search through possible denominators ( 1 to denlimit)
I realscale * I realscale 1- * ?DO \ search within integer limits bounding the real
I s>f J s>f F/ \ e.g. for 3.1419e search only between 3 and 4
thereal F- FABS to temperror
temperror besterror F< IF
temperror to besterror I to numtor J to denom
THEN
LOOP
LOOP
neg? IF numtor NEGATE to numtor THEN
numtor denom
;
(run)
1.618033988e 100 RealToRational swap . . 144 89
3.14159e 1000 RealToRational swap . . 355 113
2.71828e 1000 RealToRational swap . . 1264 465
0.9054054e 100 RealToRational swap . . 67 74
</syntaxhighlight>
=={{header|Fortran}}==
This sort of calculation works best in base ten and with an input scheme that recognises a protocol for specifying that the decimal value has a recurring sequence, whereupon the methods taught in school could be employed. Such a protocol often conflicts with presenting a digit sequence with the last digit correctly rounded. Although many early computers worked in base ten, these days a binary base is usual, be it 2, 4, 8, or 16. Alas, five is not a factor of two nor any of its powers, and many decimal fractions, even a brief one such as 0·1, convert to a recurring sequence in binary so for example, 10·15 is in binary 1010·0010011001100110011... However, 203/20 when evaluated in binary floating-point will generate the same binary sequence as 10·15, granted that the compiler does its arithmetic correctly. Nevertheless, such a decimal value is ''not'' the value that a binary computer will work with, except for restricted fractions such as 0·5, 0·25, 0·75, and so forth up to the precision of the arithmetic.
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.
The source uses the MODULE facility of F90 merely to avoid the annoyance of having to declare the type of integer function GCD. The T ("tab") format code is employed to facilitate the alignment of output, given that P/Q is presented with I0 format so that there are no spaces (as in " 22/ 7" for example), the latter being standard in F90 but an extension in earlier Fortrans. <syntaxhighlight lang="fortran"> MODULE PQ !Plays with some integer arithmetic.
INTEGER MSG !Output unit number.
CONTAINS !One good routine.
INTEGER FUNCTION GCD(I,J) !Greatest common divisor.
INTEGER I,J !Of these two integers.
INTEGER N,M,R !Workers.
N = MAX(I,J) !Since I don't want to damage I or J,
M = MIN(I,J) !These copies might as well be the right way around.
1 R = MOD(N,M) !Divide N by M to get the remainder R.
IF (R.GT.0) THEN !Remainder zero?
N = M !No. Descend a level.
M = R !M-multiplicity has been removed from N.
IF (R .GT. 1) GO TO 1 !No point dividing by one.
END IF !If R = 0, M divides N.
GCD = M !There we are.
END FUNCTION GCD !Euclid lives on!
SUBROUTINE RATIONAL10(X)!By contrast, this is rather crude.
DOUBLE PRECISION X !The number.
DOUBLE PRECISION R !Its latest rational approach.
INTEGER P,Q !For R = P/Q.
INTEGER F,WHACK !Assistants.
PARAMETER (WHACK = 10**8) !The rescale...
P = X*WHACK + 0.5 !Multiply by WHACK/WHACK = 1 and round to integer.
Q = WHACK !Thus compute X/1, sortof.
F = GCD(P,Q) !Perhaps there is a common factor.
P = P/F !Divide it out.
Q = Q/F !For a proper rational number.
R = DBLE(P)/DBLE(Q) !So, where did we end up?
WRITE (MSG,1) P,Q,X - R,WHACK !Details.
1 FORMAT ("x - ",I0,"/",I0,T28," = ",F18.14,
1 " via multiplication by ",I0)
END SUBROUTINE RATIONAL10 !Enough of this.
SUBROUTINE RATIONAL(X) !Use brute force in a different way.
DOUBLE PRECISION X !The number.
DOUBLE PRECISION R,E,BEST !Assistants.
INTEGER P,Q !For R = P/Q.
INTEGER TRY,F !Floundering.
P = 1 + X !Prevent P = 0.
Q = 1 !So, X/1, sortof.
BEST = X*6 !A largeish value for the first try.
DO TRY = 1,10000000 !Pound away.
R = DBLE(P)/DBLE(Q) !The current approximation.
E = X - R !Deviation.
IF (ABS(E) .LE. BEST) THEN !Significantly better than before?
BEST = ABS(E)*0.125 !Yes. Demand eightfold improvement to notice.
F = GCD(P,Q) !We may land on a multiple.
IF (BEST.LT.0.1D0) WRITE (MSG,1) P/F,Q/F,E !Skip early floundering.
1 FORMAT ("x - ",I0,"/",I0,T28," = ",F18.14) !Try to align columns.
IF (F.NE.1) WRITE (MSG,*) "Common factor!",F !A surprise!
IF (E.EQ.0) EXIT !Perhaps we landed a direct hit?
END IF !So much for possible announcements.
IF (E.GT.0) THEN !Is R too small?
P = P + CEILING(E*Q) !Yes. Make P bigger by the shortfall.
ELSE IF (E .LT. 0) THEN !But perhaps R is too big?
Q = Q + 1 !If so, use a smaller interval.
END IF !So much for adjustments.
END DO !Try again.
END SUBROUTINE RATIONAL !Limited integers, limited sense.
SUBROUTINE RATIONALISE(X,WOT) !Run the tests.
DOUBLE PRECISION X !The value.
CHARACTER*(*) WOT !Some blather.
WRITE (MSG,*) X,WOT !Explanations can help.
CALL RATIONAL10(X) !Try a crude method.
CALL RATIONAL(X) !Try a laborious method.
WRITE (MSG,*) !Space off.
END SUBROUTINE RATIONALISE !That wasn't much fun.
END MODULE PQ !But computer time is cheap.
PROGRAM APPROX
USE PQ
DOUBLE PRECISION PI,E
MSG = 6
WRITE (MSG,*) "Rational numbers near to decimal values."
WRITE (MSG,*)
PI = 1 !Thus get a double precision conatant.
PI = 4*ATAN(PI) !That will determine the precision of ATAN.
E = DEXP(1.0D0) !Rather than blabber on about 1 in double precision.
CALL RATIONALISE(0.1D0,"1/10 Repeating in binary..")
CALL RATIONALISE(3.14159D0,"Pi approx.")
CALL RATIONALISE(PI,"Pi approximated better.")
CALL RATIONALISE(E,"e: rational approximations aren't much use.")
CALL RATIONALISE(10.15D0,"Exact in decimal, recurring in binary.")
WRITE (MSG,*)
WRITE (MSG,*) "Variations on 67/74"
CALL RATIONALISE(0.9054D0,"67/74 = 0·9(054) repeating in base 10")
CALL RATIONALISE(0.9054054D0,"Two repeats.")
CALL RATIONALISE(0.9054054054D0,"Three repeats.")
WRITE (MSG,*)
WRITE (MSG,*) "Variations on 14/27"
CALL RATIONALISE(0.518D0,"14/27 = 0·(518) repeating in decimal.")
CALL RATIONALISE(0.519D0,"Rounded.")
CALL RATIONALISE(0.518518D0,"Two repeats, truncated.")
CALL RATIONALISE(0.518519D0,"Two repeats, rounded.")
END</syntaxhighlight>
Some examples. Each rational value is followed by X - P/Q. Notice that 0·(518) repeating, presented as 0·518518, is ''not'' correctly rounded.
<pre>
Rational numbers near to decimal values.
0.100000000000000 1/10 Repeating in binary..
x - 1/10 = 0.00000000000000 via multiplication by 100000000
x - 1/2 = -0.40000000000000
x - 1/7 = -0.04285714285714
x - 1/10 = 0.00000000000000
3.14159000000000 Pi approx.
x - 314159/100000 = 0.00000000000000 via multiplication by 100000000
x - 16/5 = -0.05841000000000
x - 22/7 = -0.00126714285714
x - 355/113 = -0.00000292035398
x - 9563/3044 = -0.00000001314060
x - 85712/27283 = -0.00000000109959
x - 238010/75761 = -0.00000000013199
x - 314159/100000 = 0.00000000000000
3.14159265358979 Pi approximated better.
x - 62831853/20000000 = 0.00000000358979 via multiplication by 100000000
x - 16/5 = -0.05840734641021
x - 22/7 = -0.00126448926735
x - 355/113 = -0.00000026676419
x - 104348/33215 = -0.00000000033163
x - 312689/99532 = -0.00000000002914
x - 1146408/364913 = -0.00000000000161
x - 5419351/1725033 = -0.00000000000002
2.71828182845905 e: rational approximations aren't much use.
x - 271828183/100000000 = -0.00000000154095 via multiplication by 100000000
x - 3/1 = -0.28171817154095
x - 11/4 = -0.03171817154095
x - 49/18 = -0.00394039376318
x - 87/32 = -0.00046817154095
x - 193/71 = -0.00002803069588
x - 1457/536 = -0.00000175363051
x - 9620/3539 = -0.00000017210609
x - 23225/8544 = -0.00000000674695
x - 49171/18089 = -0.00000000027665
x - 566827/208524 = -0.00000000001154
x - 3820276/1405401 = -0.00000000000130
x - 11411657/4198114 = -0.00000000000012
10.1500000000000 Exact in decimal, recurring in binary.
x - 203/20 = 0.00000000000000 via multiplication by 100000000
x - 41/4 = -0.10000000000000
x - 132/13 = -0.00384615384615
x - 203/20 = 0.00000000000000
Variations on 67/74
0.905400000000000 67/74 = 0·9(054) repeating in base 10
x - 4527/5000 = 0.00000000000000 via multiplication by 100000000
x - 1/1 = -0.09460000000000
x - 9/10 = 0.00540000000000
x - 19/21 = 0.00063809523810
x - 67/74 = -0.00000540540541
x - 2029/2241 = 0.00000062472111
x - 4527/5000 = 0.00000000000000
0.905405400000000 Two repeats.
x - 4527027/5000000 = 0.00000000000000 via multiplication by 100000000
x - 1/1 = -0.09459460000000
x - 9/10 = 0.00540540000000
x - 19/21 = 0.00064349523810
x - 67/74 = -0.00000000540541
x - 2012029/2222241 = 0.00000000067562
x - 2228707/2461557 = 0.00000000008442
x - 2259125/2495153 = 0.00000000001050
x - 2263011/2499445 = 0.00000000000120
x - 2263480/2499963 = 0.00000000000008
x - 4527027/5000000 = 0.00000000000000
0.905405405400000 Three repeats.
x - 90540541/100000000 = -0.00000000460000 via multiplication by 100000000
x - 1/1 = -0.09459459460000
x - 9/10 = 0.00540540540000
x - 19/21 = 0.00064350063810
x - 67/74 = -0.00000000000541
Variations on 14/27
0.518000000000000 14/27 = 0·(518) repeating in decimal.
x - 259/500 = 0.00000000000000 via multiplication by 100000000
x - 1/1 = -0.48200000000000
x - 1/2 = 0.01800000000000
x - 13/25 = -0.00200000000000
x - 29/56 = 0.00014285714286
x - 72/139 = 0.00001438848921
x - 259/500 = 0.00000000000000
0.519000000000000 Rounded.
x - 519/1000 = 0.00000000000000 via multiplication by 100000000
x - 1/1 = -0.48100000000000
x - 1/2 = 0.01900000000000
x - 13/25 = -0.00100000000000
x - 41/79 = 0.00001265822785
x - 478/921 = -0.00000108577633
x - 519/1000 = 0.00000000000000
0.518518000000000 Two repeats, truncated.
x - 259259/500000 = 0.00000000000000 via multiplication by 100000000
x - 1/1 = -0.48148200000000
x - 1/2 = 0.01851800000000
x - 13/25 = -0.00148200000000
x - 14/27 = -0.00000051851852
x - 32929/63506 = 0.00000006468680
x - 36471/70337 = 0.00000000804697
x - 36975/71309 = 0.00000000086946
x - 37031/71417 = 0.00000000008401
x - 185183/357139 = 0.00000000000560
x - 259259/500000 = 0.00000000000000
0.518519000000000 Two repeats, rounded.
x - 518519/1000000 = 0.00000000000000 via multiplication by 100000000
x - 1/1 = -0.48148100000000
x - 1/2 = 0.01851900000000
x - 13/25 = -0.00148100000000
x - 14/27 = 0.00000048148148
x - 35461/68389 = -0.00000006008276
x - 39283/75760 = -0.00000000739176
x - 39815/76786 = -0.00000000085953
x - 39885/76921 = -0.00000000001300
x - 478634/923079 = 0.00000000000108
x - 518519/1000000 = 0.00000000000000
</pre>
=={{header|FreeBASIC}}==
<
'' (no error checking, limited to 64-bit signed math)
type number as longint
Line 564 ⟶ 1,265:
if n = "" then exit do
print n & ":", parserational(n)
loop</
=={{header|Frink}}==
<syntaxhighlight lang="frink">println[toRational[0.9054054]]
println[toRational[0.518518]]
println[toRational[0.75]]</syntaxhighlight>
{{out}}
<pre>
4527027/5000000 (exactly 0.9054054)
259259/500000 (exactly 0.518518)
3/4 (exactly 0.75)
</pre>
=={{header|Fōrmulæ}}==
{{FormulaeEntry|page=https://formulae.org/?script=examples/Convert_decimal_number_to_rational}}
'''Solution'''
=== Without repeating digits===
The Fōrmulæ '''Rationalize''' expression converts from decimal numberts to rational in lower terms.
[[File:Fōrmulæ - Convert decimal number to rational 01.png]]
[[File:Fōrmulæ - Convert decimal number to rational 02.png]]
[[File:Fōrmulæ - Convert decimal number to rational 03.png]]
[[File:Fōrmulæ - Convert decimal number to rational 04.png]]
=== With repeating digits ===
Fōrmulæ '''Rationalize''' expression can convert from decimal numbers with infinite number of repeating digits. The second arguments specifies the number of repeating digits.
[[File:Fōrmulæ - Convert decimal number to rational 05.png]]
[[File:Fōrmulæ - Convert decimal number to rational 06.png]]
In the following example, a conversion of the resulting rational back to decimal is provided in order to prove that it was correct:
[[File:Fōrmulæ - Convert decimal number to rational 07.png]]
[[File:Fōrmulæ - Convert decimal number to rational 08.png]]
[[File:Fōrmulæ - Convert decimal number to rational 09.png]]
[[File:Fōrmulæ - Convert decimal number to rational 10.png]]
[https://en.wikipedia.org/wiki/0.999... 0.999... is actually 1]:
[[File:Fōrmulæ - Convert decimal number to rational 11.png]]
[[File:Fōrmulæ - Convert decimal number to rational 12.png]]
=== Programatically ===
Even when rationalization expressions are intrinsic in Fōrmulæ, we can write explicit functions:
[[File:Fōrmulæ - Convert decimal number to rational 13.png]]
[[File:Fōrmulæ - Convert decimal number to rational 14.png]]
[[File:Fōrmulæ - Convert decimal number to rational 15.png]]
[[File:Fōrmulæ - Convert decimal number to rational 16.png]]
[[File:Fōrmulæ - Convert decimal number to rational 17.png]]
[[File:Fōrmulæ - Convert decimal number to rational 18.png]]
[[File:Fōrmulæ - Convert decimal number to rational 19.png]]
[[File:Fōrmulæ - Convert decimal number to rational 20.png]]
=={{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 (
Line 583 ⟶ 1,362:
}
}
}</
Output:
<pre>
Line 590 ⟶ 1,369:
0.75 = 3/4
</pre>
=={{header|Groovy}}==
'''Solution:'''<br>
Uses the ''Rational'' number class and ''RationalCategory'' helper class created in the solution to the [[Arithmetic/Rational#Groovy|Arithmetic/Rational]] task.
'''Test:'''
<syntaxhighlight lang="groovy">Number.metaClass.mixin RationalCategory
[
0.9054054, 0.518518, 0.75, Math.E, -0.423310825, Math.PI, 0.111111111111111111111111
].each{
printf "%30.27f %s\n", it, it as Rational
}</syntaxhighlight>
'''Output:'''
<pre> 0.905405400000000000000000000 4527027//5000000
0.518518000000000000000000000 259259//500000
0.750000000000000000000000000 3//4
2.718281828459045000000000000 6121026514868073//2251799813685248
-0.423310825000000000000000000 -16932433//40000000
3.141592653589793000000000000 884279719003555//281474976710656
0.111111111111111111111111000 111111111111111111111111//1000000000000000000000000</pre>
Clearly this solution does not make any assumptions or approximations regarding repeating decimals.
=={{header|Haskell}}==
Line 595 ⟶ 1,397:
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]
[4527027 % 5000000,259259 % 500000,3 % 4]</
=={{header|J}}==
J's <code>x:</code> built-in will find a rational number which "best matches" a floating point number.
<
127424481939351r140737488355328 866492568306r1671094481399 3r4</
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
Line 622 ⟶ 1,424:
67r74 14r27
x: 0.9054054054054054 0.5185185185185185
67r74 14r27</
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:
<
67r74 14r27</
(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.)
Line 633 ⟶ 1,435:
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
67r74 14r27 3r4</
=={{header|Java}}==
<syntaxhighlight lang="java">
double fractionToDecimal(String string) {
int indexOf = string.indexOf(' ');
int integer = 0;
int numerator, denominator;
if (indexOf != -1) {
integer = Integer.parseInt(string.substring(0, indexOf));
string = string.substring(indexOf + 1);
}
indexOf = string.indexOf('/');
numerator = Integer.parseInt(string.substring(0, indexOf));
denominator = Integer.parseInt(string.substring(indexOf + 1));
return integer + ((double) numerator / denominator);
}
String decimalToFraction(double value) {
String string = String.valueOf(value);
string = string.substring(string.indexOf('.') + 1);
int numerator = Integer.parseInt(string);
int denominator = (int) Math.pow(10, string.length());
int gcf = gcf(numerator, denominator);
if (gcf != 0) {
numerator /= gcf;
denominator /= gcf;
}
int integer = (int) value;
if (integer != 0)
return "%d %d/%d".formatted(integer, numerator, denominator);
return "%d/%d".formatted(numerator, denominator);
}
int gcf(int valueA, int valueB) {
if (valueB == 0) return valueA;
else return gcf(valueB, valueA % valueB);
}
</syntaxhighlight>
<pre>
67/74 = 0.9054054054054054
14/27 = 0.5185185185185185
0.9054054 = 4527027/5000000
0.518518 = 259259/500000
</pre>
<br />
<syntaxhighlight lang="java">import org.apache.commons.math3.fraction.BigFraction;
public class Test {
Line 652 ⟶ 1,497:
System.out.printf("%-12s : %s%n", d, new BigFraction(d, 0.00000002D, 10000));
}
}</
<pre>0.75 : 3 / 4
0.518518 : 37031 / 71417
Line 661 ⟶ 1,506:
-0.423310825 : -1253 / 2960
31.415926536 : 208696 / 6643</pre>
=={{header|JavaScript}}==
Deriving an approximation within a specified tolerance:
<syntaxhighlight lang="javascript">(() => {
"use strict";
// ---------------- APPROXIMATE RATIO ----------------
// approxRatio :: Real -> Real -> Ratio
const approxRatio = epsilon =>
n => {
const
c = gcdApprox(
0 < epsilon
? epsilon
: (1 / 10000)
)(1, n);
return Ratio(
Math.floor(n / c),
Math.floor(1 / c)
);
};
// gcdApprox :: Real -> (Real, Real) -> Real
const gcdApprox = epsilon =>
(x, y) => {
const _gcd = (a, b) =>
b < epsilon
? a
: _gcd(b, a % b);
return _gcd(Math.abs(x), Math.abs(y));
};
// ---------------------- TEST -----------------------
// main :: IO ()
const main = () =>
// Using a tolerance of 1/10000
[0.9054054, 0.518518, 0.75]
.map(
compose(
showRatio,
approxRatio(0.0001)
)
)
.join("\n");
// ---------------- GENERIC FUNCTIONS ----------------
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (...fs) =>
// A function defined by the right-to-left
// composition of all the functions in fs.
fs.reduce(
(f, g) => x => f(g(x)),
x => x
);
// Ratio :: Int -> Int -> Ratio
const Ratio = (n, d) => ({
type: "Ratio",
n,
d
});
// showRatio :: Ratio -> String
const showRatio = nd =>
`${nd.n.toString()}/${nd.d.toString()}`;
// MAIN ---
return main();
})();</syntaxhighlight>
{{Out}}
<pre>67/74
14/27
3/4</pre>
=={{header|jq}}==
'''Works with [[#jq|jq]]''' (*)
'''Works with gojq, the Go implementation of jq'''
The following definition of `number_to_r` can be used for JSON numbers in general, as illustrated by the examples.
This entry assumes the availability
of two functions (`r/2` and `power/1`) as defined in
the "rational" module
at [[Arithmetic/Rational#jq]], which also provides a definition
for `rpp/0`, a pretty-printer used in the examples.
(*) With the caveat that the C implementation of jq does not support unlimited-precision integer arithmetic.
<syntaxhighlight lang="jq"># include "rational"; # a reminder that r/2 and power/1 are required
# Input: any JSON number, not only a decimal
# Output: a rational, constructed using r/2
# Requires power/1 (to take advantage of gojq's support for integer arithmetic)
# and r/2 (for rational number constructor)
def number_to_r:
# input: a decimal string
# $in - either null or the original input
# $e - the integer exponent of the original number, or 0
def dtor($in; $e):
index(".") as $ix
| if $in and ($ix == null) then $in
else (if $ix then sub("[.]"; "") else . end | tonumber) as $n
| (if $ix then ((length - ($ix+1)) - $e) else - $e end) as $p
| if $p >= 0
then r( $n; 10|power($p))
else r( $n * (10|power(-$p)); 1)
end
end;
. as $in
| tostring
| if (test("[Ee]")|not) then dtor($in; 0)
else capture("^(?<s>[^eE]*)[Ee](?<e>.*$)")
| (.e | if length > 0 then tonumber else 0 end) as $e
| .s | dtor(null; $e)
end ;</syntaxhighlight>
'''Examples'''
<syntaxhighlight lang="jq">0.9054054,
0.518518,
0.75,
1e308
| "\(.) → \(number_to_r | rpp)"</syntaxhighlight>
{{out}}
<pre>
0.9054054 → 4527027 // 5000000
0.518518 → 259259 // 500000
0.75 → 3 // 4
1e+308 → 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 // 1
</pre>
=={{header|Julia}}==
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.
<
4527027//5000000
Line 675 ⟶ 1,661:
Since Julia by default uses its [[wp:Double-precision floating-point format#IEEE 754 double-precision binary floating-point format: binary64|Float64]] type to represent floating-point numbers, if enough decimal digits are provided (so that the difference between the floating-point representation of the resulting fraction and the original number is smaller than the machine epsilon) the smaller fraction is returned, which in this case is the exact result:
julia>
14//27
julia>
67//74
Here is the core algorithm written in Julia 1.0, without handling various data types and corner cases:
<syntaxhighlight lang="julia">function rat(x::AbstractFloat, tol::Real=eps(x))::Rational
p, q, pp, qq = copysign(1,x), 0, 0, 1
x, y = abs(x), 1.0
r, a = modf(x)
nt, t, tt = tol, 0.0, tol
while r > nt # convergents of the continued fraction: np//nq = (p*a + pp) // (q*a + qq)
np, nq = Int(a).*(p,q) .+ (pp,qq)
p, pp, q, qq = np, p, nq, q
x, y = y, r # instead of the inexact 1/r...
a, r = divrem(x,y)
t, tt = nt, t # maintain x = (p + (-1)^i * r) / q
nt = a*t+tt
end
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)
end</syntaxhighlight>
=={{header|Kotlin}}==
<syntaxhighlight lang="scala">// version 1.1.2
class Rational(val num: Long, val den: Long) {
override fun toString() = "$num/$den"
}
fun decimalToRational(d: Double): Rational {
val ds = d.toString().trimEnd('0').trimEnd('.')
val index = ds.indexOf('.')
if (index == -1) return Rational(ds.toLong(), 1L)
var num = ds.replace(".", "").toLong()
var den = 1L
for (n in 1..(ds.length - index - 1)) den *= 10L
while (num % 2L == 0L && den % 2L == 0L) {
num /= 2L
den /= 2L
}
while (num % 5L == 0L && den % 5L == 0L) {
num /= 5L
den /= 5L
}
return Rational(num, den)
}
fun main(args: Array<String>) {
val decimals = doubleArrayOf(0.9054054, 0.518518, 2.405308, .75, 0.0, -0.64, 123.0, -14.6)
for (decimal in decimals)
println("${decimal.toString().padEnd(9)} = ${decimalToRational(decimal)}")
}</syntaxhighlight>
{{out}}
<pre>
0.9054054 = 4527027/5000000
0.518518 = 259259/500000
2.405308 = 601327/250000
0.75 = 3/4
0.0 = 0/1
-0.64 = -16/25
123.0 = 123/1
-14.6 = -73/5
</pre>
=={{header|Liberty BASIC}}==
<syntaxhighlight lang="lb">
' Uses convention that one repeating sequence implies infinitely repeating sequence..
' Non-recurring fractions are limited to nd number of digits in nuerator & denominator
Line 757 ⟶ 1,809:
if length /2 =int( length /2) then if mid$( i$, 3, length /2) =mid$( i$, 3 +length /2, length /2) then check$ ="recurring"
end function
</syntaxhighlight>
<pre>
0.5 is non-recurring
Line 790 ⟶ 1,842:
END.
</pre>
=={{header|Lua}}==
Brute force, and why not?
<syntaxhighlight lang="lua">for _,v in ipairs({ 0.9054054, 0.518518, 0.75, math.pi }) do
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))
end</syntaxhighlight>
{{out}}
<pre>0.9054054000000 --> 4527027 / 5000000
0.5185180000000 --> 259259 / 500000
0.7500000000000 --> 3 / 4
3.1415926535898 --> 31415926 / 10000000</pre>
=={{header|M2000 Interpreter}}==
<syntaxhighlight lang="m2000 interpreter">
module Convert_decimal_number_to_rational{
Function Rational(numerator as decimal, denominator as decimal=1) {
if denominator==0 then denominator=1
while frac(numerator)<>0 {
numerator*=10@
denominator*=10@
}
sgn=Sgn(numerator)*Sgn(denominator)
denominator<=abs(denominator)
numerator<=abs(numerator)*sgn
gcd1=lambda (a as decimal, b as decimal) -> {
if a<b then swap a,b
g=a mod b
while g {a=b:b=g: g=a mod b}
=abs(b)
}
gdcval=gcd1(abs(numerator), denominator)
if gdcval<denominator and gdcval<>0 then
denominator/=gdcval
numerator/=gdcval
end if
=(numerator,denominator)
}
Print Rational(0.9054054)#str$(" / ")="4527027 / 5000000" ' true
Print Rational(0.518518)#str$(" / ")="259259 / 500000" ' true
Print Rational(0.75)#str$(" / ")="3 / 4" ' true
}
Convert_decimal_number_to_rational
</syntaxhighlight>
=={{header|Maple}}==
<syntaxhighlight lang="maple">
> map( convert, [ 0.9054054, 0.518518, 0.75 ], 'rational', 'exact' );
4527027 259259
[-------, ------, 3/4]
5000000 500000
</syntaxhighlight>
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<
-> {4527027/5000000,259259/500000,3/4}</
=={{header|MATLAB}} / {{header|Octave}}==
<syntaxhighlight lang="matlab">
[a,b]=rat(.75)
[a,b]=rat(.518518)
[a,b]=rat(.9054054)
</syntaxhighlight>
Output:
Line 824 ⟶ 1,923:
=={{header|МК-61/52}}==
{{improve|МК-61/52|This example is not clear, please improve it or delete it!}}
<syntaxhighlight lang="text">П0 П1 ИП1 {x} x#0 14 КИП4 ИП1 1 0
* П1 БП 02 ИП4 10^x П0 ПA ИП1 ПB
ИПA ИПB / П9 КИП9 ИПA ИПB ПA ИП9 *
- ПB x=0 20 ИПA ИП0 ИПA / П0 ИП1
ИПA / П1 ИП0 ИП1 С/П</
=={{header|NetRexx}}==
Now the nearly equivalent program.
<
/*NetRexx program to convert decimal numbers to fractions *************
* 16.08.2012 Walter Pachl derived from Rexx Version 2
Line 894 ⟶ 1,995:
end
If den=1 Then Return nom /* an integer */
Else Return nom'/'den /* otherwise a fraction */</
Output is the same as
=={{header|Nim}}==
Similar to the Rust and Julia implementations.
<syntaxhighlight lang="nim">import math
import fenv
type
Rational = object
numerator: int
denominator: int
proc `$`(self: Rational): string =
if self.denominator == 1:
$self.numerator
else:
$self.numerator & "//" & $self.denominator
func rationalize(x: float, tol: float = epsilon(float)): Rational =
var xx = x
let flagNeg = xx < 0.0
if flagNeg:
xx = -xx
if xx < minimumPositiveValue(float):
return Rational(numerator: 0, denominator: 1)
if abs(xx - round(xx)) < tol:
return Rational(numerator: int(round(xx)), denominator: 1)
var a = 0
var b = 1
var c = int(ceil(xx))
var d = 1
var aux1 = high(int) div 2
while c < aux1 and d < aux1:
var aux2 = (float(a) + float(c)) / (float(b) + float(d))
if abs(xx - aux2) < tol:
break
if xx > aux2:
inc a, c
inc b, d
else:
inc c, a
inc d, b
var gcd = gcd(a + c, b + d)
if flagNeg:
Rational(numerator: -(a + c) div gcd, denominator: (b + d) div gcd)
else:
Rational(numerator: (a + c) div gcd, denominator: (b + d) div gcd)
echo rationalize(0.9054054054)
echo rationalize(0.9054054054, 0.0001)
echo rationalize(0.5185185185)
echo rationalize(0.5185185185, 0.0001)
echo rationalize(0.75)
echo rationalize(0.1428571428, 0.001)
echo rationalize(35.000)
echo rationalize(35.001)
echo rationalize(0.9)
echo rationalize(0.99)
echo rationalize(0.909)
echo rationalize(0.909, 0.001)</syntaxhighlight>
{{out}}
<pre>
2263456195//2499936693
67//74
1037026019//1999978751
14//27
3//4
1//7
35
35001//1000
9//10
99//100
909//1000
10//11
</pre>
Nim also has a rationals library for this, though it does not allow you to set tolerances like the code above.
<syntaxhighlight lang="nim">import rationals
echo toRational(0.9054054054)
echo toRational(0.5185185185)
echo toRational(0.75)
echo toRational(0.1428571428)
echo toRational(35.000)
echo toRational(35.001)
echo toRational(0.9)
echo toRational(0.99)
echo toRational(0.909)</syntaxhighlight>
{{out}}
<pre>
67/74
1037036407/1999998785
3/4
1/7
35/1
35001/1000
9/10
99/100
909/1000
</pre>
=={{header|Ol}}==
Any number in Ol by default is exact. It means that any number automatically converted into rational form. If you want to use real floating point numbers, use "#i" prefix.
Function `exact` creates exact number (possibly rational) from any inexact.
<syntaxhighlight lang="scheme">
(print (exact #i0.9054054054))
(print (exact #i0.5185185185))
(print (exact #i0.75))
(print (exact #i35.000))
(print (exact #i35.001))
(print (exact #i0.9))
(print (exact #i0.99))
(print (exact #i0.909))
</syntaxhighlight>
{{out}}
<pre>
4077583446378673/4503599627370496
4670399613402603/9007199254740992
3/4
35
4925952829924835/140737488355328
8106479329266893/9007199254740992
4458563631096791/4503599627370496
4093772061279781/4503599627370496
</pre>
=={{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.
Line 909 ⟶ 2,138:
=={{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;
Line 1,037 ⟶ 2,266:
for (map { 10 ** $_ } 1 .. 10) {
printf "approx below %g: %d / %d\n", $_, best_approx($x, $_)
}</
Output: <pre>3/8 = 0.375:
machine: 3/8
Line 1,072 ⟶ 2,301:
approx below 1e+10: -9978066541 / 3176117225</pre>
=={{header|
<!--<syntaxhighlight lang="phix">(phixonline)-->
<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: #004080;">integer</span> <span style="color: #000000;">nom</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">denom</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #7060A8;">assert</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">"0."</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">3</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">ch</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #7060A8;">assert</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ch</span><span style="color: #0000FF;">>=</span><span style="color: #008000;">'0'</span> <span style="color: #008080;">and</span> <span style="color: #000000;">ch</span><span style="color: #0000FF;"><=</span><span style="color: #008000;">'9'</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">nom</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">nom</span><span style="color: #0000FF;">*</span><span style="color: #000000;">10</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">ch</span><span style="color: #0000FF;">-</span><span style="color: #008000;">'0'</span>
<span style="color: #000000;">denom</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">10</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">sq_div</span><span style="color: #0000FF;">({</span><span style="color: #000000;">nom</span><span style="color: #0000FF;">,</span><span style="color: #000000;">denom</span><span style="color: #0000FF;">},</span><span style="color: #7060A8;">gcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nom</span><span style="color: #0000FF;">,</span><span style="color: #000000;">denom</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #0000FF;">?</span><span style="color: #000000;">decrat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"0.9054054"</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>
<!--</syntaxhighlight>-->
{{out}}
<pre>
{4527027,5000000}
{259259,500000}
{3,4}
</pre>
=={{header|PHP}}==
{{works with|PHP|5.3+}}
<
{
if ($val == (int) $val) {
Line 1,140 ⟶ 2,361:
echo asRational(1/4)."\n"; // "1/4"
echo asRational(1/3)."\n"; // "1/3"
echo asRational(5)."\n"; // "5"</
=={{header|PL/I}}==
<syntaxhighlight lang="text">(size, fofl):
Convert_Decimal_To_Rational: procedure options (main); /* 14 January 2014, from Ada */
Line 1,196 ⟶ 2,417:
end;
end;
end Convert_Decimal_To_Rational;</
Output:
<pre>
Line 1,213 ⟶ 2,434:
=={{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
Line 1,236 ⟶ 2,457:
DataSection
Data.d 0.9054054,0.518518,0.75,0.0
EndDataSection</
<pre>
0.9054054 -> 4527027/5000000
Line 1,248 ⟶ 2,469:
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))
Line 1,259 ⟶ 2,480:
0.518518 259259/500000
0.75 3/4
>>> </
Or, writing our own '''approxRatio''' function:
{{Works with|Python|3.7}}
<syntaxhighlight lang="python">'''Approximate rationals from decimals'''
from math import (floor, gcd)
import sys
# approxRatio :: Float -> Float -> Ratio
def approxRatio(epsilon):
'''The simplest rational approximation to
n within the margin given by epsilon.
'''
def gcde(e, x, y):
def _gcd(a, b):
return a if b < e else _gcd(b, a % b)
return _gcd(abs(x), abs(y))
return lambda n: (lambda c=(
gcde(epsilon if 0 < epsilon else (0.0001), 1, n)
): ratio(floor(n / c))(floor(1 / c)))()
# main :: IO ()
def main():
'''Conversions at different levels of precision.'''
xs = [0.9054054, 0.518518, 0.75]
print(
fTable(__doc__ + ' (epsilon of 1/10000):\n')(str)(
lambda r: showRatio(r) + ' -> ' + repr(fromRatio(r))
)(
approxRatio(1 / 10000)
)(xs)
)
print('\n')
e = minBound(float)
print(
fTable(__doc__ + ' (epsilon of ' + repr(e) + '):\n')(str)(
lambda r: showRatio(r) + ' -> ' + repr(fromRatio(r))
)(
approxRatio(e)
)(xs)
)
# GENERIC -------------------------------------------------
# fromRatio :: Ratio Int -> Float
def fromRatio(r):
'''A floating point value derived from a
a rational value.
'''
return r.get('numerator') / r.get('denominator')
# minBound :: Bounded Type -> a
def minBound(t):
'''Minimum value for a bounded type.'''
maxsize = sys.maxsize
float_infomin = sys.float_info.min
return {
int: (-maxsize - 1),
float: float_infomin,
bool: False,
str: chr(0)
}[t]
# ratio :: Int -> Int -> Ratio Int
def ratio(n):
'''Rational value constructed
from a numerator and a denominator.
'''
def go(n, d):
g = gcd(n, d)
return {
'type': 'Ratio',
'numerator': n // g, 'denominator': d // g
}
return lambda d: go(n * signum(d), abs(d))
# showRatio :: Ratio -> String
def showRatio(r):
'''String representation of the ratio r.'''
d = r.get('denominator')
return str(r.get('numerator')) + (
' / ' + str(d) if 1 != d else ''
)
# signum :: Num -> Num
def signum(n):
'''The sign of n.'''
return -1 if 0 > n else (1 if 0 < n else 0)
# DISPLAY -------------------------------------------------
# fTable :: String -> (a -> String) ->
# (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
'''Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
'''
def go(xShow, fxShow, f, xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
return s + '\n' + '\n'.join(map(
lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
xs, ys
))
return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
# MAIN ---
if __name__ == '__main__':
main()</syntaxhighlight>
{{Out}}
<pre>Approximate rationals from decimals (epsilon of 1/10000):
0.9054054 -> 67 / 74 -> 0.9054054054054054
0.518518 -> 14 / 27 -> 0.5185185185185185
0.75 -> 3 / 4 -> 0.75
Approximate rationals from decimals (epsilon of 2.2250738585072014e-308):
0.9054054 -> 4077583422059235 / 4503599627370496 -> 0.9054054
0.518518 -> 2335197471584895 / 4503599627370496 -> 0.518518
0.75 -> 3 / 4 -> 0.75</pre>
=={{header|R}}==
<syntaxhighlight lang="r">
ratio<-function(decimal){
denominator=1
while(nchar(decimal*denominator)!=nchar(round(decimal*denominator))){
denominator=denominator+1
}
str=paste(decimal*denominator,"/",sep="")
str=paste(str,denominator,sep="")
return(str)
}
</syntaxhighlight>
{{Out}}
<pre>
> ratio(.75)
[1] "3/4"
> ratio(0.9054054)
[1] "4527027/5e+06"
> ratio(3.14)
[1] "157/50"
</pre>
=={{header|Quackery}}==
Using the Quackery big number rational arithmetic library <code>bigrat.qky</code>.
<syntaxhighlight lang="quackery">[ $ "bigrat.qky" loadfile ] now!
[ dup echo$
say " is "
$->v drop
dup 100 > if
[ say "approximately "
proper 100 round
improper ]
vulgar$ echo$
say "." cr ] is task ( $ --> )
$ "0.9054054 0.518518 0.75" nest$ witheach task</syntaxhighlight>
{{out}}
<pre>0.9054054 is approximately 67/74.
0.518518 is approximately 14/27.
0.75 is 3/4.</pre>
=={{header|Racket}}==
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
Line 1,270 ⟶ 2,673:
(exact->inexact 67/74) ; -> 0.9054054054054054
(inexact->exact 0.9054054054054054) ;-> 8155166892806033/9007199254740992</
=={{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:
<syntaxhighlight lang="raku" line>say .nude.join('/') for 0.9054054, 0.518518, 0.75;</syntaxhighlight>
{{out}}
<pre>4527027/5000000
259259/500000
3/4</pre>
However, if we want to take repeating decimals into account, then we can get a bit fancier.
<syntaxhighlight lang="raku" line>sub decimal_to_fraction ( Str $n, Int $rep_digits = 0 ) returns Str {
my ( $int, $dec ) = ( $n ~~ /^ (\d+) \. (\d+) $/ )».Str or die;
my ( $numer, $denom ) = ( $dec, 10 ** $dec.chars );
if $rep_digits {
my $to_move = $dec.chars - $rep_digits;
$numer -= $dec.substr(0, $to_move);
$denom -= 10 ** $to_move;
}
my $rat = Rat.new( $numer.Int, $denom.Int ).nude.join('/');
return $int > 0 ?? "$int $rat" !! $rat;
}
my @a = ['0.9054', 3], ['0.518', 3], ['0.75', 0], | (^4).map({['12.34567', $_]});
for @a -> [ $n, $d ] {
say "$n with $d repeating digits = ", decimal_to_fraction( $n, $d );
}</syntaxhighlight>
{{out}}
<pre>0.9054 with 3 repeating digits = 67/74
0.518 with 3 repeating digits = 14/27
0.75 with 0 repeating digits = 3/4
12.34567 with 0 repeating digits = 12 34567/100000
12.34567 with 1 repeating digits = 12 31111/90000
12.34567 with 2 repeating digits = 12 17111/49500
12.34567 with 3 repeating digits = 12 1279/3700</pre>
=={{header|REXX}}==
===version 1===
This REXX example supports almost any form of numeric input, some examples are:
::* ±nnn
::* ±nnn<b>.
::* ±nnn<b>.</
::* ±
::* ±nnnE±ppp
::* <b>.</b>fffE±ppp
::* ±nnn<b>.</b>fffE±ppp (with an uppercase exponent signifier)
::* ±nnn<b>.</b>fffe±ppp (with an lowercase exponent signifier)
::* numeratorNumber/denominatorNumber
::* denominator is optional (but if a <big>'''/'''</big> is used, it must be present)
::* superfluous blanks are permitted (for whitespace)
::* leading zeroes are permitted
::* leading signs are permitted
::* improper fractions are permitted
─── where:
:* '''nnn'''
:* '''fff''' represent decimal digits after the decimal point (if there is one)
:* '''ppp''' represent decimal digits of the (base ten) exponent
:* '''<big>±</big>''' represent an optional (leading) sign, either '''+''' or '''-'''
:* '''<b>.</b>''' if it's trailing, the decimal point is optional
<br>REXX can support almost any number of decimal digits, but '''10''' was chosen for practicality for this task.
<syntaxhighlight lang="rexx">/*REXX program converts a rational fraction [n/m] (or nnn.ddd) to it's lowest terms.*/
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.*/
if n.1='' then call er 'no argument specified.'
do j=1 for 2; if \datatype(n.j, 'N') then call er "argument isn't numeric:" n.j
end /*j*/ /* [↑] validate arguments: n.1 n.2 */
if n.2=0 then call
say 'old =' space(orig)
say 'new =' rat(n.1/n.2)
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
er: say; say '***error
/*──────────────────────────────────────────────────────────────────────────────────────*/
rat: procedure; parse arg x 1 _x,y; if y=='' then y = 10**(digits()-1)
b=0; g=0; a=1; h=1 /* [↑] Y is the tolerance.*/
do while a<=y & g<=y;
_=a; a=n*a+b; b=_
_=g; g=n*g+h; h=_
Line 1,309 ⟶ 2,761:
end
_x=1/(_x-n)
end /*while
if h==1 then return b /*don't
return b'/'h /*
'''output''' when using various inputs (which are displayed as part of the output):
<br>(Multiple runs are shown, outputs are separated by a blank line.)
Line 1,335 ⟶ 2,787:
===version 2===
<
* 15.08.2012 Walter Pachl derived from above for readability
* It took me time to understand :-) I need descriptive variable names
Line 1,404 ⟶ 2,856:
if den=1 then return nom /* denominator 1: integer */
return nom'/'den /* otherwise a fraction */
</syntaxhighlight>
Output:
<pre>
Line 1,433 ⟶ 2,885:
===version 3===
<
* 13.02.2014 Walter Pachl
* specify the number as xxx.yyy(pqr) pqr is the period
Line 1,485 ⟶ 2,937:
tag='should be' soll
say zin '=' fract tag
Return
GCD: procedure
Line 1,494 ⟶ 2,945:
Parse Arg a,b
if b = 0 then return abs(a)
return GCD(b,a//b)</
'''Output:'''
<pre>5.55555 = 111111/20000 ok
Line 1,511 ⟶ 2,962:
0.00000000001 = 1/100000000000 ok
0.000001000001 = 1000001/1000000000000 ok</pre>
=={{header|RPL}}==
Starting with HP-48 versions, the conversion can be performed directly with the <code>→Q</code> instruction. Earlier versions must instead use the following code, which is based on continued fractions.
{{works with|Halcyon Calc|4.2.8}}
{| class="wikitable"
! RPL code
! Comment
|-
|
≪
ABS LAST SIGN 1E6 → sign dmax
≪ (0,1) (1,0)
'''DO'''
SWAP OVER 4 ROLL INV IP
LAST FP 5 ROLLD
* +
'''UNTIL''' DUP2 IM SWAP IM * dmax >
4 PICK dmax INV < OR '''END'''
ROT ROT DROP2
"'" OVER RE sign * →STR + "/" +
SWAP IM →STR + STR→
≫ ≫ '<span style="color:blue">→PQ</span>' STO
|
<span style="color:blue">'''→PQ'''</span> ''( x → 'p/q' ) ''
store sign(x) and max denominator
a(0) = x ; hk(-2) = (0,1) ; hk(-1) = (1,0)
loop
reorder int(a(n)), hk(n-1) and hk(n-2) in stack
a(n+1)=1/frac(a(n)) back to top of stack
hk(n) = a(n)*hk(n-1) + hk(n-2)
until k(n)*(kn-1) > max denominator or
a(n+1) > max denominator
clean stack
convert a(n) from (p,q) to 'p/q' format
return 'p/q'
|}
.518518 <span style="color:blue">'''→PQ'''</span>
.905405405405 <span style="color:blue">'''→PQ'''</span>
-3.875 <span style="color:blue">'''→PQ'''</span>
'''Output:'''
<span style="color:grey"> 3:</span> '37031/71417'
<span style="color:grey"> 2:</span> '67/74'
<span style="color:grey"> 1:</span> '-31/8'
=={{header|Ruby}}==
Line 1,517 ⟶ 3,012:
This converts the string representation of the given values directly to fractions.
<
0.9054054 4527027/5000000
0.518518 259259/500000
0.75 3/4
=> ["0.9054054", "0.518518", "0.75"]</
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
</syntaxhighlight>
{{works with|Ruby|2.1.0+}}
A suffix for integer and float literals was introduced:
Line 1,541 ⟶ 3,035:
=> (3/4)
</pre>
=={{header|Rust}}==
<syntaxhighlight lang="text">
extern crate rand;
extern crate num;
use num::Integer;
use rand::Rng;
fn decimal_to_rational (mut n : f64) -> [isize;2] {
//Based on Farey sequences
assert!(n.is_finite());
let flag_neg = n < 0.0;
if flag_neg { n = n*(-1.0) }
if n < std::f64::MIN_POSITIVE { return [0,1] }
if (n - n.round()).abs() < std::f64::EPSILON { return [n.round() as isize, 1] }
let mut a : isize = 0;
let mut b : isize = 1;
let mut c : isize = n.ceil() as isize;
let mut d : isize = 1;
let aux1 = isize::max_value()/2;
while c < aux1 && d < aux1 {
let aux2 : f64 = (a as f64 + c as f64)/(b as f64 + d as f64);
if (n - aux2).abs() < std::f64::EPSILON { break }
if n > aux2 {
a = a + c;
b = b + d;
} else {
c = a + c;
d = b + d;
}
}
// Make sure that the fraction is irreducible
let gcd = (a+c).gcd(&(b+d));
if flag_neg { [-(a + c)/gcd, (b + d)/gcd] } else { [(a + c)/gcd, (b + d)/gcd] }
}
#[test]
fn test1 () {
// Test the function with 1_000_000 random decimal numbers
let mut rng = rand::thread_rng();
for _i in 1..1_000_000 {
let number = rng.gen::<f64>();
let result = decimal_to_rational(number);
assert!((number - (result[0] as f64)/(result[1] as f64)).abs() < std::f64::EPSILON);
assert!(result[0].gcd(&result[1]) == 1);
}
}
fn main () {
let mut rng = rand::thread_rng();
for _i in 1..10 {
let number = rng.gen::<f64>();
let result = decimal_to_rational(number);
if result[1] == 1 { println!("{} -> {}", number, result[0]) } else { println!("{} -> {}/{}", number, result[0], result[1]) }
}
for i in [-0.9054054, 0.518518, -0.75, 0.5185185185185185, -0.9054054054054054, 0.0, 1.0, 2.0].iter() {
let result = decimal_to_rational(*i as f64);
if result[1] == 1 { println!("{} = {}",*i, result[0]) } else { println!("{} = {}/{}", *i, result[0], result[1]) }
}
}
</syntaxhighlight>
First test the function with 1_000_000 random double floats :
<pre>
running 1 test
test test1 ... ok
test result: ok. 1 passed; 0 failed; 0 ignored; 0 measured; 0 filtered out
</pre>
Now run it with 10 random double floats and some selected ones printing the results
<pre>
0.47783621261626297 -> 36036136/75415247
0.29135687639237284 -> 164756020/565478399
0.04962905490905656 -> 4105111/82715881
0.33418703921783965 -> 16612678/49710719
0.07284921759788943 -> 6252865/85832974
0.783712619202954 -> 62096920/79234299
0.3788902482801324 -> 30401287/80237713
0.03780115715370047 -> 1522201/40268635
0.9975233883406127 -> 79858287/80056556
-0.9054054 -> -4527027/5000000
0.518518 -> 259259/500000
-0.75 -> -3/4
0.5185185185185185 -> 14/27
-0.9054054054054054 -> -67/74
0 -> 0
1 -> 1
2 -> 2
</pre>
=={{header|Scala}}==
{{Out}}Best seen running in your browser [https://scastie.scala-lang.org/rrlFnuTURgirBiTsH3Kqrg Scastie (remote JVM)].
<syntaxhighlight lang="scala">import org.apache.commons.math3.fraction.BigFraction
object Number2Fraction extends App {
val n = Array(0.750000000, 0.518518000, 0.905405400,
0.142857143, 3.141592654, 2.718281828, -0.423310825, 31.415926536)
for (d <- n)
println(f"$d%-12s : ${new BigFraction(d, 0.00000002D, 10000)}%s")
}</syntaxhighlight>
=={{header|Seed7}}==
The library [http://seed7.sourceforge.net/libraries/bigrat.htm bigrat.s7i]
defines the
which creates a [https://seed7.sourceforge.net/manual/types.htm#bigRational bigRational] number from a string.
This function accepts, besides fractions, also a decimal number with repeating decimals.
Internally a bigRational uses numerator and denominator to represent a rational number.
Writing a bigRational does not create a fraction with numerator and denominator, but a decimal number with possibly repeating decimals.
The function [https://seed7.sourceforge.net/libraries/bigrat.htm#fraction(in_bigRational) fraction]
uses numerator and denominator from the bigRational number to get a string with the fraction.
<syntaxhighlight lang="seed7">$ include "seed7_05.s7i";
include "bigrat.s7i";
Line 1,565 ⟶ 3,163:
writeln(bigRational parse "31.415926536");
writeln(bigRational parse "0.000000000");
writeln;
writeln(fraction(bigRational("0.9(054)")));
writeln(fraction(bigRational("0.(518)")));
writeln(fraction(bigRational("0.75")));
writeln(fraction(bigRational("3.(142857)")));
writeln(fraction(bigRational("0.(8867924528301)")));
writeln(fraction(bigRational("0.(846153)")));
writeln(fraction(bigRational("0.9054054")));
writeln(fraction(bigRational("0.518518")));
writeln(fraction(bigRational("0.14285714285714")));
writeln(fraction(bigRational("3.14159265358979")));
writeln(fraction(bigRational("2.718281828")));
writeln(fraction(bigRational("31.415926536")));
writeln(fraction(bigRational("0.000000000")));
end func;</syntaxhighlight>
{{out}}
<pre>
0.9(054)
0.(518)
0.75
3.(142857)
0.(8867924528301)
0.(846153)
0.9054054
0.518518
0.14285714285714
3.14159265358979
2.718281828
31.415926536
0.0
67/74
14/27
Line 1,585 ⟶ 3,212:
=={{header|Sidef}}==
By default, literal numbers are represented in rational form:
<syntaxhighlight lang="ruby">say 0.75.as_frac #=> 3/4
say 0.518518.as_frac #=> 259259/500000
say 0.9054054.as_frac #=> 4527027/5000000</syntaxhighlight>
Additionally, '''Num(str)''' can be used for parsing a decimal expansion into rational form:
<
say
}</syntaxhighlight>
{{out}}
Line 1,601 ⟶ 3,228:
3/4
</pre>
For rational approximations, the Number '''.rat_approx''' method can be used:
<syntaxhighlight lang="ruby">say 0.518518.rat_approx.as_frac #=> 14/27
say 0.9054054.rat_approx.as_frac #=> 67/74</syntaxhighlight>
=={{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}} {
Line 1,626 ⟶ 3,257:
}
}
}</
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
dbl2frac 0.518518 -> 14 27, expected 42 81
~ $
=={{header|TI SR-56}}==
{| class="wikitable"
|+ Texas Instruments SR-56 Program Listing for "Convert decimal number to rational"
|-
! Display !! Key !! Display !! Key !! Display !! Key !! Display !! Key
|-
| 00 33 || STO || 25 00 || 0 || 50 || || 75 ||
|-
| 01 00 || 0 || 26 64 || * || 51 || || 76 ||
|-
| 02 00 || 0 || 27 34 || RCL || 52 || || 77 ||
|-
| 03 33 || STO || 28 01 || 1 || 53 || || 78 ||
|-
| 04 01 || 1 || 29 94 || = || 54 || || 79 ||
|-
| 05 56 || *CP || 30 41 || R/S || 55 || || 80 ||
|-
| 06 01 || 1 || 31 || || 56 || || 81 ||
|-
| 07 35 || SUM || 32 || || 57 || || 82 ||
|-
| 08 01 || 1 || 33 || || 58 || || 83 ||
|-
| 09 34 || RCL || 34 || || 59 || || 84 ||
|-
| 10 01 || 1 || 35 || || 60 || || 85 ||
|-
| 11 64 || * || 36 || || 61 || || 86 ||
|-
| 12 34 || RCL || 37 || || 62 || || 87 ||
|-
| 13 00 || 0 || 38 || || 63 || || 88 ||
|-
| 14 94 || = || 39 || || 64 || || 89 ||
|-
| 15 12 || Inv || 40 || || 65 || || 90 ||
|-
| 16 29 || *Int || 41 || || 66 || || 91 ||
|-
| 17 12 || Inv || 42 || || 67 || || 92 ||
|-
| 18 37 || *x=t || 43 || || 68 || || 93 ||
|-
| 19 00 || 0 || 44 || || 69 || || 94 ||
|-
| 20 06 || 6 || 45 || || 70 || || 95 ||
|-
| 21 34 || RCL || 46 || || 71 || || 96 ||
|-
| 22 01 || 1 || 47 || || 72 || || 97 ||
|-
| 23 32 || x<>t || 48 || || 73 || || 98 ||
|-
| 24 34 || RCL || 49 || || 74 || || 99 ||
|}
Asterisk denotes 2nd function key.
{| class="wikitable"
|+ Register allocation
|-
| 0: Decimal || 1: Denominator|| 2: Unused || 3: Unused || 4: Unused
|-
| 5: Unused || 6: Unused || 7: Unused || 8: Unused || 9: Unused
|}
Annotated listing:
<syntaxhighlight lang="text">
STO 0 // Decimal := User Input
0 STO 1 // Denominator := 0
*CP // RegT := 0
1 SUM 1 // Denominator += 1
RCL 1 * RCL 0 = Inv *Int // Find fractional part of Decimal * Denominator
Inv *x=t 0 6 // If it is nonzero loop back to instruction 6
RCL 1 x<>t // Report denominator
RCL 0 * RCL 1 = // Report numerator
R/S // End
</syntaxhighlight>
'''Usage:'''
Enter the decimal and then press RST R/S. Once a suitable rational is found, the numerator will be displayed. Press x<>t to toggle between the numerator and denominator.
{{in}}
0.875 RST R/S
{{out}}
After about five seconds, '''3''' will appear on the screen. Press x<>t and '''8''' will appear on the screen. The fraction is 3/8.
Any fraction with denominator <=100 is found in 1 minute or less. 100 denominators are checked per minute.
=={{header|Vala}}==
{{trans|C}}
<syntaxhighlight lang="vala">struct Fraction {
public long d;
public long n;
}
Fraction rat_approx(double f, long md) {
long a;
long[] h = {0, 1, 0};
long[] k = {1, 0, 0};
long x, d, n = 1;
bool neg = false;
if (md <= 1) return {1, (long)f};
if (f < 0) {
neg = true;
f = -f;
}
while (f != Math.floor(f)) {
n <<= 1;
f *= 2;
}
d = (long)f;
for (int i = 0; i < 64; i++) {
a = (n != 0) ? d / n : 0;
if (i != 0 && a == 0) break;
x = d; d = n; n = x %n;
x = a;
if (k[1] * a + k[0] >= md) {
x = (md - k[0]) / k[1];
if (x * 2 >= a || k[1] >= md)
i = 65;
else
break;
}
h[2] = x * h[1] + h[0]; h[0] = h[1]; h[1] = h[2];
k[2] = x * k[1] + k[0]; k[0] = k[1]; k[1] = k[2];
}
return {k[1], neg ? -h[1] : h[1]};
}
void main() {
double f;
print("f = %16.14f\n", f = 1.0/7);
for (int i = 1; i < 20000000; i *= 16) {
print("denom <= %11d: ", i);
var r = rat_approx(f, i);
print("%11ld/%ld\n", r.n, r.d);
}
print("f = %16.14f\n", f = Math.atan2(1,1) * 4);
for (int i = 1; i < 20000000; i *= 16) {
print("denom <= %11d: ", i);
var r = rat_approx(f, i);
print("%11ld/%ld\n", r.n, r.d);
}
}</syntaxhighlight>
{{out}}
<pre>
f = 0.14285714285714
denom <= 1: 0/1
denom <= 16: 1/7
denom <= 256: 1/7
denom <= 4096: 1/7
denom <= 65536: 1/7
denom <= 1048576: 1/7
denom <= 16777216: 1/7
f = 3.14159265358979
denom <= 1: 3/1
denom <= 16: 22/7
denom <= 256: 355/113
denom <= 4096: 355/113
denom <= 65536: 104348/33215
denom <= 1048576: 3126535/995207
denom <= 16777216: 47627751/15160384
</pre>
=={{header|VBA}}==
{{trans|D}}
<syntaxhighlight lang="vb">Function Real2Rational(r As Double, bound As Long) As String
If r = 0 Then
Real2Rational = "0/1"
ElseIf r < 0 Then
Result = Real2Rational(-r, bound)
Real2Rational = "-" & Result
Else
best = 1
bestError = 1E+99
For i = 1 To bound + 1
currentError = Abs(i * r - Round(i * r))
If currentError < bestError Then
best = i
bestError = currentError
If bestError < 1 / bound Then GoTo SkipLoop
End If
Next i
SkipLoop:
Real2Rational = Round(best * r) & "/" & best
End If
End Function
Sub TestReal2Rational()
Debug.Print "0.75" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(0.75, CLng(Order));
Next i
Debug.Print
Debug.Print "0.518518" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(0.518518, CLng(Order));
Next i
Debug.Print
Debug.Print "0.9054054" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(0.9054054, CLng(Order));
Next i
Debug.Print
Debug.Print "0.142857143" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(0.142857143, CLng(Order));
Next i
Debug.Print
Debug.Print "3.141592654" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(3.141592654, CLng(Order));
Next i
Debug.Print
Debug.Print "2.718281828" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(2.718281828, CLng(Order));
Next i
Debug.Print
Debug.Print "-0.423310825" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(-0.423310825, CLng(Order));
Next i
Debug.Print
Debug.Print "31.415926536" & ":";
For i = 0 To 5
Order = CDbl(10) ^ CDbl(i)
Debug.Print " " & Real2Rational(31.415926536, CLng(Order));
Next i
End Sub
</syntaxhighlight>
{{out}}
<pre>
0.75: 1/1 3/4 3/4 3/4 3/4 3/4
0.518518: 1/1 1/2 14/27 14/27 14/27 37031/71417
0.9054054: 1/1 1/1 67/74 67/74 67/74 67/74
0.142857143: 0/1 1/7 1/7 1/7 1/7 1/7
3.141592654: 3/1 22/7 22/7 355/113 355/113 104348/33215
2.718281828: 3/1 19/7 193/71 1457/536 23225/8544 173459/63812
-0.423310825: -0/1 -3/7 -11/26 -69/163 -1253/2960 -10093/23843
31.415926536: 31/1 157/5 377/12 3550/113 208696/6643 1563445/49766
</pre>
=={{header|Wren}}==
{{libheader|Wren-rat}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./rat" for Rat
import "./fmt" for Fmt
var tests = [0.9054054, 0.518518, 0.75]
for (test in tests) {
var r = Rat.fromFloat(test)
System.print("%(Fmt.s(-9, test)) -> %(r)")
}</syntaxhighlight>
{{out}}
<pre>
0.9054054 -> 4527027/5000000
0.518518 -> 259259/500000
0.75 -> 3/4
</pre>
=={{header|zkl}}==
{{trans|D}}
<
if (r == 0.0) return(0,1);
if (r < 0.0){
Line 1,647 ⟶ 3,564:
return((r*best).round().toInt(),best);
}
}</
<
0.142857143, 3.141592654, 2.718281828,
-0.423310825, 31.415926536);
Line 1,656 ⟶ 3,573:
{ print(" %d/%d".fmt(real2Rational(r,(10).pow(i)).xplode())) }
println();
}</
{{out}}
<pre>
| |||