How to convert floats to human-readable fractions?
Let's say we have 0.33, we need to output 1/3.
If we have 0.4, we need to output 2/5.
The idea is to make it human-readable to make the user understand "x parts out of y" as a better way of understanding data.
I know that percentages is a good substitute but I was wondering if there was a simple way to do this?
Solution 1:
I have found David Eppstein's find rational approximation to given real number C code to be exactly what you are asking for. Its based on the theory of continued fractions and very fast and fairly compact.
I have used versions of this customized for specific numerator and denominator limits.
/*
** find rational approximation to given real number
** David Eppstein / UC Irvine / 8 Aug 1993
**
** With corrections from Arno Formella, May 2008
**
** usage: a.out r d
** r is real number to approx
** d is the maximum denominator allowed
**
** based on the theory of continued fractions
** if x = a1 + 1/(a2 + 1/(a3 + 1/(a4 + ...)))
** then best approximation is found by truncating this series
** (with some adjustments in the last term).
**
** Note the fraction can be recovered as the first column of the matrix
** ( a1 1 ) ( a2 1 ) ( a3 1 ) ...
** ( 1 0 ) ( 1 0 ) ( 1 0 )
** Instead of keeping the sequence of continued fraction terms,
** we just keep the last partial product of these matrices.
*/
#include <stdio.h>
main(ac, av)
int ac;
char ** av;
{
double atof();
int atoi();
void exit();
long m[2][2];
double x, startx;
long maxden;
long ai;
/* read command line arguments */
if (ac != 3) {
fprintf(stderr, "usage: %s r d\n",av[0]); // AF: argument missing
exit(1);
}
startx = x = atof(av[1]);
maxden = atoi(av[2]);
/* initialize matrix */
m[0][0] = m[1][1] = 1;
m[0][1] = m[1][0] = 0;
/* loop finding terms until denom gets too big */
while (m[1][0] * ( ai = (long)x ) + m[1][1] <= maxden) {
long t;
t = m[0][0] * ai + m[0][1];
m[0][1] = m[0][0];
m[0][0] = t;
t = m[1][0] * ai + m[1][1];
m[1][1] = m[1][0];
m[1][0] = t;
if(x==(double)ai) break; // AF: division by zero
x = 1/(x - (double) ai);
if(x>(double)0x7FFFFFFF) break; // AF: representation failure
}
/* now remaining x is between 0 and 1/ai */
/* approx as either 0 or 1/m where m is max that will fit in maxden */
/* first try zero */
printf("%ld/%ld, error = %e\n", m[0][0], m[1][0],
startx - ((double) m[0][0] / (double) m[1][0]));
/* now try other possibility */
ai = (maxden - m[1][1]) / m[1][0];
m[0][0] = m[0][0] * ai + m[0][1];
m[1][0] = m[1][0] * ai + m[1][1];
printf("%ld/%ld, error = %e\n", m[0][0], m[1][0],
startx - ((double) m[0][0] / (double) m[1][0]));
}
Solution 2:
From Python 2.6 on there is the fractions module.
(Quoting from the docs.)
>>> from fractions import Fraction
>>> Fraction('3.1415926535897932').limit_denominator(1000)
Fraction(355, 113)
>>> from math import pi, cos
>>> Fraction.from_float(cos(pi/3))
Fraction(4503599627370497, 9007199254740992)
>>> Fraction.from_float(cos(pi/3)).limit_denominator()
Fraction(1, 2)
Solution 3:
If the the output is to give a human reader a fast impression of the order of the result, it makes no sense return something like "113/211", so the output should limit itself to using one-digit numbers (and maybe 1/10 and 9/10). If so, you can observe that there are only 27 different fractions.
Since the underlying math for generating the output will never change, a solution could be to simply hard-code a binary search tree, so that the function would perform at most log(27) ~= 4 3/4 comparisons. Here is a tested C version of the code
char *userTextForDouble(double d, char *rval)
{
if (d == 0.0)
return "0";
// TODO: negative numbers:if (d < 0.0)...
if (d >= 1.0)
sprintf(rval, "%.0f ", floor(d));
d = d-floor(d); // now only the fractional part is left
if (d == 0.0)
return rval;
if( d < 0.47 )
{
if( d < 0.25 )
{
if( d < 0.16 )
{
if( d < 0.12 ) // Note: fixed from .13
{
if( d < 0.11 )
strcat(rval, "1/10"); // .1
else
strcat(rval, "1/9"); // .1111....
}
else // d >= .12
{
if( d < 0.14 )
strcat(rval, "1/8"); // .125
else
strcat(rval, "1/7"); // .1428...
}
}
else // d >= .16
{
if( d < 0.19 )
{
strcat(rval, "1/6"); // .1666...
}
else // d > .19
{
if( d < 0.22 )
strcat(rval, "1/5"); // .2
else
strcat(rval, "2/9"); // .2222...
}
}
}
else // d >= .25
{
if( d < 0.37 ) // Note: fixed from .38
{
if( d < 0.28 ) // Note: fixed from .29
{
strcat(rval, "1/4"); // .25
}
else // d >=.28
{
if( d < 0.31 )
strcat(rval, "2/7"); // .2857...
else
strcat(rval, "1/3"); // .3333...
}
}
else // d >= .37
{
if( d < 0.42 ) // Note: fixed from .43
{
if( d < 0.40 )
strcat(rval, "3/8"); // .375
else
strcat(rval, "2/5"); // .4
}
else // d >= .42
{
if( d < 0.44 )
strcat(rval, "3/7"); // .4285...
else
strcat(rval, "4/9"); // .4444...
}
}
}
}
else
{
if( d < 0.71 )
{
if( d < 0.60 )
{
if( d < 0.55 ) // Note: fixed from .56
{
strcat(rval, "1/2"); // .5
}
else // d >= .55
{
if( d < 0.57 )
strcat(rval, "5/9"); // .5555...
else
strcat(rval, "4/7"); // .5714
}
}
else // d >= .6
{
if( d < 0.62 ) // Note: Fixed from .63
{
strcat(rval, "3/5"); // .6
}
else // d >= .62
{
if( d < 0.66 )
strcat(rval, "5/8"); // .625
else
strcat(rval, "2/3"); // .6666...
}
}
}
else
{
if( d < 0.80 )
{
if( d < 0.74 )
{
strcat(rval, "5/7"); // .7142...
}
else // d >= .74
{
if(d < 0.77 ) // Note: fixed from .78
strcat(rval, "3/4"); // .75
else
strcat(rval, "7/9"); // .7777...
}
}
else // d >= .8
{
if( d < 0.85 ) // Note: fixed from .86
{
if( d < 0.83 )
strcat(rval, "4/5"); // .8
else
strcat(rval, "5/6"); // .8333...
}
else // d >= .85
{
if( d < 0.87 ) // Note: fixed from .88
{
strcat(rval, "6/7"); // .8571
}
else // d >= .87
{
if( d < 0.88 ) // Note: fixed from .89
{
strcat(rval, "7/8"); // .875
}
else // d >= .88
{
if( d < 0.90 )
strcat(rval, "8/9"); // .8888...
else
strcat(rval, "9/10"); // .9
}
}
}
}
}
}
return rval;
}
Solution 4:
Here's a link explaining the math behind converting a decimal to a fraction:
http://www.webmath.com/dec2fract.html
And here's an example function for how to actually do it using VB (from www.freevbcode.com/ShowCode.asp?ID=582):
Public Function Dec2Frac(ByVal f As Double) As String
Dim df As Double
Dim lUpperPart As Long
Dim lLowerPart As Long
lUpperPart = 1
lLowerPart = 1
df = lUpperPart / lLowerPart
While (df <> f)
If (df < f) Then
lUpperPart = lUpperPart + 1
Else
lLowerPart = lLowerPart + 1
lUpperPart = f * lLowerPart
End If
df = lUpperPart / lLowerPart
Wend
Dec2Frac = CStr(lUpperPart) & "/" & CStr(lLowerPart)
End Function
(From google searches: convert decimal to fraction, convert decimal to fraction code)