Converting float decimal to fraction

Continued fractions can be used to find rational approximations to real numbers that are "best" in a strict sense. Here's a PHP function that finds a rational approximation to a given (positive) floating point number with a relative error less than $tolerance:

<?phpfunction float2rat($n, $tolerance = 1.e-6) {    $h1=1; $h2=0;    $k1=0; $k2=1;    $b = 1/$n;    do {        $b = 1/$b;        $a = floor($b);        $aux = $h1; $h1 = $a*$h1+$h2; $h2 = $aux;        $aux = $k1; $k1 = $a*$k1+$k2; $k2 = $aux;        $b = $b-$a;    } while (abs($n-$h1/$k1) > $n*$tolerance);    return "$h1/$k1";}printf("%s\n", float2rat(66.66667)); # 200/3printf("%s\n", float2rat(sqrt(2)));  # 1393/985printf("%s\n", float2rat(0.43212));  # 748/1731

I have written more about this algorithm and why it works, and even a JavaScript demo here: https://web.archive.org/web/20180731235708/http://jonisalonen.com/2012/converting-decimal-numbers-to-ratios/

Farey fractions can be quite useful in this case.

They can be used to convert any decimal into a fraction with the lowest possible denominator.

Sorry - I don't have a prototype in PHP, so here's one in Python:

def farey(v, lim):    """No error checking on args.  lim = maximum denominator.        Results are (numerator, denominator); (1, 0) is 'infinity'."""    if v < 0:        n, d = farey(-v, lim)        return (-n, d)    z = lim - lim   # Get a "zero of the right type" for the denominator    lower, upper = (z, z+1), (z+1, z)    while True:        mediant = (lower[0] + upper[0]), (lower[1] + upper[1])        if v * mediant[1] > mediant[0]:            if lim < mediant[1]:                return upper            lower = mediant        elif v * mediant[1] == mediant[0]:            if lim >= mediant[1]:                return mediant            if lower[1] < upper[1]:                return lower            return upper        else:            if lim < mediant[1]:                return lower            upper = mediant

Converted Python code in answer from @APerson241 to PHP

<?phpfunction farey($v, $lim) {    // No error checking on args.  lim = maximum denominator.    // Results are array(numerator, denominator); array(1, 0) is 'infinity'.    if($v < 0) {        list($n, $d) = farey(-$v, $lim);        return array(-$n, $d);    }    $z = $lim - $lim;   // Get a "zero of the right type" for the denominator    list($lower, $upper) = array(array($z, $z+1), array($z+1, $z));    while(true) {        $mediant = array(($lower[0] + $upper[0]), ($lower[1] + $upper[1]));        if($v * $mediant[1] > $mediant[0]) {            if($lim < $mediant[1])                 return $upper;            $lower = $mediant;        }        else if($v * $mediant[1] == $mediant[0]) {            if($lim >= $mediant[1])                return $mediant;            if($lower[1] < $upper[1])                return $lower;            return $upper;        }        else {            if($lim < $mediant[1])                return $lower;            $upper = $mediant;        }    }}// Example use:$f = farey(66.66667, 10);echo $f[0], '/', $f[1], "\n"; # 200/3$f = farey(sqrt(2), 1000);echo $f[0], '/', $f[1], "\n";  # 1393/985$f = farey(0.43212, 2000);echo $f[0], '/', $f[1], "\n";  # 748/1731