How do you find the best approximations? You could do a brute force search. For example, if the maximum denominator size is N, you could try all fractions with denominators less than or equal to N. But there’s a much more efficient algorithm. The algorithm is related to the Farey sequence named after John Farey, though I don’t know whether he invented the algorithm.
The idea is to start with two fractions, a/b = 0/1 and c/d = 1/1. We update either a/b or c/d at each step so that a/b will be the best lower bound of x with denominator no bigger than b, and c/d will be the best upper bound with denominator no bigger than d. At each step we do a sort of binary search by introducing the mediant of the upper and lower bounds. The mediant of a/b and c/d is the fraction (a+c)/(b+d) which always lies between a/b and c/d.
Here is an implementation of the algorithm in Python. The code takes a number x between 0 and 1 and a maximum denominator size N. It returns the numerator and denominator of the best rational approximation to x using denominators no larger than N.
def farey(x, N):
a, b = 0, 1
c, d = 1, 1
while (b <= N and d <= N):
mediant = float(a+c)/(b+d)
if x == mediant:
if b + d <= N:
return a+c, b+d
elif d > b:
return c, d
else:
return a, b
elif x > mediant:
a, b = a+c, b+d
else:
c, d = a+c, b+d
if (b > N):
return c, d
else:
return a, bHere is a translation into C++ with some minor updates. You cannot really determine equality between two floating point numbers, and besides, I need accuracy to some epsilon as well as a maximum size for the numerator. Anyway, here is a C++ offering.
#include <utility>
/**
* Farey algorithm
*
* Translated from John D. Cook's Python implementation in
* http://www.johndcook.com/blog/2010/10/20/best-rational-approximation/
*
* @param x A value between 0.0 and 1.0 to be approximated by two integers
* @param eps The required precision such that abs(x-a/b) < eps. Eps > 0.
* @param N The maximum size of the numerator allowed
*/
pair<unsigned int, unsigned int> farley(double x, double eps, unsigned int N)
{
unsigned int a(0);
unsigned int b(1);
unsigned int c(1);
unsigned int d(1);
double mediant(0.0F);
while ((b <= N) and (d <= N))
{
mediant = static_cast<float> (a + c) / static_cast<float> (b + d);
if (abs(x - mediant) < eps)
{
if (b + d <= N)
{
return pair<unsigned int, unsigned int> (a + c, b + d);
}
else if (d > b)
{
return pair<unsigned int, unsigned int> (c, d);
}
else
{
return pair<unsigned int, unsigned int> (a, b);
}
}
else if (x > mediant)
{
a = a + c;
b = b + d;
}
else
{
c = a + c;
d = b + d;
}
}
if (b > N)
{
return pair<unsigned int, unsigned int> (c, d);
}
else
{
return pair<unsigned int, unsigned int> (a, b);
}
}
Falls das ein wenig besser verständlich ist 
Thema ist damit gelöst.
[autoit][/autoit][autoit][/autoit][autoit]Global $e = 2.71828182846
Global $pi = 3.14159265359
Global $ap = 1.20205690316
Local $1 = FareyApprox($e, 255)
Local $2 = FareyApprox($pi, 255)
Local $3 = FareyApprox($ap, 255)
ConsoleWrite('e : ' & $1[0] & '/' & $1[1] & ' Err: ' & $1[0]/$1[1] - $e & @CRLF)
ConsoleWrite('pi : ' & $2[0] & '/' & $2[1] & ' Err: ' & $2[0]/$2[1] - $pi & @CRLF)
ConsoleWrite('ap : ' & $3[0] & '/' & $3[1] & ' Err: ' & $3[0]/$3[1] - $ap & @CRLF)
Func FareyApprox($x, $N)
If $x > 1 Then
Local $t = FareyApprox(1 / $x, $N)
Return Array($t[1], $t[0])
EndIf
Local $a = 0, $b = 1, $c = 1, $d = 1, $m
While ($b <= $N And $d <= $N)
$m = ($a + $c) / ($b + $d)
If $x = $m Then
Return $b + $d <= $N ? Array($a + $c, $b + $d) : $d > $b ? Array($c, $d) : Array($a, $b)
ElseIf $x > $m Then
$a += $c
$b += $d
Else
$c += $a
$d += $b
EndIf
WEnd
Return $b > $N ? Array($c, $d) : Array($a, $b)
EndFunc
Func Array($x1, $x2) ; Keiner weiß warum es das nicht nativ in AutoIt gibt... ein Mysterium 
Local $aRet[2] = [$x1, $x2]
Return $aRet
EndFunc
$dez1 = ATan(1) * 4 ;pi
ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $dez1 = ' & $dez1 & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
$dez2 = 600007 / 31
ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $dez2 = ' & $dez2 & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
$ret = dezi2bruch($dez1, 10)
ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $ret = ' & $ret & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
$ret = dezi2bruch($dez2, -2)
ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $ret = ' & $ret & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
$ret = dezi2bruch($dez2, 8)
ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $ret = ' & $ret & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
$ret = dezi2bruch($dez2, 2)
ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $ret = ' & $ret & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
Func dezi2bruch($dez, $anzahl = 2) ; by andy@autoit.de
;~ If $anzahl = -1 Then $anzahl = StringLen(StringRegExpReplace($dez, "\d+\.(\d*)", "\1")) ;anzahl dezimalstellen
;~ ConsoleWrite('@@ Debug(' & @ScriptLineNumber & ') : $anzahl = ' & $anzahl & @CRLF & '>Error code: ' & @error & @CRLF) ;### Debug Console
$b = 10 ^ $anzahl
$c = 1 / $b
$r = Int($dez)
If $anzahl <= 0 Then Return Round($dez) & " / 1 = " & Round($dez)
For $i = $dez To $dez * $b Step $dez
$zaehler1 = Int($i)
$zaehler2 = Int($i) + 1
$nenner1 = Floor($i / $dez)
$nenner2 = Ceiling($i / $dez)
;~ consolewrite ($zaehler1 & " / " & $nenner1 & " = " & abs(($zaehler1/$nenner1)-$dez) & @crlf)
;~ consolewrite ($zaehler2 & " / " & $nenner1 & " = " & abs(($zaehler2/$nenner1)-$dez) & @crlf)
;~ consolewrite ($zaehler1 & " / " & $nenner2 & " = " & abs(($zaehler1/$nenner2)-$dez) & @crlf)
;~ consolewrite ($zaehler2 & " / " & $nenner2 & " = " & abs(($zaehler2/$nenner2)-$dez) & @crlf)
If $nenner1 <> 1 And Abs(($zaehler1 / $nenner1) - $dez) < $c Then Return $zaehler1 & " / " & $nenner1 & " = " & $zaehler1 / $nenner1
If $nenner1 <> 1 And Abs(($zaehler2 / $nenner1) - $dez) < $c Then Return $zaehler2 & " / " & $nenner1 & " = " & $zaehler2 / $nenner1
If $nenner1 <> 1 And Abs(($zaehler1 / $nenner2) - $dez) < $c Then Return $zaehler1 & " / " & $nenner2 & " = " & $zaehler1 / $nenner2
If $nenner1 <> 1 And Abs(($zaehler2 / $nenner2) - $dez) < $c Then Return $zaehler2 & " / " & $nenner2 & " = " & $zaehler2 / $nenner2
Next
Return -1
EndFunc ;==>dezi2bruch