Calculate Biggest Rational Fraction Within Some Bounds

I am trying to place currency trades that match an exact rate on a market that only accepts integral bid/offer amounts. I want to make the largest trade possible at a specific rate. This is a toy program, not a real trading bot, so I am using C#.

I need an algorithm that returns an answer in a reasonable amount of time even when the numerator and denominator can be large (100000+).

static bool CalcBiggestRationalFraction(float target_real, float epsilon, int numerator_max, int denominator_max, out int numerator, out in开发者_JS百科t denominator)
{
    // target_real is the ratio we are tryig to achieve in our output fraction (numerator / denominator)
    // epsilon is the largest difference abs(target_real - (numerator / denominator)) we are willing to tolerate in the answer
    // numerator_max, denominator_max are the upper bounds on the numerator and the denominator in the answer
    //
    // in the case where there are multiple answers, we want to return the largest one
    //
    // in the case where an answer is found that is within epsilon, we return true and the answer.
    // in the case where an answer is not found that is within epsilon, we return false and the closest answer that we have found.
    //
    // ex: CalcBiggestRationalFraction(.5, .001, 4, 4, num, denom) returns (2/4) instead of (1/2).


}

I asked a previous question that is similar (http://stackoverflow.com/questions/4385580/finding-the-closest-integer-fraction-to-a-given-random-real) before I thought about what I was actually trying to accomplish and it turns out that I am trying to solve a different, but related problem.

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The canonical way to solve your problem is with continued fraction expansion. In particular, see this section.

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If you want the unreduced fraction, then here's one optimization you can do: Since you'll never be interested in n/2, because you want 2n/4, 4n/8, or 1024n/2048, we only need to check some of the numbers. As soon as we check any multiple of 2, we never need to check 2. Therefore, I believe you can try denominators denominator_max through denominator_max/2, and you'll have implicitly checked all of the factors of those numbers, which would be everything 2 through denominator_max/2.

I'm not at a compiler at the moment, so I haven't checked this code for correctness, or even that it compiles, but it should be close.

static bool CalcBiggestRationalFraction(float target_real, float epsilon, 
    int numerator_max, int denominator_max, 
    out int numerator, out int denominator)
{
    if((int)Math.Round(target_real * denominator_max) > numerator_max)
    {
        // We were given values that don't match up.
        // For example, target real = 0.5, but max_num / max_den = 0.3
        denominator_max = (int)(numerator_max / target_real);
    }

    float bestEpsilon = float.MAX_VALUE;
    for(int den = denominator_max; den >= denominator_max/2, den--)
    {
        int num = (int)Math.Round(target_real * den);
        float thisEpsilon = Math.abs(((float)num / den) - target_real);
        if(thisEpsilon < bestEpsilon)
        {
            numerator = num;
            denominator = den;
            bestEpsilon = thisEpsilon;
        }
    }

    return bestEpsilon < epsilon;
}

---

Let's try this:

First, we need to turn the float into a fraction. Easiest way I can think to do this is to find the order of magnitude of the epsilon, multiply the float by that order, and truncate to get the numerator.

long orderOfMagnitude = 1
while(epsilon * orderOfMagnitude <1)
   orderOfMagnitude *= 10;

numerator = (int)(target_real*orderOfMagnitude);
denominator = orderOfMagnitude;

//sanity check; if the initial fraction isn't within the epsilon, then add sig figs until it is
while(target_real - (float)numerator / denominator > epsilon)
{
   orderOfMagnitude *= 10;
   numerator = (int)(target_real*orderOfMagnitude);
   denominator = orderOfMagnitude;
}

Now, we can break the fraction down into least terms. The most efficient way I know of is to attempt to divide by all prime numbers less than or equal to the square root of the smaller of the numerator and denominator.

var primes = new List<int>{2,3,5,7,11,13,17,19,23}; //to start us off

var i = 0;
while (true)
{
   if(Math.Sqrt(numerator) < primes[i] || Math.Sqrt(denominator) < primes[i]) break;

   if(numerator % primes[i] == 0 && denominator % primes[i] == 0)
   {
      numerator /= primes[i];
      denominator /= primes[i];
      i=0;
   }
   else
   {
      i++;
      if(i > primes.Count)
      {
        //Find the next prime number by looking for the first number not divisible
        //by any prime < sqrt(number).
        //We are actually unlikely to have to use this, because the denominator
        //is a power of 10, so its prime factorization will be 2^x*5^x
        var next = primes.Last() + 2;
        bool add;
        do
        {
           add = true;
           for(var x=0; primes[x] <= Math.Sqrt(next); x++)
              if(next % primes[x] == 0)
              {
                add = false;
                break;
              }

           if(add)
              primes.Add(next);
           else
              next+=2;   
        } while(!add);
      }
   }
}