C++: Simple 1d equation solver

gistfile1.cpp

#include <cstdio>
#include <string>
#include <sstream>


template<typename T>
class Fraction{
public:
  typedef T number_type;

	Fraction():
	_numerator(0),
	_denominator(0),
	_requiresStringification(false),
	_stringified(){

	}

	Fraction(T numerator, T denominator = 1):
		_numerator(numerator),
		 _denominator(denominator),	
		  _requiresStringification(true),
		  _stringified(){
	 	reduce();
	}

	std::string to_s(){
		if(_requiresStringification){
			stringify();
			_requiresStringification = false;
		}
		return _stringified;
	}

	void setNumerator(T numerator){
		_numerator = numerator;
		_requiresStringification = true;
		reduce();
	}

	void setDenominator(T denominator){
		_denominator = denominator;
		_requiresStringification = true;
		reduce();
	}


protected:
	T _numerator;
	T _denominator;
	bool _requiresStringification;
	std::string _stringified;


	void reduce(){

		//transit sign to the numerator
		if(_denominator * _numerator < 0){
			_numerator = -abs(_numerator);
			_denominator = abs(_denominator);
		}else{
			_numerator = abs(_numerator);
			_denominator = abs(_denominator);
		}

		if(_numerator==0 || _denominator==1){ //no needs for reduction
			return;
		}else if(_denominator ==0){//this is nan
			return;
		}else{//reduce
			T _gcd = gcd(abs(_numerator), abs(_denominator));
			_numerator /= _gcd;
			_denominator /= _gcd;
		}
	}


	void stringify(){
		std::ostringstream ost;
		if(_denominator==1){
			ost << _numerator;
		}else if(_denominator==0){
			ost << "NaN";
		}else{
			ost << _numerator << "/" << _denominator;
		}
		_stringified = ost.str();
	}


	//Taken from https://en.wikipedia.org/wiki/Binary_GCD_algorithm
	static T gcd(T u, T v) 
{
  // simple cases (termination)
  if (u == v)
    return u;
  if (u == 0)
    return v;
  if (v == 0)
    return u;
 
  // look for factors of 2
  if (~u & 1) // u is even
    if (v & 1) // v is odd
      return gcd(u >> 1, v);
    else // both u and v are even
      return gcd(u >> 1, v >> 1) << 1;
  if (~v & 1) // u is odd, v is even
    return gcd(u, v >> 1);
 
  // reduce larger argument
  if (u > v)
    return gcd((u - v) >> 1, v);
  return gcd((v - u) >> 1, u);
}
};

//Solve ax + b = cx + d
static std::string solve(int a, int b, int c, int d){
	if(a==0 && c!=0){
		Fraction<int> frac(b-d, c);
		return "x = " + frac.to_s();
	}else if(a!=0 && c==0){
		Fraction<int> frac(d-b, a);
		return "x = " + frac.to_s();
	}else if(a==0 && c ==0){
		return "impossible";
	}else if(a == c){
		return "impossible";
	}else{
		Fraction<int> frac(b - d, c - a);
		return "x = " + frac.to_s();
	}
}