library FixedPoint {
    // range: [0, 2**112 - 1]
    // resolution: 1 / 2**112
    struct uq112x112 {
        uint224 _x;
    }

    // returns a uq112x112 which represents the ratio of the numerator to the denominator
    // equivalent to encode(numerator).div(denominator)
    function fraction(uint112 numerator, uint112 denominator) internal pure returns (uq112x112 memory) {
        require(denominator > 0, "FixedPoint: DIV_BY_ZERO");
        return uq112x112((uint224(numerator) << 112) / denominator);
    }

    // decode a uq112x112 into a uint with 18 decimals of precision
    function decode112with18(uq112x112 memory self) internal pure returns (uint) {
        // we only have 256 - 224 = 32 bits to spare, so scaling up by ~60 bits is dangerous
        // instead, get close to:
        //  (x * 1e18) >> 112
        // without risk of overflowing, e.g.:
        //  (x) / 2 ** (112 - lg(1e18))
        return uint(self._x) / 5192296858534827;
    }
}
// Based on code from https://github.com/Uniswap/uniswap-v2-periphery
// a library for handling binary fixed point numbers (https://en.wikipedia.org/wiki/Q_(number_format))
function decode112with18(uq112x112 memory self) internal pure returns (uint) {
        // we only have 256 - 224 = 32 bits to spare, so scaling up by ~60 bits is dangerous
        // instead, get close to:
        //  (x * 1e18) >> 112
        // without risk of overflowing, e.g.:
        //  (x) / 2 ** (112 - lg(1e18))
        return uint(self._x) / 5192296858534827;
    }
// decode a uq112x112 into a uint with 18 decimals of precision