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

    // range: [0, 2**144 - 1]
    // resolution: 1 / 2**112
    struct uq144x112 {
        uint256 _x;
    }

    uint8 private constant RESOLUTION = 112;
    uint256 private constant Q112 = uint256(1) << RESOLUTION;
    uint256 private constant Q224 = Q112 << RESOLUTION;
    uint256 private constant LOWER_MASK = 0xffffffffffffffffffffffffffff; // decimal of UQ*x112 (lower 112 bits)

    // encode a uint112 as a UQ112x112
    function encode(uint112 x) internal pure returns (uq112x112 memory) {
        return uq112x112(uint224(x) << RESOLUTION);
    }

    // encodes a uint144 as a UQ144x112
    function encode144(uint144 x) internal pure returns (uq144x112 memory) {
        return uq144x112(uint256(x) << RESOLUTION);
    }

    // decode a UQ112x112 into a uint112 by truncating after the radix point
    function decode(uq112x112 memory self) internal pure returns (uint112) {
        return uint112(self._x >> RESOLUTION);
    }

    // decode a UQ144x112 into a uint144 by truncating after the radix point
    function decode144(uq144x112 memory self) internal pure returns (uint144) {
        return uint144(self._x >> RESOLUTION);
    }

    // multiply a UQ112x112 by a uint, returning a UQ144x112
    // reverts on overflow
    function mul(uq112x112 memory self, uint256 y) internal pure returns (uq144x112 memory) {
        uint256 z = 0;
        require(y == 0 || (z = self._x * y) / y == self._x, 'FixedPoint: MUL_OVERFLOW');
        return uq144x112(z);
    }

    // multiply a UQ112x112 by an int and decode, returning an int
    // reverts on overflow
    function muli(uq112x112 memory self, int256 y) internal pure returns (int256) {
        uint144 z = decode144(mul(self, uint256(y < 0 ? -y : y)));
        return y < 0 ? -int256(z) : z;
    }

    // multiply a UQ112x112 by a UQ112x112, returning a UQ112x112
    // lossy
    function muluq(uq112x112 memory self, uq112x112 memory other) internal pure returns (uq112x112 memory) {
        if (self._x == 0 || other._x == 0) {
            return uq112x112(0);
        }
        uint112 upper_self = uint112(self._x >> RESOLUTION); // * 2^0
        uint112 lower_self = uint112(self._x & LOWER_MASK); // * 2^-112
        uint112 upper_other = uint112(other._x >> RESOLUTION); // * 2^0
        uint112 lower_other = uint112(other._x & LOWER_MASK); // * 2^-112

        // partial products
        uint224 upper = uint224(upper_self) * upper_other; // * 2^0
        uint224 lower = uint224(lower_self) * lower_other; // * 2^-224
        uint224 uppers_lowero = uint224(upper_self) * lower_other; // * 2^-112
        uint224 uppero_lowers = uint224(upper_other) * lower_self; // * 2^-112

        // so the bit shift does not overflow
        require(upper <= uint112(-1), 'FixedPoint: MULUQ_OVERFLOW_UPPER');

        // this cannot exceed 256 bits, all values are 224 bits
        uint256 sum = uint256(upper << RESOLUTION) + uppers_lowero + uppero_lowers + (lower >> RESOLUTION);

        // so the cast does not overflow
        require(sum <= uint224(-1), 'FixedPoint: MULUQ_OVERFLOW_SUM');

        return uq112x112(uint224(sum));
    }

    // divide a UQ112x112 by a UQ112x112, returning a UQ112x112
    function divuq(uq112x112 memory self, uq112x112 memory other) internal pure returns (uq112x112 memory) {
        require(other._x > 0, 'FixedPoint: DIV_BY_ZERO_DIVUQ');
        if (self._x == other._x) {
            return uq112x112(uint224(Q112));
        }
        if (self._x <= uint144(-1)) {
            uint256 value = (uint256(self._x) << RESOLUTION) / other._x;
            require(value <= uint224(-1), 'FixedPoint: DIVUQ_OVERFLOW');
            return uq112x112(uint224(value));
        }

        uint256 result = FullMath.mulDiv(Q112, self._x, other._x);
        require(result <= uint224(-1), 'FixedPoint: DIVUQ_OVERFLOW');
        return uq112x112(uint224(result));
    }

    // returns a UQ112x112 which represents the ratio of the numerator to the denominator
    // lossy
    function fraction(uint112 numerator, uint112 denominator) internal pure returns (uq112x112 memory) {
        require(denominator > 0, 'FixedPoint: DIV_BY_ZERO_FRACTION');
        return uq112x112((uint224(numerator) << RESOLUTION) / denominator);
    }

    // take the reciprocal of a UQ112x112
    // reverts on overflow
    // lossy
    function reciprocal(uq112x112 memory self) internal pure returns (uq112x112 memory) {
        require(self._x > 1, 'FixedPoint: DIV_BY_ZERO_RECIPROCAL_OR_OVERFLOW');
        return uq112x112(uint224(Q224 / self._x));
    }

    // square root of a UQ112x112
    // lossy to 40 bits
    function sqrt(uq112x112 memory self) internal pure returns (uq112x112 memory) {
        return uq112x112(uint224(Babylonian.sqrt(uint256(self._x) << 32) << 40));
    }
}
function decode(uq112x112 memory self) internal pure returns (uint112) {
        return uint112(self._x >> RESOLUTION);
    }
// decode a UQ112x112 into a uint112 by truncating after the radix point