library MathExpandLibrary {

    uint256 internal constant BONE = 10**18;

    // Add two numbers together checking for overflows
    function badd(uint256 a, uint256 b) internal pure returns (uint256) {
        uint256 c = a + b;
        require(c >= a, "ERR_ADD_OVERFLOW");
        return c;
    }

    // subtract two numbers and return diffecerence when it underflows
    function bsubSign(uint256 a, uint256 b) internal pure returns (uint256, bool) {
        if (a >= b) {
            return (a - b, false);
        } else {
            return (b - a, true);
        }
    }

    // Subtract two numbers checking for underflows
    function bsub(uint256 a, uint256 b) internal pure returns (uint256) {
        (uint256 c, bool flag) = bsubSign(a, b);
        require(!flag, "ERR_SUB_UNDERFLOW");
        return c;
    }

    // Multiply two 18 decimals numbers
    function bmul(uint256 a, uint256 b) internal pure returns (uint256) {
        uint256 c0 = a * b;
        require(a == 0 || c0 / a == b, "ERR_MUL_OVERFLOW");
        uint256 c1 = c0 + (BONE / 2);
        require(c1 >= c0, "ERR_MUL_OVERFLOW");
        uint256 c2 = c1 / BONE;
        return c2;
    }

    // Divide two 18 decimals numbers
    function bdiv(uint256 a, uint256 b) internal pure returns (uint256) {
        require(b != 0, "ERR_DIV_ZERO");
        uint256 c0 = a * BONE;
        require(a == 0 || c0 / a == BONE, "ERR_DIV_INTERNAL"); // bmul overflow
        uint256 c1 = c0 + (b / 2);
        require(c1 >= c0, "ERR_DIV_INTERNAL"); //  badd require
        uint256 c2 = c1 / b;
        return c2;
    }

    function min(uint x, uint y) internal pure returns (uint z) {
        z = x < y ? x : y;
    }

    // babylonian method (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method)
    function sqrt(uint y) internal pure returns (uint z) {
        if (y > 3) {
            z = y;
            uint x = y / 2 + 1;
            while (x < z) {
                z = x;
                x = (y / x + x) / 2;
            }
        } else if (y != 0) {
            z = 1;
        }
    }
}
// a library for performing various math operations
function bsub(uint256 a, uint256 b) internal pure returns (uint256) {
        (uint256 c, bool flag) = bsubSign(a, b);
        require(!flag, "ERR_SUB_UNDERFLOW");
        return c;
    }
// Subtract two numbers checking for underflows