library BalancerSafeMath {
    /**
     * @notice Safe addition
     * @param a - first operand
     * @param b - second operand
     * @dev if we are adding b to a, the resulting sum must be greater than a
     * @return - sum of operands; throws if overflow
     */
    function badd(uint256 a, uint256 b) internal pure returns (uint256) {
        uint256 c = a + b;
        require(c >= a, "ERR_ADD_OVERFLOW");
        return c;
    }

    /**
     * @notice Safe unsigned subtraction
     * @param a - first operand
     * @param b - second operand
     * @dev Do a signed subtraction, and check that it produces a positive value
     *      (i.e., a - b is valid if b <= a)
     * @return - a - b; throws if underflow
     */
    function bsub(uint256 a, uint256 b) internal pure returns (uint256) {
        (uint256 c, bool negativeResult) = bsubSign(a, b);
        require(!negativeResult, "ERR_SUB_UNDERFLOW");
        return c;
    }

    /**
     * @notice Safe signed subtraction
     * @param a - first operand
     * @param b - second operand
     * @dev Do a signed subtraction
     * @return - difference between a and b, and a flag indicating a negative result
     *           (i.e., a - b if a is greater than or equal to b; otherwise b - a)
     */
    function bsubSign(uint256 a, uint256 b)
        internal
        pure
        returns (uint256, bool)
    {
        if (b <= a) {
            return (a - b, false);
        } else {
            return (b - a, true);
        }
    }

    /**
     * @notice Safe multiplication
     * @param a - first operand
     * @param b - second operand
     * @dev Multiply safely (and efficiently), rounding down
     * @return - product of operands; throws if overflow or rounding error
     */
    function bmul(uint256 a, uint256 b) internal pure returns (uint256) {
        // Gas optimization (see github.com/OpenZeppelin/openzeppelin-contracts/pull/522)
        if (a == 0) {
            return 0;
        }

        // Standard overflow check: a/a*b=b
        uint256 c0 = a * b;
        require(c0 / a == b, "ERR_MUL_OVERFLOW");

        // Round to 0 if x*y < BONE/2?
        uint256 c1 = c0 + (BalancerConstants.BONE / 2);
        require(c1 >= c0, "ERR_MUL_OVERFLOW");
        uint256 c2 = c1 / BalancerConstants.BONE;
        return c2;
    }

    /**
     * @notice Safe division
     * @param dividend - first operand
     * @param divisor - second operand
     * @dev Divide safely (and efficiently), rounding down
     * @return - quotient; throws if overflow or rounding error
     */
    function bdiv(uint256 dividend, uint256 divisor)
        internal
        pure
        returns (uint256)
    {
        require(divisor != 0, "ERR_DIV_ZERO");

        // Gas optimization
        if (dividend == 0) {
            return 0;
        }

        uint256 c0 = dividend * BalancerConstants.BONE;
        require(c0 / dividend == BalancerConstants.BONE, "ERR_DIV_INTERNAL"); // bmul overflow

        uint256 c1 = c0 + (divisor / 2);
        require(c1 >= c0, "ERR_DIV_INTERNAL"); //  badd require

        uint256 c2 = c1 / divisor;
        return c2;
    }

    /**
     * @notice Safe unsigned integer modulo
     * @dev Returns the remainder of dividing two unsigned integers.
     *      Reverts when dividing by zero.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * @param dividend - first operand
     * @param divisor - second operand -- cannot be zero
     * @return - quotient; throws if overflow or rounding error
     */
    function bmod(uint256 dividend, uint256 divisor)
        internal
        pure
        returns (uint256)
    {
        require(divisor != 0, "ERR_MODULO_BY_ZERO");

        return dividend % divisor;
    }

    /**
     * @notice Safe unsigned integer max
     * @dev Returns the greater of the two input values
     *
     * @param a - first operand
     * @param b - second operand
     * @return - the maximum of a and b
     */
    function bmax(uint256 a, uint256 b) internal pure returns (uint256) {
        return a >= b ? a : b;
    }

    /**
     * @notice Safe unsigned integer min
     * @dev returns b, if b < a; otherwise returns a
     *
     * @param a - first operand
     * @param b - second operand
     * @return - the lesser of the two input values
     */
    function bmin(uint256 a, uint256 b) internal pure returns (uint256) {
        return a < b ? a : b;
    }

    /**
     * @notice Safe unsigned integer average
     * @dev Guard against (a+b) overflow by dividing each operand separately
     *
     * @param a - first operand
     * @param b - second operand
     * @return - the average of the two values
     */
    function baverage(uint256 a, uint256 b) internal pure returns (uint256) {
        // (a + b) / 2 can overflow, so we distribute
        return (a / 2) + (b / 2) + (((a % 2) + (b % 2)) / 2);
    }

    /**
     * @notice Babylonian square root implementation
     * @dev (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method)
     * @param y - operand
     * @return z - the square root result
     */
    function sqrt(uint256 y) internal pure returns (uint256 z) {
        if (y > 3) {
            z = y;
            uint256 x = y / 2 + 1;
            while (x < z) {
                z = x;
                x = (y / x + x) / 2;
            }
        } else if (y != 0) {
            z = 1;
        }
    }
}
/**
 * @author Balancer Labs
 * @title SafeMath - wrap Solidity operators to prevent underflow/overflow
 * @dev badd and bsub are basically identical to OpenZeppelin SafeMath; mul/div have extra checks
 */
function bmin(uint256 a, uint256 b) internal pure returns (uint256) {
        return a < b ? a : b;
    }
/**
 * @notice Safe unsigned integer min
 * @dev returns b, if b < a; otherwise returns a
 *
 * @param a - first operand
 * @param b - second operand
 * @return - the lesser of the two input values
 */