library Math {
    /**
     * @dev Returns the largest of two numbers.
     */
    function max(uint256 a, uint256 b) internal pure returns (uint256) {
        return a >= b ? a : b;
    }

    /**
     * @dev Returns the smallest of two numbers.
     */
    function min(uint256 a, uint256 b) internal pure returns (uint256) {
        return a < b ? a : b;
    }

    /**
     * @dev Returns the average of two numbers. The result is rounded towards
     * zero.
     */
    function average(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);
    }

    // https://github.com/abdk-consulting/abdk-libraries-solidity/blob/master/ABDKMath64x64.sol#L687
    function sqrt(uint256 x) internal pure returns (uint256) {
        if (x == 0) return 0;
        // this block is equivalent to r = uint256(1) << (BitMath.mostSignificantBit(x) / 2);
        // however that code costs significantly more gas
        uint256 xx = x;
        uint256 r = 1;
        if (xx >= 0x100000000000000000000000000000000) {
            xx >>= 128;
            r <<= 64;
        }
        if (xx >= 0x10000000000000000) {
            xx >>= 64;
            r <<= 32;
        }
        if (xx >= 0x100000000) {
            xx >>= 32;
            r <<= 16;
        }
        if (xx >= 0x10000) {
            xx >>= 16;
            r <<= 8;
        }
        if (xx >= 0x100) {
            xx >>= 8;
            r <<= 4;
        }
        if (xx >= 0x10) {
            xx >>= 4;
            r <<= 2;
        }
        if (xx >= 0x8) {
            r <<= 1;
        }
        r = (r + x / r) >> 1;
        r = (r + x / r) >> 1;
        r = (r + x / r) >> 1;
        r = (r + x / r) >> 1;
        r = (r + x / r) >> 1;
        r = (r + x / r) >> 1;
        r = (r + x / r) >> 1; // Seven iterations should be enough
        uint256 r1 = x / r;
        return (r < r1 ? r : r1);
    }
}
/**
 * @dev Standard math utilities missing in the Solidity language.
 */
function average(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);
}

/**
 * @dev Returns the average of two numbers. The result is rounded towards
 * zero.
 */