// Contract: StakingV2
// Target function: function log_2(int128 x) internal pure returns (int128)
4 fixed point number into unsigned 64-bit integer
     * number rounding down.  Revert on underflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return unsigned 64-bit integer number
     */
    function toUInt(int128 x) internal pure returns (uint64) {
        require(x >= 0);
        return uint64(x >> 64);
    }

    /**
     * Convert signed 128.128 fixed point number into signed 64.64-bit fixed point
     * number rounding down.  Revert on overflow.
     *
     * @param x signed 128.128-bin fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function from128x128(int256 x) internal pure returns (int128) {
        int256 result = x >> 64;
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Convert signed 64.64 fixed point number into signed 128.128 fixed point
     * number.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 128.128 fixed point number
     */
    function to128x128(int128 x) internal pure returns (int256) {
        return int256(x) << 64;
    }

    /**
     * Calculate x + y.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function add(int128 x, int128 y) internal pure returns (int128) {
        int256 result = int256(x) + y;
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Calculate x - y.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function sub(int128 x, int128 y) internal pure returns (int128) {
        int256 result = int256(x) - y;
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Calculate x * y rounding down.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function mul(int128 x, int128 y) internal pure returns (int128) {
        int256 result = (int256(x) * y) >> 64;
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Calculate x * y rounding towards zero, where x is signed 64.64 fixed point
     * number and y is signed 256-bit integer number.  Revert on overflow.
     *
     * @param x signed 64.64 fixed point number
     * @param y signed 256-bit integer number
     * @return signed 256-bit integer number
     */
    function muli(int128 x, int256 y) internal pure returns (int256) {
        if (x == MIN_64x64) {
            require(
                y >= -0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF &&
                    y <= 0x1000000000000000000000000000000000000000000000000
            );
            return -y << 63;
        } else {
            bool negativeResult = false;
            if (x < 0) {
                x = -x;
                negativeResult = true;
            }
            if (y < 0) {
                y = -y; // We rely on overflow behavior here
                negativeResult = !negativeResult;
            }
            uint256 absoluteResult = mulu(x, uint256(y));
            if (negativeResult) {
                require(absoluteResult <= 0x8000000000000000000000000000000000000000000000000000000000000000);
                return -int256(absoluteResult); // We rely on overflow behavior here
            } else {
                require(absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
                return int256(absoluteResult);
            }
        }
    }

    /**
     * Calculate x * y rounding down, where x is signed 64.64 fixed point number
     * and y is unsigned 256-bit integer number.  Revert on overflow.
     *
     * @param x signed 64.64 fixed point number
     * @param y unsigned 256-bit integer number
     * @return unsigned 256-bit integer number
     */
    function mulu(int128 x, uint256 y) internal pure returns (uint256) {
        if (y == 0) return 0;

        require(x >= 0);

        uint256 lo = (uint256(x) * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) >> 64;
        uint256 hi = uint256(x) * (y >> 128);

        require(hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
        hi <<= 64;

        require(hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF - lo);
        return hi + lo;
    }

    /**
     * Calculate x / y rounding towards zero.  Revert on overflow or when y is
     * zero.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function div(int128 x, int128 y) internal pure returns (int128) {
        require(y != 0);
        int256 result = (int256(x) << 64) / y;
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Calculate x / y rounding towards zero, where x and y are signed 256-bit
     * integer numbers.  Revert on overflow or when y is zero.
     *
     * @param x signed 256-bit integer number
     * @param y signed 256-bit integer number
     * @return signed 64.64-bit fixed point number
     */
    function divi(int256 x, int256 y) internal pure returns (int128) {
        require(y != 0);

        bool negativeResult = false;
        if (x < 0) {
            x = -x; // We rely on overflow behavior here
            negativeResult = true;
        }
        if (y < 0) {
            y = -y; // We rely on overflow behavior here
            negativeResult = !negativeResult;
        }
        uint128 absoluteResult = divuu(uint256(x), uint256(y));
        if (negativeResult) {
            require(absoluteResult <= 0x80000000000000000000000000000000);
            return -int128(absoluteResult); // We rely on overflow behavior here
        } else {
            require(absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
            return int128(absoluteResult); // We rely on overflow behavior here
        }
    }

    /**
     * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
     * integer numbers.  Revert on overflow or when y is zero.
     *
     * @param x unsigned 256-bit integer number
     * @param y unsigned 256-bit integer number
     * @return signed 64.64-bit fixed point number
     */
    function divu(uint256 x, uint256 y) internal pure returns (int128) {
        require(y != 0);
        uint128 result = divuu(x, y);
        require(result <= uint128(MAX_64x64));
        return int128(result);
    }

    /**
     * Calculate -x.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function neg(int128 x) internal pure returns (int128) {
        require(x != MIN_64x64);
        return -x;
    }

    /**
     * Calculate |x|.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function abs(int128 x) internal pure returns (int128) {
        require(x != MIN_64x64);
        return x < 0 ? -x : x;
    }

    /**
     * Calculate 1 / x rounding towards zero.  Revert on overflow or when x is
     * zero.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function inv(int128 x) internal pure returns (int128) {
        require(x != 0);
        int256 result = int256(0x100000000000000000000000000000000) / x;
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Calculate arithmetics average of x and y, i.e. (x + y) / 2 rounding down.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function avg(int128 x, int128 y) internal pure returns (int128) {
        return int128((int256(x) + int256(y)) >> 1);
    }

    /**
     * Calculate geometric average of x and y, i.e. sqrt (x * y) rounding down.
     * Revert on overflow or in case x * y is negative.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function gavg(int128 x, int128 y) internal pure returns (int128) {
        int256 m = int256(x) * int256(y);
        require(m >= 0);
        require(m < 0x4000000000000000000000000000000000000000000000000000000000000000);
        return int128(sqrtu(uint256(m)));
    }

    /**
     * Calculate x^y assuming 0^0 is 1, where x is signed 64.64 fixed point number
     * and y is unsigned 256-bit integer number.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y uint256 value
     * @return signed 64.64-bit fixed point number
     */
    function pow(int128 x, uint256 y) internal pure returns (int128) {
        bool negative = x < 0 && y & 1 == 1;

        uint256 absX = uint128(x < 0 ? -x : x);
        uint256 absResult;
        absResult = 0x100000000000000000000000000000000;

        if (absX <= 0x10000000000000000) {
            absX <<= 63;
            while (y != 0) {
                if (y & 0x1 != 0) {
                    absResult = (absResult * absX) >> 127;
                }
                absX = (absX * absX) >> 127;

                if (y & 0x2 != 0) {
                    absResult = (absResult * absX) >> 127;
                }
                absX = (absX * absX) >> 127;

                if (y & 0x4 != 0) {
                    absResult = (absResult * absX) >> 127;
                }
                absX = (absX * absX) >> 127;

                if (y & 0x8 != 0) {
                    absResult = (absResult * absX) >> 127;
                }
                absX = (absX * absX) >> 127;

                y >>= 4;
            }

            absResult >>= 64;
        } else {
            uint256 absXShift = 63;
            if (absX < 0x1000000000000000000000000) {
                absX <<= 32;
                absXShift -= 32;
            }
            if (absX < 0x10000000000000000000000000000) {
                absX <<= 16;
                absXShift -= 16;
            }
            if (absX < 0x1000000000000000000000000000000) {
                absX <<= 8;
                absXShift -= 8;
            }
            if (absX < 0x10000000000000000000000000000000) {
                absX <<= 4;
                absXShift -= 4;
            }
            if (absX < 0x40000000000000000000000000000000) {
                absX <<= 2;
                absXShift -= 2;
            }
            if (absX < 0x80000000000000000000000000000000) {
                absX <<= 1;
                absXShift -= 1;
            }

            uint256 resultShift = 0;
            while (y != 0) {
                require(absXShift < 64);

                if (y & 0x1 != 0) {
                    absResult = (absResult * absX) >> 127;
                    resultShift += absXShift;
                    if (absResult > 0x100000000000000000000000000000000) {
                        absResult >>= 1;
                        resultShift += 1;
                    }
                }
                absX = (absX * absX) >> 127;
                absXShift <<= 1;
                if (absX >= 0x100000000000000000000000000000000) {
                    absX >>= 1;
                    absXShift += 1;
                }

                y >>= 1;
            }

            require(resultShift < 64);
            absResult >>= 64 - resultShift;
        }
        int256 result = negative ? -int256(absResult) : int256(absResult);
        require(result >= MIN_64x64 && result <= MAX_64x64);
        return int128(result);
    }

    /**
     * Calculate sqrt (x) rounding down.  Revert if x < 0.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function sqrt(int128 x) internal pure returns (int128) {
        require(x >= 0);
        return int128(sqrtu(uint256(x) << 64));
    }