// File: contracts/GalleryAuction.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.4;

import "./interfaces/IGallery.sol";
import "prb-math/contracts/PRBMathUD60x18Typed.sol";

contract GalleryAuction {
    using PRBMathUD60x18Typed for PRBMath.UD60x18;

    uint256 MINIMUM_AUCTION_PRICE = (1e15);
    uint256 constant ORIGINAL_SALE_PRICE = (1 ether) / 8;
    CurrentAuction auction;
    uint256[5]  lastSalePrices = [ORIGINAL_SALE_PRICE, ORIGINAL_SALE_PRICE, ORIGINAL_SALE_PRICE, ORIGINAL_SALE_PRICE, ORIGINAL_SALE_PRICE];
    uint256 currentSaleIndex = 0;
    address galleryAddress;
    address owner;
    bool finished;

    constructor(address newgalleryAddress, address newOwner)  {
        owner = newOwner;
        galleryAddress = newgalleryAddress;
        finished = false;
    }

    struct CurrentAuction {
        uint256 tokenid;
        uint256 auctionStartTime;
        uint256 startPrice;
    }



    function updateMinimumAuctionPrice(uint256 newPrice)
    external
    {
        require(msg.sender == owner);
        MINIMUM_AUCTION_PRICE = newPrice;
    }

    function withdraw()
    external {
        require(msg.sender == owner);
        payable(owner).transfer(address(this).balance);
    }

    function getCurrentAuctionDetails() external view returns (uint256, uint256){
        return (auction.tokenid, getCurrentAuctionPrice());
    }

    function kickOff() external {
        require(auction.tokenid == 0);
        startAuction();
    }

    function startAuction() private {
        require(!finished);

        IGallery gallery = IGallery(galleryAddress);

        auction.tokenid = gallery.mintNFT(address(this));
        uint256 totalPrices = lastSalePrices[0];
        for (uint256 i = 1; i < 5; i++) {
            totalPrices += lastSalePrices[i];
        }
        auction.startPrice = PRBMath.mulDiv(totalPrices, 3, 10);
        auction.auctionStartTime = block.timestamp;
    }


    function buyAuction(uint256 tokenId) payable external {
        require(!finished && tokenId == auction.tokenid && auction.auctionStartTime <= block.timestamp && auction.startPrice > 0);

        uint256 currentPrice = getCurrentAuctionPrice();
        require(msg.value >= currentPrice);
        IGallery gallery = IGallery(galleryAddress);
        uint256 currentAuctionToken = auction.tokenid;

        lastSalePrices[currentSaleIndex] = currentPrice;
        currentSaleIndex = (currentSaleIndex + 1) % 5;
        finished = gallery.totalSupply() >= 10000;

        if (!finished)
            startAuction();

        gallery.safeTransferFrom(address(this), msg.sender, currentAuctionToken);

    }

    function getCurrentAuctionPrice() public view returns (uint256)
    {
        require(!finished && auction.auctionStartTime <= block.timestamp && auction.startPrice > 0);



        //Time in seconds
        uint256 timeDiff = block.timestamp - auction.auctionStartTime;

        //Time in mins
        uint256 timeDiffInCounts = timeDiff / (60 minutes);
        if (timeDiffInCounts == 0) {
            return auction.startPrice;
        }
        //Amount of price reductions
        PRBMath.UD60x18 memory fixedPrice = PRBMath.UD60x18({value : auction.startPrice * 1e18});
        PRBMath.UD60x18 memory decrease = PRBMath.UD60x18({value : 98 * 1e16});
        decrease = decrease.powu(timeDiffInCounts);

        uint256 price = decrease.mul(fixedPrice).value / 1e18;

        if (price < MINIMUM_AUCTION_PRICE)
        {
            price = MINIMUM_AUCTION_PRICE;
        }

        return price;
    }
    function getFinished() external view returns (bool) {
        return finished;
    }
}

// File: contracts/interfaces/IGallery.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.4;
import "@openzeppelin/contracts/interfaces/IERC721.sol";

interface IGallery is IERC721 {
    function mintNFT(address recipient) external returns (uint256);
    function totalSupply()  external returns (uint256);
}

// File: prb-math/contracts/PRBMathUD60x18Typed.sol



// SPDX-License-Identifier: Unlicense
pragma solidity >=0.8.4;

import "./PRBMath.sol";

/// @title PRBMathUD60x18Typed
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math that works with uint256 numbers considered to have 18
/// trailing decimals. We call this number representation unsigned 60.18-decimal fixed-point, since there can be up to 60
/// digits in the integer part and up to 18 decimals in the fractional part. The numbers are bound by the minimum and the
/// maximum values permitted by the Solidity type uint256.
/// @dev This is the same as PRBMathUD59x18, except that it works with structs instead of raw uint256s.
library PRBMathUD60x18Typed {
    /// STORAGE ///

    /// @dev Half the SCALE number.
    uint256 internal constant HALF_SCALE = 5e17;

    /// @dev log2(e) as an unsigned 60.18-decimal fixed-point number.
    uint256 internal constant LOG2_E = 1_442695040888963407;

    /// @dev The maximum value an unsigned 60.18-decimal fixed-point number can have.
    uint256 internal constant MAX_UD60x18 =
        115792089237316195423570985008687907853269984665640564039457_584007913129639935;

    /// @dev The maximum whole value an unsigned 60.18-decimal fixed-point number can have.
    uint256 internal constant MAX_WHOLE_UD60x18 =
        115792089237316195423570985008687907853269984665640564039457_000000000000000000;

    /// @dev How many trailing decimals can be represented.
    uint256 internal constant SCALE = 1e18;

    /// @notice Adds two unsigned 60.18-decimal fixed-point numbers together, returning a new unsigned 60.18-decimal
    /// fixed-point number.
    /// @param x The first summand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The second summand as an unsigned 60.18-decimal fixed-point number.
    /// @param result The sum as an unsigned 59.18 decimal fixed-point number.
    function add(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        unchecked {
            uint256 rValue = x.value + y.value;
            if (rValue < x.value) {
                revert PRBMathUD60x18__AddOverflow(x.value, y.value);
            }
            result = PRBMath.UD60x18({ value: rValue });
        }
    }

    /// @notice Calculates the arithmetic average of x and y, rounding down.
    /// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
    /// @return result The arithmetic average as an unsigned 60.18-decimal fixed-point number.
    function avg(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        // The operations can never overflow.
        unchecked {
            // The last operand checks if both x and y are odd and if that is the case, we add 1 to the result. We need
            // to do this because if both numbers are odd, the 0.5 remainder gets truncated twice.
            uint256 rValue = (x.value >> 1) + (y.value >> 1) + (x.value & y.value & 1);
            result = PRBMath.UD60x18({ value: rValue });
        }
    }

    /// @notice Yields the least unsigned 60.18 decimal fixed-point number greater than or equal to x.
    ///
    /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
    /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
    ///
    /// Requirements:
    /// - x must be less than or equal to MAX_WHOLE_UD60x18.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number to ceil.
    /// @param result The least integer greater than or equal to x, as an unsigned 60.18-decimal fixed-point number.
    function ceil(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        uint256 xValue = x.value;
        if (xValue > MAX_WHOLE_UD60x18) {
            revert PRBMathUD60x18__CeilOverflow(xValue);
        }

        uint256 rValue;
        assembly {
            // Equivalent to "x % SCALE" but faster.
            let remainder := mod(xValue, SCALE)

            // Equivalent to "SCALE - remainder" but faster.
            let delta := sub(SCALE, remainder)

            // Equivalent to "x + delta * (remainder > 0 ? 1 : 0)" but faster.
            rValue := add(xValue, mul(delta, gt(remainder, 0)))
        }
        result = PRBMath.UD60x18({ value: rValue });
    }

    /// @notice Divides two unsigned 60.18-decimal fixed-point numbers, returning a new unsigned 60.18-decimal fixed-point number.
    ///
    /// @dev Uses mulDiv to enable overflow-safe multiplication and division.
    ///
    /// Requirements:
    /// - The denominator cannot be zero.
    ///
    /// @param x The numerator as an unsigned 60.18-decimal fixed-point number.
    /// @param y The denominator as an unsigned 60.18-decimal fixed-point number.
    /// @param result The quotient as an unsigned 60.18-decimal fixed-point number.
    function div(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        result = PRBMath.UD60x18({ value: PRBMath.mulDiv(x.value, SCALE, y.value) });
    }

    /// @notice Returns Euler's number as an unsigned 60.18-decimal fixed-point number.
    /// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
    function e() internal pure returns (PRBMath.UD60x18 memory result) {
        result = PRBMath.UD60x18({ value: 2_718281828459045235 });
    }

    /// @notice Calculates the natural exponent of x.
    ///
    /// @dev Based on the insight that e^x = 2^(x * log2(e)).
    ///
    /// Requirements:
    /// - All from "log2".
    /// - x must be less than 88.722839111672999628.
    ///
    /// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        uint256 xValue = x.value;

        // Without this check, the value passed to "exp2" would be greater than 192.
        if (xValue >= 133_084258667509499441) {
            revert PRBMathUD60x18__ExpInputTooBig(xValue);
        }

        // Do the fixed-point multiplication inline to save gas.
        unchecked {
            uint256 doubleScaleProduct = x.value * LOG2_E;
            PRBMath.UD60x18 memory exponent = PRBMath.UD60x18({ value: (doubleScaleProduct + HALF_SCALE) / SCALE });
            result = exp2(exponent);
        }
    }

    /// @notice Calculates the binary exponent of x using the binary fraction method.
    ///
    /// @dev See https://ethereum.stackexchange.com/q/79903/24693.
    ///
    /// Requirements:
    /// - x must be 192 or less.
    /// - The result must fit within MAX_UD60x18.
    ///
    /// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp2(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        // 2^192 doesn't fit within the 192.64-bit format used internally in this function.
        if (x.value >= 192e18) {
            revert PRBMathUD60x18__Exp2InputTooBig(x.value);
        }

        unchecked {
            // Convert x to the 192.64-bit fixed-point format.
            uint256 x192x64 = (x.value << 64) / SCALE;

            // Pass x to the PRBMath.exp2 function, which uses the 192.64-bit fixed-point number representation.
            result = PRBMath.UD60x18({ value: PRBMath.exp2(x192x64) });
        }
    }

    /// @notice Yields the greatest unsigned 60.18 decimal fixed-point number less than or equal to x.
    /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
    /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
    /// @param x The unsigned 60.18-decimal fixed-point number to floor.
    /// @param result The greatest integer less than or equal to x, as an unsigned 60.18-decimal fixed-point number.
    function floor(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        uint256 xValue = x.value;
        uint256 rValue;
        assembly {
            // Equivalent to "x % SCALE" but faster.
            let remainder := mod(xValue, SCALE)

            // Equivalent to "x - remainder * (remainder > 0 ? 1 : 0)" but faster.
            rValue := sub(xValue, mul(remainder, gt(remainder, 0)))
        }
        result = PRBMath.UD60x18({ value: rValue });
    }

    /// @notice Yields the excess beyond the floor of x.
    /// @dev Based on the odd function definition https://en.wikipedia.org/wiki/Fractional_part.
    /// @param x The unsigned 60.18-decimal fixed-point number to get the fractional part of.
    /// @param result The fractional part of x as an unsigned 60.18-decimal fixed-point number.
    function frac(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        uint256 xValue = x.value;
        uint256 rValue;
        assembly {
            rValue := mod(xValue, SCALE)
        }
        result = PRBMath.UD60x18({ value: rValue });
    }

    /// @notice Converts a number from basic integer form to unsigned 60.18-decimal fixed-point representation.
    ///
    /// @dev Requirements:
    /// - x must be less than or equal to MAX_UD60x18 divided by SCALE.
    ///
    /// @param x The basic integer to convert.
    /// @param result The same number in unsigned 60.18-decimal fixed-point representation.
    function fromUint(uint256 x) internal pure returns (PRBMath.UD60x18 memory result) {
        unchecked {
            if (x > MAX_UD60x18 / SCALE) {
                revert PRBMathUD60x18__FromUintOverflow(x);
            }
            result = PRBMath.UD60x18({ value: x * SCALE });
        }
    }

    /// @notice Calculates geometric mean of x and y, i.e. sqrt(x * y), rounding down.
    ///
    /// @dev Requirements:
    /// - x * y must fit within MAX_UD60x18, lest it overflows.
    ///
    /// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function gm(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        if (x.value == 0) {
            return PRBMath.UD60x18({ value: 0 });
        }

        unchecked {
            // Checking for overflow this way is faster than letting Solidity do it.
            uint256 xy = x.value * y.value;
            if (xy / x.value != y.value) {
                revert PRBMathUD60x18__GmOverflow(x.value, y.value);
            }

            // We don't need to multiply by the SCALE here because the x*y product had already picked up a factor of SCALE
            // during multiplication. See the comments within the "sqrt" function.
            result = PRBMath.UD60x18({ value: PRBMath.sqrt(xy) });
        }
    }

    /// @notice Calculates 1 / x, rounding toward zero.
    ///
    /// @dev Requirements:
    /// - x cannot be zero.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the inverse.
    /// @return result The inverse as an unsigned 60.18-decimal fixed-point number.
    function inv(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        unchecked {
            // 1e36 is SCALE * SCALE.
            result = PRBMath.UD60x18({ value: 1e36 / x.value });
        }
    }

    /// @notice Calculates the natural logarithm of x.
    ///
    /// @dev Based on the insight that ln(x) = log2(x) / log2(e).
    ///
    /// Requirements:
    /// - All from "log2".
    ///
    /// Caveats:
    /// - All from "log2".
    /// - This doesn't return exactly 1 for 2.718281828459045235, for that we would need more fine-grained precision.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the natural logarithm.
    /// @return result The natural logarithm as an unsigned 60.18-decimal fixed-point number.
    function ln(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        // Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
        // can return is 196205294292027477728.
        unchecked {
            uint256 rValue = (log2(x).value * SCALE) / LOG2_E;
            result = PRBMath.UD60x18({ value: rValue });
        }
    }

    /// @notice Calculates the common logarithm of x.
    ///
    /// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common
    /// logarithm based on the insight that log10(x) = log2(x) / log2(10).
    ///
    /// Requirements:
    /// - All from "log2".
    ///
    /// Caveats:
    /// - All from "log2".
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the common logarithm.
    /// @return result The common logarithm as an unsigned 60.18-decimal fixed-point number.
    function log10(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        uint256 xValue = x.value;
        if (xValue < SCALE) {
            revert PRBMathUD60x18__LogInputTooSmall(xValue);
        }

        // Note that the "mul" in this block is the assembly multiplication operation, not the "mul" function defined
        // in this contract.
        uint256 rValue;

        // prettier-ignore
        assembly {
            switch xValue
            case 1 { rValue := mul(SCALE, sub(0, 18)) }
            case 10 { rValue := mul(SCALE, sub(1, 18)) }
            case 100 { rValue := mul(SCALE, sub(2, 18)) }
            case 1000 { rValue := mul(SCALE, sub(3, 18)) }
            case 10000 { rValue := mul(SCALE, sub(4, 18)) }
            case 100000 { rValue := mul(SCALE, sub(5, 18)) }
            case 1000000 { rValue := mul(SCALE, sub(6, 18)) }
            case 10000000 { rValue := mul(SCALE, sub(7, 18)) }
            case 100000000 { rValue := mul(SCALE, sub(8, 18)) }
            case 1000000000 { rValue := mul(SCALE, sub(9, 18)) }
            case 10000000000 { rValue := mul(SCALE, sub(10, 18)) }
            case 100000000000 { rValue := mul(SCALE, sub(11, 18)) }
            case 1000000000000 { rValue := mul(SCALE, sub(12, 18)) }
            case 10000000000000 { rValue := mul(SCALE, sub(13, 18)) }
            case 100000000000000 { rValue := mul(SCALE, sub(14, 18)) }
            case 1000000000000000 { rValue := mul(SCALE, sub(15, 18)) }
            case 10000000000000000 { rValue := mul(SCALE, sub(16, 18)) }
            case 100000000000000000 { rValue := mul(SCALE, sub(17, 18)) }
            case 1000000000000000000 { rValue := 0 }
            case 10000000000000000000 { rValue := SCALE }
            case 100000000000000000000 { rValue := mul(SCALE, 2) }
            case 1000000000000000000000 { rValue := mul(SCALE, 3) }
            case 10000000000000000000000 { rValue := mul(SCALE, 4) }
            case 100000000000000000000000 { rValue := mul(SCALE, 5) }
            case 1000000000000000000000000 { rValue := mul(SCALE, 6) }
            case 10000000000000000000000000 { rValue := mul(SCALE, 7) }
            case 100000000000000000000000000 { rValue := mul(SCALE, 8) }
            case 1000000000000000000000000000 { rValue := mul(SCALE, 9) }
            case 10000000000000000000000000000 { rValue := mul(SCALE, 10) }
            case 100000000000000000000000000000 { rValue := mul(SCALE, 11) }
            case 1000000000000000000000000000000 { rValue := mul(SCALE, 12) }
            case 10000000000000000000000000000000 { rValue := mul(SCALE, 13) }
            case 100000000000000000000000000000000 { rValue := mul(SCALE, 14) }
            case 1000000000000000000000000000000000 { rValue := mul(SCALE, 15) }
            case 10000000000000000000000000000000000 { rValue := mul(SCALE, 16) }
            case 100000000000000000000000000000000000 { rValue := mul(SCALE, 17) }
            case 1000000000000000000000000000000000000 { rValue := mul(SCALE, 18) }
            case 10000000000000000000000000000000000000 { rValue := mul(SCALE, 19) }
            case 100000000000000000000000000000000000000 { rValue := mul(SCALE, 20) }
            case 1000000000000000000000000000000000000000 { rValue := mul(SCALE, 21) }
            case 10000000000000000000000000000000000000000 { rValue := mul(SCALE, 22) }
            case 100000000000000000000000000000000000000000 { rValue := mul(SCALE, 23) }
            case 1000000000000000000000000000000000000000000 { rValue := mul(SCALE, 24) }
            case 10000000000000000000000000000000000000000000 { rValue := mul(SCALE, 25) }
            case 100000000000000000000000000000000000000000000 { rValue := mul(SCALE, 26) }
            case 1000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 27) }
            case 10000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 28) }
            case 100000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 29) }
            case 1000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 30) }
            case 10000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 31) }
            case 100000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 32) }
            case 1000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 33) }
            case 10000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 34) }
            case 100000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 35) }
            case 1000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 36) }
            case 10000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 37) }
            case 100000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 38) }
            case 1000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 39) }
            case 10000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 40) }
            case 100000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 41) }
            case 1000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 42) }
            case 10000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 43) }
            case 100000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 44) }
            case 1000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 45) }
            case 10000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 46) }
            case 100000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 47) }
            case 1000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 48) }
            case 10000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 49) }
            case 100000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 50) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 51) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 52) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 53) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 54) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 55) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 56) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 57) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 58) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000000 { rValue := mul(SCALE, 59) }
            default {
                rValue := MAX_UD60x18
            }
        }

        if (rValue != MAX_UD60x18) {
            result = PRBMath.UD60x18({ value: rValue });
        } else {
            // Do the fixed-point division inline to save gas. The denominator is log2(10).
            unchecked {
                rValue = (log2(x).value * SCALE) / 3_321928094887362347;
                result = PRBMath.UD60x18({ value: rValue });
            }
        }
    }

    /// @notice Calculates the binary logarithm of x.
    ///
    /// @dev Based on the iterative approximation algorithm.
    /// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation
    ///
    /// Requirements:
    /// - x must be greater than or equal to SCALE, otherwise the result would be negative.
    ///
    /// Caveats:
    /// - The results are nor perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the binary logarithm.
    /// @return result The binary logarithm as an unsigned 60.18-decimal fixed-point number.
    function log2(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        uint256 xValue = x.value;
        if (xValue < SCALE) {
            revert PRBMathUD60x18__LogInputTooSmall(xValue);
        }
        unchecked {
            // Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
            uint256 n = PRBMath.mostSignificantBit(xValue / SCALE);

            // The integer part of the logarithm as an unsigned 60.18-decimal fixed-point number. The operation can't overflow
            // because n is maximum 255 and SCALE is 1e18.
            uint256 rValue = n * SCALE;

            // This is y = x * 2^(-n).
            uint256 y = xValue >> n;

            // If y = 1, the fractional part is zero.
            if (y == SCALE) {
                return PRBMath.UD60x18({ value: rValue });
            }

            // Calculate the fractional part via the iterative approximation.
            // The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster.
            for (uint256 delta = HALF_SCALE; delta > 0; delta >>= 1) {
                y = (y * y) / SCALE;

                // Is y^2 > 2 and so in the range [2,4)?
                if (y >= 2 * SCALE) {
                    // Add the 2^(-m) factor to the logarithm.
                    rValue += delta;

                    // Corresponds to z/2 on Wikipedia.
                    y >>= 1;
                }
            }
            result = PRBMath.UD60x18({ value: rValue });
        }
    }

    /// @notice Multiplies two unsigned 60.18-decimal fixed-point numbers together, returning a new unsigned 60.18-decimal
    /// fixed-point number.
    /// @dev See the documentation for the "PRBMath.mulDivFixedPoint" function.
    /// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
    /// @return result The product as an unsigned 60.18-decimal fixed-point number.
    function mul(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        result = PRBMath.UD60x18({ value: PRBMath.mulDivFixedPoint(x.value, y.value) });
    }

    /// @notice Returns PI as an unsigned 60.18-decimal fixed-point number.
    function pi() internal pure returns (PRBMath.UD60x18 memory result) {
        result = PRBMath.UD60x18({ value: 3_141592653589793238 });
    }

    /// @notice Raises x to the power of y.
    ///
    /// @dev Based on the insight that x^y = 2^(log2(x) * y).
    ///
    /// Requirements:
    /// - All from "exp2", "log2" and "mul".
    ///
    /// Caveats:
    /// - All from "exp2", "log2" and "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x Number to raise to given power y, as an unsigned 60.18-decimal fixed-point number.
    /// @param y Exponent to raise x to, as an unsigned 60.18-decimal fixed-point number.
    /// @return result x raised to power y, as an unsigned 60.18-decimal fixed-point number.
    function pow(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        if (x.value == 0) {
            return PRBMath.UD60x18({ value: y.value == 0 ? SCALE : uint256(0) });
        } else {
            result = exp2(mul(log2(x), y));
        }
    }

    /// @notice Raises x (unsigned 60.18-decimal fixed-point number) to the power of y (basic unsigned integer) using the
    /// famous algorithm "exponentiation by squaring".
    ///
    /// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring
    ///
    /// Requirements:
    /// - The result must fit within MAX_UD60x18.
    ///
    /// Caveats:
    /// - All from "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x The base as an unsigned 60.18-decimal fixed-point number.
    /// @param y The exponent as an uint256.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function powu(PRBMath.UD60x18 memory x, uint256 y) internal pure returns (PRBMath.UD60x18 memory result) {
        // Calculate the first iteration of the loop in advance.
        uint256 xValue = x.value;
        uint256 rValue = y & 1 > 0 ? xValue : SCALE;

        // Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster.
        for (y >>= 1; y > 0; y >>= 1) {
            xValue = PRBMath.mulDivFixedPoint(xValue, xValue);

            // Equivalent to "y % 2 == 1" but faster.
            if (y & 1 > 0) {
                rValue = PRBMath.mulDivFixedPoint(rValue, xValue);
            }
        }
        result = PRBMath.UD60x18({ value: rValue });
    }

    /// @notice Returns 1 as an unsigned 60.18-decimal fixed-point number.
    function scale() internal pure returns (PRBMath.UD60x18 memory result) {
        result = PRBMath.UD60x18({ value: SCALE });
    }

    /// @notice Calculates the square root of x, rounding down.
    /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
    ///
    /// Requirements:
    /// - x must be less than MAX_UD60x18 / SCALE.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the square root.
    /// @return result The result as an unsigned 60.18-decimal fixed-point .
    function sqrt(PRBMath.UD60x18 memory x) internal pure returns (PRBMath.UD60x18 memory result) {
        unchecked {
            if (x.value > MAX_UD60x18 / SCALE) {
                revert PRBMathUD60x18__SqrtOverflow(x.value);
            }
            // Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two unsigned
            // 60.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
            result = PRBMath.UD60x18({ value: PRBMath.sqrt(x.value * SCALE) });
        }
    }

    /// @notice Subtracts one unsigned 60.18-decimal fixed-point number from another one, returning a new unsigned 60.18-decimal
    /// fixed-point number.
    /// @param x The minuend as an unsigned 60.18-decimal fixed-point number.
    /// @param y The subtrahend as an unsigned 60.18-decimal fixed-point number.
    /// @param result The difference as an unsigned 60.18 decimal fixed-point number.
    function sub(PRBMath.UD60x18 memory x, PRBMath.UD60x18 memory y)
        internal
        pure
        returns (PRBMath.UD60x18 memory result)
    {
        unchecked {
            if (x.value < y.value) {
                revert PRBMathUD60x18__SubUnderflow(x.value, y.value);
            }
            result = PRBMath.UD60x18({ value: x.value - y.value });
        }
    }

    /// @notice Converts a unsigned 60.18-decimal fixed-point number to basic integer form, rounding down in the process.
    /// @param x The unsigned 60.18-decimal fixed-point number to convert.
    /// @return result The same number in basic integer form.
    function toUint(PRBMath.UD60x18 memory x) internal pure returns (uint256 result) {
        unchecked {
            result = x.value / SCALE;
        }
    }
}

// File: @openzeppelin/contracts/interfaces/IERC721.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../token/ERC721/IERC721.sol";

// File: @openzeppelin/contracts/token/ERC721/IERC721.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../../utils/introspection/IERC165.sol";

/**
 * @dev Required interface of an ERC721 compliant contract.
 */
interface IERC721 is IERC165 {
    /**
     * @dev Emitted when `tokenId` token is transferred from `from` to `to`.
     */
    event Transfer(address indexed from, address indexed to, uint256 indexed tokenId);

    /**
     * @dev Emitted when `owner` enables `approved` to manage the `tokenId` token.
     */
    event Approval(address indexed owner, address indexed approved, uint256 indexed tokenId);

    /**
     * @dev Emitted when `owner` enables or disables (`approved`) `operator` to manage all of its assets.
     */
    event ApprovalForAll(address indexed owner, address indexed operator, bool approved);

    /**
     * @dev Returns the number of tokens in ``owner``'s account.
     */
    function balanceOf(address owner) external view returns (uint256 balance);

    /**
     * @dev Returns the owner of the `tokenId` token.
     *
     * Requirements:
     *
     * - `tokenId` must exist.
     */
    function ownerOf(uint256 tokenId) external view returns (address owner);

    /**
     * @dev Safely transfers `tokenId` token from `from` to `to`, checking first that contract recipients
     * are aware of the ERC721 protocol to prevent tokens from being forever locked.
     *
     * Requirements:
     *
     * - `from` cannot be the zero address.
     * - `to` cannot be the zero address.
     * - `tokenId` token must exist and be owned by `from`.
     * - If the caller is not `from`, it must be have been allowed to move this token by either {approve} or {setApprovalForAll}.
     * - If `to` refers to a smart contract, it must implement {IERC721Receiver-onERC721Received}, which is called upon a safe transfer.
     *
     * Emits a {Transfer} event.
     */
    function safeTransferFrom(
        address from,
        address to,
        uint256 tokenId
    ) external;

    /**
     * @dev Transfers `tokenId` token from `from` to `to`.
     *
     * WARNING: Usage of this method is discouraged, use {safeTransferFrom} whenever possible.
     *
     * Requirements:
     *
     * - `from` cannot be the zero address.
     * - `to` cannot be the zero address.
     * - `tokenId` token must be owned by `from`.
     * - If the caller is not `from`, it must be approved to move this token by either {approve} or {setApprovalForAll}.
     *
     * Emits a {Transfer} event.
     */
    function transferFrom(
        address from,
        address to,
        uint256 tokenId
    ) external;

    /**
     * @dev Gives permission to `to` to transfer `tokenId` token to another account.
     * The approval is cleared when the token is transferred.
     *
     * Only a single account can be approved at a time, so approving the zero address clears previous approvals.
     *
     * Requirements:
     *
     * - The caller must own the token or be an approved operator.
     * - `tokenId` must exist.
     *
     * Emits an {Approval} event.
     */
    function approve(address to, uint256 tokenId) external;

    /**
     * @dev Returns the account approved for `tokenId` token.
     *
     * Requirements:
     *
     * - `tokenId` must exist.
     */
    function getApproved(uint256 tokenId) external view returns (address operator);

    /**
     * @dev Approve or remove `operator` as an operator for the caller.
     * Operators can call {transferFrom} or {safeTransferFrom} for any token owned by the caller.
     *
     * Requirements:
     *
     * - The `operator` cannot be the caller.
     *
     * Emits an {ApprovalForAll} event.
     */
    function setApprovalForAll(address operator, bool _approved) external;

    /**
     * @dev Returns if the `operator` is allowed to manage all of the assets of `owner`.
     *
     * See {setApprovalForAll}
     */
    function isApprovedForAll(address owner, address operator) external view returns (bool);

    /**
     * @dev Safely transfers `tokenId` token from `from` to `to`.
     *
     * Requirements:
     *
     * - `from` cannot be the zero address.
     * - `to` cannot be the zero address.
     * - `tokenId` token must exist and be owned by `from`.
     * - If the caller is not `from`, it must be approved to move this token by either {approve} or {setApprovalForAll}.
     * - If `to` refers to a smart contract, it must implement {IERC721Receiver-onERC721Received}, which is called upon a safe transfer.
     *
     * Emits a {Transfer} event.
     */
    function safeTransferFrom(
        address from,
        address to,
        uint256 tokenId,
        bytes calldata data
    ) external;
}

// File: @openzeppelin/contracts/utils/introspection/IERC165.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC165 standard, as defined in the
 * https://eips.ethereum.org/EIPS/eip-165[EIP].
 *
 * Implementers can declare support of contract interfaces, which can then be
 * queried by others ({ERC165Checker}).
 *
 * For an implementation, see {ERC165}.
 */
interface IERC165 {
    /**
     * @dev Returns true if this contract implements the interface defined by
     * `interfaceId`. See the corresponding
     * https://eips.ethereum.org/EIPS/eip-165#how-interfaces-are-identified[EIP section]
     * to learn more about how these ids are created.
     *
     * This function call must use less than 30 000 gas.
     */
    function supportsInterface(bytes4 interfaceId) external view returns (bool);
}

// File: prb-math/contracts/PRBMath.sol



// SPDX-License-Identifier: Unlicense
pragma solidity >=0.8.4;

/// @notice Emitted when the result overflows uint256.
error PRBMath__MulDivFixedPointOverflow(uint256 prod1);

/// @notice Emitted when the result overflows uint256.
error PRBMath__MulDivOverflow(uint256 prod1, uint256 denominator);

/// @notice Emitted when one of the inputs is type(int256).min.
error PRBMath__MulDivSignedInputTooSmall();

/// @notice Emitted when the intermediary absolute result overflows int256.
error PRBMath__MulDivSignedOverflow(uint256 rAbs);

/// @notice Emitted when the input is MIN_SD59x18.
error PRBMathSD59x18__AbsInputTooSmall();

/// @notice Emitted when ceiling a number overflows SD59x18.
error PRBMathSD59x18__CeilOverflow(int256 x);

/// @notice Emitted when one of the inputs is MIN_SD59x18.
error PRBMathSD59x18__DivInputTooSmall();

/// @notice Emitted when one of the intermediary unsigned results overflows SD59x18.
error PRBMathSD59x18__DivOverflow(uint256 rAbs);

/// @notice Emitted when the input is greater than 133.084258667509499441.
error PRBMathSD59x18__ExpInputTooBig(int256 x);

/// @notice Emitted when the input is greater than 192.
error PRBMathSD59x18__Exp2InputTooBig(int256 x);

/// @notice Emitted when flooring a number underflows SD59x18.
error PRBMathSD59x18__FloorUnderflow(int256 x);

/// @notice Emitted when converting a basic integer to the fixed-point format overflows SD59x18.
error PRBMathSD59x18__FromIntOverflow(int256 x);

/// @notice Emitted when converting a basic integer to the fixed-point format underflows SD59x18.
error PRBMathSD59x18__FromIntUnderflow(int256 x);

/// @notice Emitted when the product of the inputs is negative.
error PRBMathSD59x18__GmNegativeProduct(int256 x, int256 y);

/// @notice Emitted when multiplying the inputs overflows SD59x18.
error PRBMathSD59x18__GmOverflow(int256 x, int256 y);

/// @notice Emitted when the input is less than or equal to zero.
error PRBMathSD59x18__LogInputTooSmall(int256 x);

/// @notice Emitted when one of the inputs is MIN_SD59x18.
error PRBMathSD59x18__MulInputTooSmall();

/// @notice Emitted when the intermediary absolute result overflows SD59x18.
error PRBMathSD59x18__MulOverflow(uint256 rAbs);

/// @notice Emitted when the intermediary absolute result overflows SD59x18.
error PRBMathSD59x18__PowuOverflow(uint256 rAbs);

/// @notice Emitted when the input is negative.
error PRBMathSD59x18__SqrtNegativeInput(int256 x);

/// @notice Emitted when the calculating the square root overflows SD59x18.
error PRBMathSD59x18__SqrtOverflow(int256 x);

/// @notice Emitted when addition overflows UD60x18.
error PRBMathUD60x18__AddOverflow(uint256 x, uint256 y);

/// @notice Emitted when ceiling a number overflows UD60x18.
error PRBMathUD60x18__CeilOverflow(uint256 x);

/// @notice Emitted when the input is greater than 133.084258667509499441.
error PRBMathUD60x18__ExpInputTooBig(uint256 x);

/// @notice Emitted when the input is greater than 192.
error PRBMathUD60x18__Exp2InputTooBig(uint256 x);

/// @notice Emitted when converting a basic integer to the fixed-point format format overflows UD60x18.
error PRBMathUD60x18__FromUintOverflow(uint256 x);

/// @notice Emitted when multiplying the inputs overflows UD60x18.
error PRBMathUD60x18__GmOverflow(uint256 x, uint256 y);

/// @notice Emitted when the input is less than 1.
error PRBMathUD60x18__LogInputTooSmall(uint256 x);

/// @notice Emitted when the calculating the square root overflows UD60x18.
error PRBMathUD60x18__SqrtOverflow(uint256 x);

/// @notice Emitted when subtraction underflows UD60x18.
error PRBMathUD60x18__SubUnderflow(uint256 x, uint256 y);

/// @dev Common mathematical functions used in both PRBMathSD59x18 and PRBMathUD60x18. Note that this shared library
/// does not always assume the signed 59.18-decimal fixed-point or the unsigned 60.18-decimal fixed-point
/// representation. When it does not, it is explicitly mentioned in the NatSpec documentation.
library PRBMath {
    /// STRUCTS ///

    struct SD59x18 {
        int256 value;
    }

    struct UD60x18 {
        uint256 value;
    }

    /// STORAGE ///

    /// @dev How many trailing decimals can be represented.
    uint256 internal constant SCALE = 1e18;

    /// @dev Largest power of two divisor of SCALE.
    uint256 internal constant SCALE_LPOTD = 262144;

    /// @dev SCALE inverted mod 2^256.
    uint256 internal constant SCALE_INVERSE =
        78156646155174841979727994598816262306175212592076161876661_508869554232690281;

    /// FUNCTIONS ///

    /// @notice Calculates the binary exponent of x using the binary fraction method.
    /// @dev Has to use 192.64-bit fixed-point numbers.
    /// See https://ethereum.stackexchange.com/a/96594/24693.
    /// @param x The exponent as an unsigned 192.64-bit fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp2(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            // Start from 0.5 in the 192.64-bit fixed-point format.
            result = 0x800000000000000000000000000000000000000000000000;

            // Multiply the result by root(2, 2^-i) when the bit at position i is 1. None of the intermediary results overflows
            // because the initial result is 2^191 and all magic factors are less than 2^65.
            if (x & 0x8000000000000000 > 0) {
                result = (result * 0x16A09E667F3BCC909) >> 64;
            }
            if (x & 0x4000000000000000 > 0) {
                result = (result * 0x1306FE0A31B7152DF) >> 64;
            }
            if (x & 0x2000000000000000 > 0) {
                result = (result * 0x1172B83C7D517ADCE) >> 64;
            }
            if (x & 0x1000000000000000 > 0) {
                result = (result * 0x10B5586CF9890F62A) >> 64;
            }
            if (x & 0x800000000000000 > 0) {
                result = (result * 0x1059B0D31585743AE) >> 64;
            }
            if (x & 0x400000000000000 > 0) {
                result = (result * 0x102C9A3E778060EE7) >> 64;
            }
            if (x & 0x200000000000000 > 0) {
                result = (result * 0x10163DA9FB33356D8) >> 64;
            }
            if (x & 0x100000000000000 > 0) {
                result = (result * 0x100B1AFA5ABCBED61) >> 64;
            }
            if (x & 0x80000000000000 > 0) {
                result = (result * 0x10058C86DA1C09EA2) >> 64;
            }
            if (x & 0x40000000000000 > 0) {
                result = (result * 0x1002C605E2E8CEC50) >> 64;
            }
            if (x & 0x20000000000000 > 0) {
                result = (result * 0x100162F3904051FA1) >> 64;
            }
            if (x & 0x10000000000000 > 0) {
                result = (result * 0x1000B175EFFDC76BA) >> 64;
            }
            if (x & 0x8000000000000 > 0) {
                result = (result * 0x100058BA01FB9F96D) >> 64;
            }
            if (x & 0x4000000000000 > 0) {
                result = (result * 0x10002C5CC37DA9492) >> 64;
            }
            if (x & 0x2000000000000 > 0) {
                result = (result * 0x1000162E525EE0547) >> 64;
            }
            if (x & 0x1000000000000 > 0) {
                result = (result * 0x10000B17255775C04) >> 64;
            }
            if (x & 0x800000000000 > 0) {
                result = (result * 0x1000058B91B5BC9AE) >> 64;
            }
            if (x & 0x400000000000 > 0) {
                result = (result * 0x100002C5C89D5EC6D) >> 64;
            }
            if (x & 0x200000000000 > 0) {
                result = (result * 0x10000162E43F4F831) >> 64;
            }
            if (x & 0x100000000000 > 0) {
                result = (result * 0x100000B1721BCFC9A) >> 64;
            }
            if (x & 0x80000000000 > 0) {
                result = (result * 0x10000058B90CF1E6E) >> 64;
            }
            if (x & 0x40000000000 > 0) {
                result = (result * 0x1000002C5C863B73F) >> 64;
            }
            if (x & 0x20000000000 > 0) {
                result = (result * 0x100000162E430E5A2) >> 64;
            }
            if (x & 0x10000000000 > 0) {
                result = (result * 0x1000000B172183551) >> 64;
            }
            if (x & 0x8000000000 > 0) {
                result = (result * 0x100000058B90C0B49) >> 64;
            }
            if (x & 0x4000000000 > 0) {
                result = (result * 0x10000002C5C8601CC) >> 64;
            }
            if (x & 0x2000000000 > 0) {
                result = (result * 0x1000000162E42FFF0) >> 64;
            }
            if (x & 0x1000000000 > 0) {
                result = (result * 0x10000000B17217FBB) >> 64;
            }
            if (x & 0x800000000 > 0) {
                result = (result * 0x1000000058B90BFCE) >> 64;
            }
            if (x & 0x400000000 > 0) {
                result = (result * 0x100000002C5C85FE3) >> 64;
            }
            if (x & 0x200000000 > 0) {
                result = (result * 0x10000000162E42FF1) >> 64;
            }
            if (x & 0x100000000 > 0) {
                result = (result * 0x100000000B17217F8) >> 64;
            }
            if (x & 0x80000000 > 0) {
                result = (result * 0x10000000058B90BFC) >> 64;
            }
            if (x & 0x40000000 > 0) {
                result = (result * 0x1000000002C5C85FE) >> 64;
            }
            if (x & 0x20000000 > 0) {
                result = (result * 0x100000000162E42FF) >> 64;
            }
            if (x & 0x10000000 > 0) {
                result = (result * 0x1000000000B17217F) >> 64;
            }
            if (x & 0x8000000 > 0) {
                result = (result * 0x100000000058B90C0) >> 64;
            }
            if (x & 0x4000000 > 0) {
                result = (result * 0x10000000002C5C860) >> 64;
            }
            if (x & 0x2000000 > 0) {
                result = (result * 0x1000000000162E430) >> 64;
            }
            if (x & 0x1000000 > 0) {
                result = (result * 0x10000000000B17218) >> 64;
            }
            if (x & 0x800000 > 0) {
                result = (result * 0x1000000000058B90C) >> 64;
            }
            if (x & 0x400000 > 0) {
                result = (result * 0x100000000002C5C86) >> 64;
            }
            if (x & 0x200000 > 0) {
                result = (result * 0x10000000000162E43) >> 64;
            }
            if (x & 0x100000 > 0) {
                result = (result * 0x100000000000B1721) >> 64;
            }
            if (x & 0x80000 > 0) {
                result = (result * 0x10000000000058B91) >> 64;
            }
            if (x & 0x40000 > 0) {
                result = (result * 0x1000000000002C5C8) >> 64;
            }
            if (x & 0x20000 > 0) {
                result = (result * 0x100000000000162E4) >> 64;
            }
            if (x & 0x10000 > 0) {
                result = (result * 0x1000000000000B172) >> 64;
            }
            if (x & 0x8000 > 0) {
                result = (result * 0x100000000000058B9) >> 64;
            }
            if (x & 0x4000 > 0) {
                result = (result * 0x10000000000002C5D) >> 64;
            }
            if (x & 0x2000 > 0) {
                result = (result * 0x1000000000000162E) >> 64;
            }
            if (x & 0x1000 > 0) {
                result = (result * 0x10000000000000B17) >> 64;
            }
            if (x & 0x800 > 0) {
                result = (result * 0x1000000000000058C) >> 64;
            }
            if (x & 0x400 > 0) {
                result = (result * 0x100000000000002C6) >> 64;
            }
            if (x & 0x200 > 0) {
                result = (result * 0x10000000000000163) >> 64;
            }
            if (x & 0x100 > 0) {
                result = (result * 0x100000000000000B1) >> 64;
            }
            if (x & 0x80 > 0) {
                result = (result * 0x10000000000000059) >> 64;
            }
            if (x & 0x40 > 0) {
                result = (result * 0x1000000000000002C) >> 64;
            }
            if (x & 0x20 > 0) {
                result = (result * 0x10000000000000016) >> 64;
            }
            if (x & 0x10 > 0) {
                result = (result * 0x1000000000000000B) >> 64;
            }
            if (x & 0x8 > 0) {
                result = (result * 0x10000000000000006) >> 64;
            }
            if (x & 0x4 > 0) {
                result = (result * 0x10000000000000003) >> 64;
            }
            if (x & 0x2 > 0) {
                result = (result * 0x10000000000000001) >> 64;
            }
            if (x & 0x1 > 0) {
                result = (result * 0x10000000000000001) >> 64;
            }

            // We're doing two things at the same time:
            //
            //   1. Multiply the result by 2^n + 1, where "2^n" is the integer part and the one is added to account for
            //      the fact that we initially set the result to 0.5. This is accomplished by subtracting from 191
            //      rather than 192.
            //   2. Convert the result to the unsigned 60.18-decimal fixed-point format.
            //
            // This works because 2^(191-ip) = 2^ip / 2^191, where "ip" is the integer part "2^n".
            result *= SCALE;
            result >>= (191 - (x >> 64));
        }
    }

    /// @notice Finds the zero-based index of the first one in the binary representation of x.
    /// @dev See the note on msb in the "Find First Set" Wikipedia article https://en.wikipedia.org/wiki/Find_first_set
    /// @param x The uint256 number for which to find the index of the most significant bit.
    /// @return msb The index of the most significant bit as an uint256.
    function mostSignificantBit(uint256 x) internal pure returns (uint256 msb) {
        if (x >= 2**128) {
            x >>= 128;
            msb += 128;
        }
        if (x >= 2**64) {
            x >>= 64;
            msb += 64;
        }
        if (x >= 2**32) {
            x >>= 32;
            msb += 32;
        }
        if (x >= 2**16) {
            x >>= 16;
            msb += 16;
        }
        if (x >= 2**8) {
            x >>= 8;
            msb += 8;
        }
        if (x >= 2**4) {
            x >>= 4;
            msb += 4;
        }
        if (x >= 2**2) {
            x >>= 2;
            msb += 2;
        }
        if (x >= 2**1) {
            // No need to shift x any more.
            msb += 1;
        }
    }

    /// @notice Calculates floor(x*y÷denominator) with full precision.
    ///
    /// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv.
    ///
    /// Requirements:
    /// - The denominator cannot be zero.
    /// - The result must fit within uint256.
    ///
    /// Caveats:
    /// - This function does not work with fixed-point numbers.
    ///
    /// @param x The multiplicand as an uint256.
    /// @param y The multiplier as an uint256.
    /// @param denominator The divisor as an uint256.
    /// @return result The result as an uint256.
    function mulDiv(
        uint256 x,
        uint256 y,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use
        // use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256
        // variables such that product = prod1 * 2^256 + prod0.
        uint256 prod0; // Least significant 256 bits of the product
        uint256 prod1; // Most significant 256 bits of the product
        assembly {
            let mm := mulmod(x, y, not(0))
            prod0 := mul(x, y)
            prod1 := sub(sub(mm, prod0), lt(mm, prod0))
        }

        // Handle non-overflow cases, 256 by 256 division.
        if (prod1 == 0) {
            unchecked {
                result = prod0 / denominator;
            }
            return result;
        }

        // Make sure the result is less than 2^256. Also prevents denominator == 0.
        if (prod1 >= denominator) {
            revert PRBMath__MulDivOverflow(prod1, denominator);
        }

        ///////////////////////////////////////////////
        // 512 by 256 division.
        ///////////////////////////////////////////////

        // Make division exact by subtracting the remainder from [prod1 prod0].
        uint256 remainder;
        assembly {
            // Compute remainder using mulmod.
            remainder := mulmod(x, y, denominator)

            // Subtract 256 bit number from 512 bit number.
            prod1 := sub(prod1, gt(remainder, prod0))
            prod0 := sub(prod0, remainder)
        }

        // Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1.
        // See https://cs.stackexchange.com/q/138556/92363.
        unchecked {
            // Does not overflow because the denominator cannot be zero at this stage in the function.
            uint256 lpotdod = denominator & (~denominator + 1);
            assembly {
                // Divide denominator by lpotdod.
                denominator := div(denominator, lpotdod)

                // Divide [prod1 prod0] by lpotdod.
                prod0 := div(prod0, lpotdod)

                // Flip lpotdod such that it is 2^256 / lpotdod. If lpotdod is zero, then it becomes one.
                lpotdod := add(div(sub(0, lpotdod), lpotdod), 1)
            }

            // Shift in bits from prod1 into prod0.
            prod0 |= prod1 * lpotdod;

            // Invert denominator mod 2^256. Now that denominator is an odd number, it has an inverse modulo 2^256 such
            // that denominator * inv = 1 mod 2^256. Compute the inverse by starting with a seed that is correct for
            // four bits. That is, denominator * inv = 1 mod 2^4.
            uint256 inverse = (3 * denominator) ^ 2;

            // Use the Newton-Raphson iteration to improve the precision. Thanks to Hensel's lifting lemma, this also works
            // in modular arithmetic, doubling the correct bits in each step.
            inverse *= 2 - denominator * inverse; // inverse mod 2^8
            inverse *= 2 - denominator * inverse; // inverse mod 2^16
            inverse *= 2 - denominator * inverse; // inverse mod 2^32
            inverse *= 2 - denominator * inverse; // inverse mod 2^64
            inverse *= 2 - denominator * inverse; // inverse mod 2^128
            inverse *= 2 - denominator * inverse; // inverse mod 2^256

            // Because the division is now exact we can divide by multiplying with the modular inverse of denominator.
            // This will give us the correct result modulo 2^256. Since the preconditions guarantee that the outcome is
            // less than 2^256, this is the final result. We don't need to compute the high bits of the result and prod1
            // is no longer required.
            result = prod0 * inverse;
            return result;
        }
    }

    /// @notice Calculates floor(x*y÷1e18) with full precision.
    ///
    /// @dev Variant of "mulDiv" with constant folding, i.e. in which the denominator is always 1e18. Before returning the
    /// final result, we add 1 if (x * y) % SCALE >= HALF_SCALE. Without this, 6.6e-19 would be truncated to 0 instead of
    /// being rounded to 1e-18.  See "Listing 6" and text above it at https://accu.org/index.php/journals/1717.
    ///
    /// Requirements:
    /// - The result must fit within uint256.
    ///
    /// Caveats:
    /// - The body is purposely left uncommented; see the NatSpec comments in "PRBMath.mulDiv" to understand how this works.
    /// - It is assumed that the result can never be type(uint256).max when x and y solve the following two equations:
    ///     1. x * y = type(uint256).max * SCALE
    ///     2. (x * y) % SCALE >= SCALE / 2
    ///
    /// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function mulDivFixedPoint(uint256 x, uint256 y) internal pure returns (uint256 result) {
        uint256 prod0;
        uint256 prod1;
        assembly {
            let mm := mulmod(x, y, not(0))
            prod0 := mul(x, y)
            prod1 := sub(sub(mm, prod0), lt(mm, prod0))
        }

        if (prod1 >= SCALE) {
            revert PRBMath__MulDivFixedPointOverflow(prod1);
        }

        uint256 remainder;
        uint256 roundUpUnit;
        assembly {
            remainder := mulmod(x, y, SCALE)
            roundUpUnit := gt(remainder, 499999999999999999)
        }

        if (prod1 == 0) {
            unchecked {
                result = (prod0 / SCALE) + roundUpUnit;
                return result;
            }
        }

        assembly {
            result := add(
                mul(
                    or(
                        div(sub(prod0, remainder), SCALE_LPOTD),
                        mul(sub(prod1, gt(remainder, prod0)), add(div(sub(0, SCALE_LPOTD), SCALE_LPOTD), 1))
                    ),
                    SCALE_INVERSE
                ),
                roundUpUnit
            )
        }
    }

    /// @notice Calculates floor(x*y÷denominator) with full precision.
    ///
    /// @dev An extension of "mulDiv" for signed numbers. Works by computing the signs and the absolute values separately.
    ///
    /// Requirements:
    /// - None of the inputs can be type(int256).min.
    /// - The result must fit within int256.
    ///
    /// @param x The multiplicand as an int256.
    /// @param y The multiplier as an int256.
    /// @param denominator The divisor as an int256.
    /// @return result The result as an int256.
    function mulDivSigned(
        int256 x,
        int256 y,
        int256 denominator
    ) internal pure returns (int256 result) {
        if (x == type(int256).min || y == type(int256).min || denominator == type(int256).min) {
            revert PRBMath__MulDivSignedInputTooSmall();
        }

        // Get hold of the absolute values of x, y and the denominator.
        uint256 ax;
        uint256 ay;
        uint256 ad;
        unchecked {
            ax = x < 0 ? uint256(-x) : uint256(x);
            ay = y < 0 ? uint256(-y) : uint256(y);
            ad = denominator < 0 ? uint256(-denominator) : uint256(denominator);
        }

        // Compute the absolute value of (x*y)÷denominator. The result must fit within int256.
        uint256 rAbs = mulDiv(ax, ay, ad);
        if (rAbs > uint256(type(int256).max)) {
            revert PRBMath__MulDivSignedOverflow(rAbs);
        }

        // Get the signs of x, y and the denominator.
        uint256 sx;
        uint256 sy;
        uint256 sd;
        assembly {
            sx := sgt(x, sub(0, 1))
            sy := sgt(y, sub(0, 1))
            sd := sgt(denominator, sub(0, 1))
        }

        // XOR over sx, sy and sd. This is checking whether there are one or three negative signs in the inputs.
        // If yes, the result should be negative.
        result = sx ^ sy ^ sd == 0 ? -int256(rAbs) : int256(rAbs);
    }

    /// @notice Calculates the square root of x, rounding down.
    /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
    ///
    /// Caveats:
    /// - This function does not work with fixed-point numbers.
    ///
    /// @param x The uint256 number for which to calculate the square root.
    /// @return result The result as an uint256.
    function sqrt(uint256 x) internal pure returns (uint256 result) {
        if (x == 0) {
            return 0;
        }

        // Set the initial guess to the closest power of two that is higher than x.
        uint256 xAux = uint256(x);
        result = 1;
        if (xAux >= 0x100000000000000000000000000000000) {
            xAux >>= 128;
            result <<= 64;
        }
        if (xAux >= 0x10000000000000000) {
            xAux >>= 64;
            result <<= 32;
        }
        if (xAux >= 0x100000000) {
            xAux >>= 32;
            result <<= 16;
        }
        if (xAux >= 0x10000) {
            xAux >>= 16;
            result <<= 8;
        }
        if (xAux >= 0x100) {
            xAux >>= 8;
            result <<= 4;
        }
        if (xAux >= 0x10) {
            xAux >>= 4;
            result <<= 2;
        }
        if (xAux >= 0x8) {
            result <<= 1;
        }

        // The operations can never overflow because the result is max 2^127 when it enters this block.
        unchecked {
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1; // Seven iterations should be enough
            uint256 roundedDownResult = x / result;
            return result >= roundedDownResult ? roundedDownResult : result;
        }
    }
}