{
  "language": "Solidity",
  "sources": {
    "contracts/banana/BuybackPool.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\nimport \"../interfaces/IERC20.sol\";\nimport \"./interfaces/IBanana.sol\";\nimport \"./interfaces/IBananaDistributor.sol\";\nimport \"./interfaces/ITWAMM.sol\";\nimport \"./interfaces/ITWAMMPair.sol\";\nimport \"../utils/Ownable.sol\";\nimport \"../utils/AnalyticMath.sol\";\nimport \"../libraries/FullMath.sol\";\nimport \"../libraries/TransferHelper.sol\";\n\ncontract BuybackPool is Ownable, AnalyticMath {\n    using FullMath for uint256;\n\n    event BuybackExecuted(uint256 orderId, uint256 amountIn, uint256 buyingRate, uint256 burned);\n    event WithdrawAndBurn(uint256 orderId, uint256 burnAmount);\n\n    address public immutable banana;\n    address public immutable usdc;\n    address public immutable twamm;\n    address public immutable bananaDistributor;\n    address public keeper;\n\n    uint256 public lastBuyingRate;\n    uint256 public priceT1;\n    uint256 public priceT2;\n    uint256 public rewardT1;\n    uint256 public rewardT2;\n    uint256 public priceIndex = 100;\n    uint256 public rewardIndex = 50;\n    uint256 public secondsPerBlock = 12;\n    uint256 public secondsOfEpoch;\n    uint256 public lastOrderId = type(uint256).max;\n    uint256 public lastExecuteTime;\n    uint256 public endTime;\n\n    bool public initialized;\n    bool public isStop;\n\n    constructor(\n        address banana_,\n        address usdc_,\n        address twamm_,\n        address bananaDistributor_,\n        address keeper_,\n        uint256 secondsOfEpoch_,\n        uint256 initPrice,\n        uint256 initReward,\n        uint256 startTime,\n        uint256 endTime_\n    ) {\n        owner = msg.sender;\n        banana = banana_;\n        usdc = usdc_;\n        twamm = twamm_;\n        bananaDistributor = bananaDistributor_;\n        keeper = keeper_;\n        secondsOfEpoch = secondsOfEpoch_;\n        priceT2 = initPrice;\n        rewardT2 = initReward;\n        lastExecuteTime = startTime;\n        endTime = endTime_;\n    }\n\n    // function initBuyingRate(uint256 amountIn) external onlyOwner {\n    //     require(!initialized, \"already initialized\");\n    //     initialized = true;\n    //     lastBuyingRate = amountIn / secondsOfEpoch;\n    // }\n\n    function updatePriceIndex(uint256 newPriceIndex) external onlyOwner {\n        priceIndex = newPriceIndex;\n    }\n\n    function updateRewardIndex(uint256 newRewardIndex) external onlyOwner {\n        rewardIndex = newRewardIndex;\n    }\n\n    function updateSecondsPerBlock(uint256 newSecondsPerBlock) external onlyOwner {\n        secondsPerBlock = newSecondsPerBlock;\n    }\n\n    function updateSecondsOfEpoch(uint256 newSecondsOfEpoch) external onlyOwner {\n        secondsOfEpoch = newSecondsOfEpoch;\n    }\n\n    function updateLastExecuteTime(uint256 newExecuteTime) external onlyOwner {\n        lastExecuteTime = newExecuteTime;\n    }\n\n    function updateEndTime(uint256 endTime_) external onlyOwner {\n        endTime = endTime_;\n    }\n\n    function updateStatus(bool isStop_) external onlyOwner {\n        isStop = isStop_;\n    }\n\n    function updateKeeper(address keeper_) external onlyOwner {\n        keeper = keeper_;\n    }\n\n    function resetPrice(uint256 newPriceT1, uint256 newPriceT2) external onlyOwner {\n        priceT1 = newPriceT1;\n        priceT2 = newPriceT2;\n    }\n\n    function resetReward(uint256 newRewardT1, uint256 newRewardT2) external onlyOwner {\n        rewardT1 = newRewardT1;\n        rewardT2 = newRewardT2;\n    }\n\n    function withdrawUsdc(address to) external onlyOwner {\n        require(isStop, \"not stop\");\n        uint256 usdcBalance = IERC20(usdc).balanceOf(address(this));\n        TransferHelper.safeTransfer(usdc, to, usdcBalance);\n    }\n\n    function withdrawAndBurn(uint256 orderId) external onlyOwner {\n        require(isStop, \"not stop\");\n        address pair = ITWAMM(twamm).obtainPairAddress(usdc, banana);\n        ITWAMMPair.Order memory order = ITWAMMPair(pair).getOrderDetails(orderId);\n        require(block.number > order.expirationBlock, \"not reach withdrawable block\");\n        ITWAMM(twamm).withdrawProceedsFromTermSwapTokenToToken(usdc, banana, orderId, block.timestamp);\n        uint256 bananaBalance = IERC20(banana).balanceOf(address(this));\n        require(bananaBalance > 0, \"nothing to burn\");\n        IBanana(banana).burn(bananaBalance);\n        emit WithdrawAndBurn(orderId, bananaBalance);\n    }\n\n    function execute() external {\n        require(!isStop, \"is stop\");\n        require(msg.sender == keeper, \"only keeper\");\n        require(block.timestamp < endTime, \"end\");\n        lastExecuteTime = lastExecuteTime + secondsOfEpoch;\n        require(block.timestamp >= lastExecuteTime, \"not reach execute time\");\n        require(lastExecuteTime + secondsOfEpoch > block.timestamp, \"over next epoch time\");\n\n        uint256 burnAmount;\n        if (lastOrderId != type(uint256).max) {\n            address pair = ITWAMM(twamm).obtainPairAddress(usdc, banana);\n            ITWAMMPair.Order memory order = ITWAMMPair(pair).getOrderDetails(lastOrderId);\n            require(block.number > order.expirationBlock, \"not reach withdrawable block\");\n\n            ITWAMM(twamm).withdrawProceedsFromTermSwapTokenToToken(usdc, banana, lastOrderId, block.timestamp);\n            burnAmount = IERC20(banana).balanceOf(address(this));\n            IBanana(banana).burn(burnAmount);\n        }\n\n        uint256 deltaTime = lastExecuteTime + secondsOfEpoch - block.timestamp;\n        uint256 usdcBalance = IERC20(usdc).balanceOf(address(this));\n        uint256 amountIn;\n        if (!initialized) {\n            amountIn = usdcBalance;\n            lastBuyingRate = amountIn / deltaTime;\n            initialized = true;\n        } else {\n            (uint256 pn, uint256 pd) = pow(priceT2, priceT1, priceIndex, 100);\n            (uint256 rn, uint256 rd) = pow(rewardT1, rewardT2, rewardIndex, 100);\n            lastBuyingRate = lastBuyingRate.mulDiv(pn, pd).mulDiv(rn, rd);\n            amountIn = deltaTime * lastBuyingRate;\n            if (amountIn > usdcBalance) {\n                amountIn = usdcBalance;\n                lastBuyingRate = amountIn / deltaTime;\n            }\n        }\n\n        require(amountIn > 0, \"buying amount is 0\");\n        IERC20(usdc).approve(twamm, amountIn);\n        lastOrderId = ITWAMM(twamm).longTermSwapTokenToToken(\n            usdc,\n            banana,\n            amountIn,\n            deltaTime / (secondsPerBlock * 5),\n            block.timestamp\n        );\n\n        uint256 lastReward = IBananaDistributor(bananaDistributor).lastReward();\n        if (rewardT1 > 0) {\n            rewardT2 = rewardT1;\n        }\n        rewardT1 = lastReward;\n\n        (uint256 reserve0, uint256 reserve1) = ITWAMM(twamm).obtainReserves(usdc, banana);\n        uint256 currentPrice = reserve0.mulDiv(1e30, reserve1); // reserve0/reserve1 * 10**(18+decimalsUSDC-decimalsBANA)\n        if (priceT1 > 0) {\n            priceT2 = priceT1;\n        }\n        priceT1 = currentPrice;\n\n        emit BuybackExecuted(lastOrderId, amountIn, lastBuyingRate, burnAmount);\n    }\n}\n"
    },
    "contracts/interfaces/IERC20.sol": {
      "content": "// SPDX-License-Identifier: MIT\npragma solidity ^0.8.0;\n\ninterface IERC20 {\n    event Approval(address indexed owner, address indexed spender, uint256 value);\n    event Transfer(address indexed from, address indexed to, uint256 value);\n\n    function allowance(address owner, address spender) external view returns (uint256);\n\n    function approve(address spender, uint256 value) external returns (bool);\n\n    function transfer(address to, uint256 value) external returns (bool);\n\n    function transferFrom(\n        address from,\n        address to,\n        uint256 value\n    ) external returns (bool);\n\n    function totalSupply() external view returns (uint256);\n\n    function balanceOf(address owner) external view returns (uint256);\n\n    function name() external view returns (string memory);\n\n    function symbol() external view returns (string memory);\n\n    function decimals() external pure returns (uint8);\n}\n"
    },
    "contracts/banana/interfaces/IBanana.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\nimport \"../../interfaces/IERC20.sol\";\n\ninterface IBanana is IERC20 {\n    event RedeemTimeChanged(uint256 oldRedeemTime, uint256 newRedeemTime);\n    event Redeem(address indexed user, uint256 burntAmount, uint256 apeXAmount);\n\n    function apeXToken() external view returns (address);\n    function redeemTime() external view returns (uint256);\n\n    function mint(address to, uint256 apeXAmount) external returns (uint256);\n    function burn(uint256 amount) external returns (bool);\n    function burnFrom(address from, uint256 amount) external returns (bool);\n    function redeem(uint256 amount) external returns (uint256);\n}"
    },
    "contracts/banana/interfaces/IBananaDistributor.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\ninterface IBananaDistributor {\n    function lastReward() external view returns (uint256);\n}"
    },
    "contracts/banana/interfaces/ITWAMM.sol": {
      "content": "// SPDX-License-Identifier: GPL-3.0-or-later\npragma solidity ^0.8.0;\n\ninterface ITWAMM {\n    function factory() external view returns (address);\n\n    function WETH() external view returns (address);\n\n    function obtainReserves(address token0, address token1) external view returns (uint256 reserve0, uint256 reserve1);\n\n    function obtainTotalSupply(address token0, address token1) external view returns (uint256);\n\n    function obtainPairAddress(address token0, address token1) external view returns (address);\n\n    function createPairWrapper(\n        address token0,\n        address token1,\n        uint256 deadline\n    ) external returns (address pair);\n\n    function addInitialLiquidity(\n        address token0,\n        address token1,\n        uint256 amount0,\n        uint256 amount1,\n        uint256 deadline\n    ) external returns (uint256 lpTokenAmount);\n\n    function addInitialLiquidityETH(\n        address token,\n        uint256 amountToken,\n        uint256 amountETH,\n        uint256 deadline\n    ) external payable returns (uint256 lpTokenAmount);\n\n    function addLiquidity(\n        address token0,\n        address token1,\n        uint256 lpTokenAmount,\n        uint256 amountIn0Max,\n        uint256 amountIn1Max,\n        uint256 deadline\n    ) external returns (uint256 amountIn0, uint256 amountIn1);\n\n    function addLiquidityETH(\n        address token,\n        uint256 lpTokenAmount,\n        uint256 amountTokenInMax,\n        uint256 amountETHInMax,\n        uint256 deadline\n    ) external payable returns (uint256 amountTokenIn, uint256 amountETHIn);\n\n    function withdrawLiquidity(\n        address token0,\n        address token1,\n        uint256 lpTokenAmount,\n        uint256 amountOut0Min,\n        uint256 amountOut1Min,\n        uint256 deadline\n    ) external returns (uint256 amountOut0, uint256 amountOut1);\n\n    function withdrawLiquidityETH(\n        address token,\n        uint256 lpTokenAmount,\n        uint256 amountTokenOutMin,\n        uint256 amountETHOutMin,\n        uint256 deadline\n    ) external returns (uint256 amountTokenOut, uint256 amountETHOut);\n\n    function instantSwapTokenToToken(\n        address token0,\n        address token1,\n        uint256 amountIn,\n        uint256 amountOutMin,\n        uint256 deadline\n    ) external returns (uint256 amountOut);\n\n    function instantSwapTokenToETH(\n        address token,\n        uint256 amountTokenIn,\n        uint256 amountETHOutMin,\n        uint256 deadline\n    ) external returns (uint256 amountETHOut);\n\n    function instantSwapETHToToken(\n        address token,\n        uint256 amountETHIn,\n        uint256 amountTokenOutMin,\n        uint256 deadline\n    ) external payable returns (uint256 amountTokenOut);\n\n    function longTermSwapTokenToToken(\n        address token0,\n        address token1,\n        uint256 amountIn,\n        uint256 numberOfBlockIntervals,\n        uint256 deadline\n    ) external returns (uint256 orderId);\n\n    function longTermSwapTokenToETH(\n        address token,\n        uint256 amountTokenIn,\n        uint256 numberOfBlockIntervals,\n        uint256 deadline\n    ) external returns (uint256 orderId);\n\n    function longTermSwapETHToToken(\n        address token,\n        uint256 amountETHIn,\n        uint256 numberOfBlockIntervals,\n        uint256 deadline\n    ) external payable returns (uint256 orderId);\n\n    function cancelTermSwapTokenToToken(\n        address token0,\n        address token1,\n        uint256 orderId,\n        uint256 deadline\n    ) external returns (uint256 unsoldAmount, uint256 purchasedAmount);\n\n    function cancelTermSwapTokenToETH(\n        address token,\n        uint256 orderId,\n        uint256 deadline\n    ) external returns (uint256 unsoldTokenAmount, uint256 purchasedETHAmount);\n\n    function cancelTermSwapETHToToken(\n        address token,\n        uint256 orderId,\n        uint256 deadline\n    ) external returns (uint256 unsoldETHAmount, uint256 purchasedTokenAmount);\n\n    function withdrawProceedsFromTermSwapTokenToToken(\n        address token0,\n        address token1,\n        uint256 orderId,\n        uint256 deadline\n    ) external returns (uint256 proceeds);\n\n    function withdrawProceedsFromTermSwapTokenToETH(\n        address token,\n        uint256 orderId,\n        uint256 deadline\n    ) external returns (uint256 proceedsETH);\n\n    function withdrawProceedsFromTermSwapETHToToken(\n        address token,\n        uint256 orderId,\n        uint256 deadline\n    ) external returns (uint256 proceedsToken);\n\n    function executeVirtualOrdersWrapper(address pair, uint256 blockNumber) external;\n}\n"
    },
    "contracts/banana/interfaces/ITWAMMPair.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\ninterface ITWAMMPair {\n    struct Order {\n        uint256 id;\n        uint256 expirationBlock;\n        uint256 saleRate;\n        address owner;\n        address sellTokenId;\n        address buyTokenId;\n    }\n\n    function getOrderDetails(uint256 orderId)\n        external\n        view\n        returns (Order memory);\n}"
    },
    "contracts/utils/Ownable.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\nabstract contract Ownable {\n    address public owner;\n    address public pendingOwner;\n\n    event NewOwner(address indexed oldOwner, address indexed newOwner);\n    event NewPendingOwner(address indexed oldPendingOwner, address indexed newPendingOwner);\n\n    modifier onlyOwner() {\n        require(msg.sender == owner, \"Ownable: REQUIRE_OWNER\");\n        _;\n    }\n\n    function setPendingOwner(address newPendingOwner) external onlyOwner {\n        require(pendingOwner != newPendingOwner, \"Ownable: ALREADY_SET\");\n        emit NewPendingOwner(pendingOwner, newPendingOwner);\n        pendingOwner = newPendingOwner;\n    }\n\n    function acceptOwner() external {\n        require(msg.sender == pendingOwner, \"Ownable: REQUIRE_PENDING_OWNER\");\n        address oldOwner = owner;\n        address oldPendingOwner = pendingOwner;\n        owner = pendingOwner;\n        pendingOwner = address(0);\n        emit NewOwner(oldOwner, owner);\n        emit NewPendingOwner(oldPendingOwner, pendingOwner);\n    }\n}\n"
    },
    "contracts/utils/AnalyticMath.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\nimport \"../libraries/IntegralMath.sol\";\n\ncontract AnalyticMath {\n    uint8 internal constant MIN_PRECISION = 32;\n    uint8 internal constant MAX_PRECISION = 127;\n\n    uint256 internal constant FIXED_1 = 1 << MAX_PRECISION;\n    uint256 internal constant FIXED_2 = 2 << MAX_PRECISION;\n\n    // Auto-generated via 'PrintLn2ScalingFactors.py'\n    uint256 internal constant LN2_NUMERATOR   = 0x3f80fe03f80fe03f80fe03f80fe03f8;\n    uint256 internal constant LN2_DENOMINATOR = 0x5b9de1d10bf4103d647b0955897ba80;\n\n    // Auto-generated via 'PrintOptimalThresholds.py'\n    uint256 internal constant OPT_LOG_MAX_VAL = 0x15bf0a8b1457695355fb8ac404e7a79e4;\n    uint256 internal constant OPT_EXP_MAX_VAL = 0x800000000000000000000000000000000;\n\n    uint256[MAX_PRECISION + 1] private maxExpArray;\n\n    /**\n      * @dev Should be executed either during construction or after construction (if too large for the constructor)\n    */\n    constructor() {\n        initMaxExpArray();\n    }\n\n    /**\n      * @dev Compute (a / b) ^ (c / d)\n    */\n    function pow(uint256 a, uint256 b, uint256 c, uint256 d) internal view returns (uint256, uint256) { unchecked {\n        if (a >= b)\n            return mulDivExp(mulDivLog(FIXED_1, a, b), c, d);\n        (uint256 q, uint256 p) = mulDivExp(mulDivLog(FIXED_1, b, a), c, d);\n        return (p, q);\n    }}\n\n    /**\n      * @dev Compute log(a / b)\n    */\n    function log(uint256 a, uint256 b) internal pure returns (uint256, uint256) { unchecked {\n        require(a >= b, \"log: a < b\");\n        return (mulDivLog(FIXED_1, a, b), FIXED_1);\n    }}\n\n    /**\n      * @dev Compute e ^ (a / b)\n    */\n    function exp(uint256 a, uint256 b) internal view returns (uint256, uint256) { unchecked {\n        return mulDivExp(FIXED_1, a, b);\n    }}\n\n    /**\n      * @dev Compute log(x / FIXED_1) * FIXED_1\n    */\n    function fixedLog(uint256 x) internal pure returns (uint256) { unchecked {\n        if (x < OPT_LOG_MAX_VAL) {\n            return optimalLog(x);\n        }\n        else {\n            return generalLog(x);\n        }\n    }}\n\n    /**\n      * @dev Compute e ^ (x / FIXED_1) * FIXED_1\n    */\n    function fixedExp(uint256 x) internal view returns (uint256, uint256) { unchecked {\n        if (x < OPT_EXP_MAX_VAL) {\n            return (optimalExp(x), 1 << MAX_PRECISION);\n        }\n        else {\n            uint8 precision = findPosition(x);\n            return (generalExp(x >> (MAX_PRECISION - precision), precision), 1 << precision);\n        }\n    }}\n\n    /**\n      * @dev Compute log(x / FIXED_1) * FIXED_1\n      * This functions assumes that x >= FIXED_1, because the output would be negative otherwise\n    */\n    function generalLog(uint256 x) internal pure returns (uint256) { unchecked {\n        uint256 res = 0;\n\n        // if x >= 2, then we compute the integer part of log2(x), which is larger than 0\n        if (x >= FIXED_2) {\n            uint8 count = IntegralMath.floorLog2(x / FIXED_1);\n            x >>= count; // now x < 2\n            res = count * FIXED_1;\n        }\n\n        // if x > 1, then we compute the fraction part of log2(x), which is larger than 0\n        if (x > FIXED_1) {\n            for (uint8 i = MAX_PRECISION; i > 0; --i) {\n                x = (x * x) / FIXED_1; // now 1 < x < 4\n                if (x >= FIXED_2) {\n                    x >>= 1; // now 1 < x < 2\n                    res += 1 << (i - 1);\n                }\n            }\n        }\n\n        return res * LN2_NUMERATOR / LN2_DENOMINATOR;\n    }}\n\n    /**\n      * @dev Approximate e ^ x as (x ^ 0) / 0! + (x ^ 1) / 1! + ... + (x ^ n) / n!\n      * Auto-generated via 'PrintFunctionGeneralExp.py'\n      * Detailed description:\n      * - This function returns \"e ^ (x / 2 ^ precision) * 2 ^ precision\", that is, the result is upshifted for accuracy\n      * - The global \"maxExpArray\" maps each \"precision\" to \"((maximumExponent + 1) << (MAX_PRECISION - precision)) - 1\"\n      * - The maximum permitted value for \"x\" is therefore given by \"maxExpArray[precision] >> (MAX_PRECISION - precision)\"\n    */\n    function generalExp(uint256 x, uint8 precision) internal pure returns (uint256) { unchecked {\n        uint256 xi = x;\n        uint256 res = 0;\n\n        xi = (xi * x) >> precision; res += xi * 0x3442c4e6074a82f1797f72ac0000000; // add x^02 * (33! / 02!)\n        xi = (xi * x) >> precision; res += xi * 0x116b96f757c380fb287fd0e40000000; // add x^03 * (33! / 03!)\n        xi = (xi * x) >> precision; res += xi * 0x045ae5bdd5f0e03eca1ff4390000000; // add x^04 * (33! / 04!)\n        xi = (xi * x) >> precision; res += xi * 0x00defabf91302cd95b9ffda50000000; // add x^05 * (33! / 05!)\n        xi = (xi * x) >> precision; res += xi * 0x002529ca9832b22439efff9b8000000; // add x^06 * (33! / 06!)\n        xi = (xi * x) >> precision; res += xi * 0x00054f1cf12bd04e516b6da88000000; // add x^07 * (33! / 07!)\n        xi = (xi * x) >> precision; res += xi * 0x0000a9e39e257a09ca2d6db51000000; // add x^08 * (33! / 08!)\n        xi = (xi * x) >> precision; res += xi * 0x000012e066e7b839fa050c309000000; // add x^09 * (33! / 09!)\n        xi = (xi * x) >> precision; res += xi * 0x000001e33d7d926c329a1ad1a800000; // add x^10 * (33! / 10!)\n        xi = (xi * x) >> precision; res += xi * 0x0000002bee513bdb4a6b19b5f800000; // add x^11 * (33! / 11!)\n        xi = (xi * x) >> precision; res += xi * 0x00000003a9316fa79b88eccf2a00000; // add x^12 * (33! / 12!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000048177ebe1fa812375200000; // add x^13 * (33! / 13!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000005263fe90242dcbacf00000; // add x^14 * (33! / 14!)\n        xi = (xi * x) >> precision; res += xi * 0x000000000057e22099c030d94100000; // add x^15 * (33! / 15!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000057e22099c030d9410000; // add x^16 * (33! / 16!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000052b6b54569976310000; // add x^17 * (33! / 17!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000004985f67696bf748000; // add x^18 * (33! / 18!)\n        xi = (xi * x) >> precision; res += xi * 0x000000000000003dea12ea99e498000; // add x^19 * (33! / 19!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000000031880f2214b6e000; // add x^20 * (33! / 20!)\n        xi = (xi * x) >> precision; res += xi * 0x000000000000000025bcff56eb36000; // add x^21 * (33! / 21!)\n        xi = (xi * x) >> precision; res += xi * 0x000000000000000001b722e10ab1000; // add x^22 * (33! / 22!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000000000001317c70077000; // add x^23 * (33! / 23!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000000000000cba84aafa00; // add x^24 * (33! / 24!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000000000000082573a0a00; // add x^25 * (33! / 25!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000000000000005035ad900; // add x^26 * (33! / 26!)\n        xi = (xi * x) >> precision; res += xi * 0x000000000000000000000002f881b00; // add x^27 * (33! / 27!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000000000000000001b29340; // add x^28 * (33! / 28!)\n        xi = (xi * x) >> precision; res += xi * 0x00000000000000000000000000efc40; // add x^29 * (33! / 29!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000000000000000000007fe0; // add x^30 * (33! / 30!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000000000000000000000420; // add x^31 * (33! / 31!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000000000000000000000021; // add x^32 * (33! / 32!)\n        xi = (xi * x) >> precision; res += xi * 0x0000000000000000000000000000001; // add x^33 * (33! / 33!)\n\n        return res / 0x688589cc0e9505e2f2fee5580000000 + x + (1 << precision); // divide by 33! and then add x^1 / 1! + x^0 / 0!\n    }}\n\n    /**\n      * @dev Compute log(x / FIXED_1) * FIXED_1\n      * Input range: FIXED_1 <= x <= OPT_LOG_MAX_VAL - 1\n      * Auto-generated via 'PrintFunctionOptimalLog.py'\n      * Detailed description:\n      * - Rewrite the input as a product of natural exponents and a single residual r, such that 1 < r < 2\n      * - The natural logarithm of each (pre-calculated) exponent is the degree of the exponent\n      * - The natural logarithm of r is calculated via Taylor series for log(1 + x), where x = r - 1\n      * - The natural logarithm of the input is calculated by summing up the intermediate results above\n      * - For example: log(250) = log(e^4 * e^1 * e^0.5 * 1.021692859) = 4 + 1 + 0.5 + log(1 + 0.021692859)\n    */\n    function optimalLog(uint256 x) internal pure returns (uint256) { unchecked {\n        uint256 res = 0;\n\n        uint256 y;\n        uint256 z;\n        uint256 w;\n\n        if (x >= 0xd3094c70f034de4b96ff7d5b6f99fcd9) {res += 0x40000000000000000000000000000000; x = x * FIXED_1 / 0xd3094c70f034de4b96ff7d5b6f99fcd9;} // add 1 / 2^1\n        if (x >= 0xa45af1e1f40c333b3de1db4dd55f29a8) {res += 0x20000000000000000000000000000000; x = x * FIXED_1 / 0xa45af1e1f40c333b3de1db4dd55f29a8;} // add 1 / 2^2\n        if (x >= 0x910b022db7ae67ce76b441c27035c6a2) {res += 0x10000000000000000000000000000000; x = x * FIXED_1 / 0x910b022db7ae67ce76b441c27035c6a2;} // add 1 / 2^3\n        if (x >= 0x88415abbe9a76bead8d00cf112e4d4a9) {res += 0x08000000000000000000000000000000; x = x * FIXED_1 / 0x88415abbe9a76bead8d00cf112e4d4a9;} // add 1 / 2^4\n        if (x >= 0x84102b00893f64c705e841d5d4064bd4) {res += 0x04000000000000000000000000000000; x = x * FIXED_1 / 0x84102b00893f64c705e841d5d4064bd4;} // add 1 / 2^5\n        if (x >= 0x8204055aaef1c8bd5c3259f4822735a3) {res += 0x02000000000000000000000000000000; x = x * FIXED_1 / 0x8204055aaef1c8bd5c3259f4822735a3;} // add 1 / 2^6\n        if (x >= 0x810100ab00222d861931c15e39b44e9a) {res += 0x01000000000000000000000000000000; x = x * FIXED_1 / 0x810100ab00222d861931c15e39b44e9a;} // add 1 / 2^7\n        if (x >= 0x808040155aabbbe9451521693554f734) {res += 0x00800000000000000000000000000000; x = x * FIXED_1 / 0x808040155aabbbe9451521693554f734;} // add 1 / 2^8\n\n        z = y = x - FIXED_1;\n        w = y * y / FIXED_1;\n        res += z * (0x100000000000000000000000000000000 - y) / 0x100000000000000000000000000000000; z = z * w / FIXED_1; // add y^01 / 01 - y^02 / 02\n        res += z * (0x0aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa - y) / 0x200000000000000000000000000000000; z = z * w / FIXED_1; // add y^03 / 03 - y^04 / 04\n        res += z * (0x099999999999999999999999999999999 - y) / 0x300000000000000000000000000000000; z = z * w / FIXED_1; // add y^05 / 05 - y^06 / 06\n        res += z * (0x092492492492492492492492492492492 - y) / 0x400000000000000000000000000000000; z = z * w / FIXED_1; // add y^07 / 07 - y^08 / 08\n        res += z * (0x08e38e38e38e38e38e38e38e38e38e38e - y) / 0x500000000000000000000000000000000; z = z * w / FIXED_1; // add y^09 / 09 - y^10 / 10\n        res += z * (0x08ba2e8ba2e8ba2e8ba2e8ba2e8ba2e8b - y) / 0x600000000000000000000000000000000; z = z * w / FIXED_1; // add y^11 / 11 - y^12 / 12\n        res += z * (0x089d89d89d89d89d89d89d89d89d89d89 - y) / 0x700000000000000000000000000000000; z = z * w / FIXED_1; // add y^13 / 13 - y^14 / 14\n        res += z * (0x088888888888888888888888888888888 - y) / 0x800000000000000000000000000000000;                      // add y^15 / 15 - y^16 / 16\n\n        return res;\n    }}\n\n    /**\n      * @dev Compute e ^ (x / FIXED_1) * FIXED_1\n      * Input range: 0 <= x <= OPT_EXP_MAX_VAL - 1\n      * Auto-generated via 'PrintFunctionOptimalExp.py'\n      * Detailed description:\n      * - Rewrite the input as a sum of binary exponents and a single residual r, as small as possible\n      * - The exponentiation of each binary exponent is given (pre-calculated)\n      * - The exponentiation of r is calculated via Taylor series for e^x, where x = r\n      * - The exponentiation of the input is calculated by multiplying the intermediate results above\n      * - For example: e^5.521692859 = e^(4 + 1 + 0.5 + 0.021692859) = e^4 * e^1 * e^0.5 * e^0.021692859\n    */\n    function optimalExp(uint256 x) internal pure returns (uint256) { unchecked {\n        uint256 res = 0;\n\n        uint256 y;\n        uint256 z;\n\n        z = y = x % 0x10000000000000000000000000000000; // get the input modulo 2^(-3)\n        z = z * y / FIXED_1; res += z * 0x10e1b3be415a0000; // add y^02 * (20! / 02!)\n        z = z * y / FIXED_1; res += z * 0x05a0913f6b1e0000; // add y^03 * (20! / 03!)\n        z = z * y / FIXED_1; res += z * 0x0168244fdac78000; // add y^04 * (20! / 04!)\n        z = z * y / FIXED_1; res += z * 0x004807432bc18000; // add y^05 * (20! / 05!)\n        z = z * y / FIXED_1; res += z * 0x000c0135dca04000; // add y^06 * (20! / 06!)\n        z = z * y / FIXED_1; res += z * 0x0001b707b1cdc000; // add y^07 * (20! / 07!)\n        z = z * y / FIXED_1; res += z * 0x000036e0f639b800; // add y^08 * (20! / 08!)\n        z = z * y / FIXED_1; res += z * 0x00000618fee9f800; // add y^09 * (20! / 09!)\n        z = z * y / FIXED_1; res += z * 0x0000009c197dcc00; // add y^10 * (20! / 10!)\n        z = z * y / FIXED_1; res += z * 0x0000000e30dce400; // add y^11 * (20! / 11!)\n        z = z * y / FIXED_1; res += z * 0x000000012ebd1300; // add y^12 * (20! / 12!)\n        z = z * y / FIXED_1; res += z * 0x0000000017499f00; // add y^13 * (20! / 13!)\n        z = z * y / FIXED_1; res += z * 0x0000000001a9d480; // add y^14 * (20! / 14!)\n        z = z * y / FIXED_1; res += z * 0x00000000001c6380; // add y^15 * (20! / 15!)\n        z = z * y / FIXED_1; res += z * 0x000000000001c638; // add y^16 * (20! / 16!)\n        z = z * y / FIXED_1; res += z * 0x0000000000001ab8; // add y^17 * (20! / 17!)\n        z = z * y / FIXED_1; res += z * 0x000000000000017c; // add y^18 * (20! / 18!)\n        z = z * y / FIXED_1; res += z * 0x0000000000000014; // add y^19 * (20! / 19!)\n        z = z * y / FIXED_1; res += z * 0x0000000000000001; // add y^20 * (20! / 20!)\n        res = res / 0x21c3677c82b40000 + y + FIXED_1; // divide by 20! and then add y^1 / 1! + y^0 / 0!\n\n        if ((x & 0x010000000000000000000000000000000) != 0) res = res * 0x1c3d6a24ed82218787d624d3e5eba95f9 / 0x18ebef9eac820ae8682b9793ac6d1e776; // multiply by e^2^(-3)\n        if ((x & 0x020000000000000000000000000000000) != 0) res = res * 0x18ebef9eac820ae8682b9793ac6d1e778 / 0x1368b2fc6f9609fe7aceb46aa619baed4; // multiply by e^2^(-2)\n        if ((x & 0x040000000000000000000000000000000) != 0) res = res * 0x1368b2fc6f9609fe7aceb46aa619baed5 / 0x0bc5ab1b16779be3575bd8f0520a9f21f; // multiply by e^2^(-1)\n        if ((x & 0x080000000000000000000000000000000) != 0) res = res * 0x0bc5ab1b16779be3575bd8f0520a9f21e / 0x0454aaa8efe072e7f6ddbab84b40a55c9; // multiply by e^2^(+0)\n        if ((x & 0x100000000000000000000000000000000) != 0) res = res * 0x0454aaa8efe072e7f6ddbab84b40a55c5 / 0x00960aadc109e7a3bf4578099615711ea; // multiply by e^2^(+1)\n        if ((x & 0x200000000000000000000000000000000) != 0) res = res * 0x00960aadc109e7a3bf4578099615711d7 / 0x0002bf84208204f5977f9a8cf01fdce3d; // multiply by e^2^(+2)\n        if ((x & 0x400000000000000000000000000000000) != 0) res = res * 0x0002bf84208204f5977f9a8cf01fdc307 / 0x0000003c6ab775dd0b95b4cbee7e65d11; // multiply by e^2^(+3)\n\n        return res;\n    }}\n\n    /**\n      * @dev The global \"maxExpArray\" is sorted in descending order, and therefore the following statements are equivalent:\n      * - This function finds the position of [the smallest value in \"maxExpArray\" larger than or equal to \"x\"]\n      * - This function finds the highest position of [a value in \"maxExpArray\" larger than or equal to \"x\"]\n      * This function supports the rational approximation of \"(a / b) ^ (c / d)\" via \"e ^ (log(a / b) * c / d)\".\n      * The value of \"log(a / b)\" is represented with an integer slightly smaller than \"log(a / b) * 2 ^ precision\".\n      * The larger \"precision\" is, the more accurately this value represents the real value.\n      * However, the larger \"precision\" is, the more bits are required in order to store this value.\n      * And the exponentiation function, which takes \"x\" and calculates \"e ^ x\", is limited to a maximum exponent (a maximum value of \"x\").\n      * This maximum exponent depends on the \"precision\" used, and it is given by \"maxExpArray[precision] >> (MAX_PRECISION - precision)\".\n      * Hence we need to determine the highest precision which can be used for the given input, before calling the exponentiation function.\n      * This allows us to compute the result with maximum accuracy and without exceeding 256 bits in any of the intermediate computations.\n    */\n    function findPosition(uint256 x) internal view returns (uint8) { unchecked {\n        uint8 lo = MIN_PRECISION;\n        uint8 hi = MAX_PRECISION;\n\n        while (lo + 1 < hi) {\n            uint8 mid = (lo + hi) / 2;\n            if (maxExpArray[mid] >= x)\n                lo = mid;\n            else\n                hi = mid;\n        }\n\n        if (maxExpArray[hi] >= x)\n            return hi;\n        if (maxExpArray[lo] >= x)\n            return lo;\n\n        revert(\"findPosition: x > max\");\n    }}\n\n    /**\n      * @dev Initialize internal data structure\n      * Auto-generated via 'PrintMaxExpArray.py'\n    */\n    function initMaxExpArray() internal {\n    //  maxExpArray[  0] = 0x6bffffffffffffffffffffffffffffffff;\n    //  maxExpArray[  1] = 0x67ffffffffffffffffffffffffffffffff;\n    //  maxExpArray[  2] = 0x637fffffffffffffffffffffffffffffff;\n    //  maxExpArray[  3] = 0x5f6fffffffffffffffffffffffffffffff;\n    //  maxExpArray[  4] = 0x5b77ffffffffffffffffffffffffffffff;\n    //  maxExpArray[  5] = 0x57b3ffffffffffffffffffffffffffffff;\n    //  maxExpArray[  6] = 0x5419ffffffffffffffffffffffffffffff;\n    //  maxExpArray[  7] = 0x50a2ffffffffffffffffffffffffffffff;\n    //  maxExpArray[  8] = 0x4d517fffffffffffffffffffffffffffff;\n    //  maxExpArray[  9] = 0x4a233fffffffffffffffffffffffffffff;\n    //  maxExpArray[ 10] = 0x47165fffffffffffffffffffffffffffff;\n    //  maxExpArray[ 11] = 0x4429afffffffffffffffffffffffffffff;\n    //  maxExpArray[ 12] = 0x415bc7ffffffffffffffffffffffffffff;\n    //  maxExpArray[ 13] = 0x3eab73ffffffffffffffffffffffffffff;\n    //  maxExpArray[ 14] = 0x3c1771ffffffffffffffffffffffffffff;\n    //  maxExpArray[ 15] = 0x399e96ffffffffffffffffffffffffffff;\n    //  maxExpArray[ 16] = 0x373fc47fffffffffffffffffffffffffff;\n    //  maxExpArray[ 17] = 0x34f9e8ffffffffffffffffffffffffffff;\n    //  maxExpArray[ 18] = 0x32cbfd5fffffffffffffffffffffffffff;\n    //  maxExpArray[ 19] = 0x30b5057fffffffffffffffffffffffffff;\n    //  maxExpArray[ 20] = 0x2eb40f9fffffffffffffffffffffffffff;\n    //  maxExpArray[ 21] = 0x2cc8340fffffffffffffffffffffffffff;\n    //  maxExpArray[ 22] = 0x2af09481ffffffffffffffffffffffffff;\n    //  maxExpArray[ 23] = 0x292c5bddffffffffffffffffffffffffff;\n    //  maxExpArray[ 24] = 0x277abdcdffffffffffffffffffffffffff;\n    //  maxExpArray[ 25] = 0x25daf6657fffffffffffffffffffffffff;\n    //  maxExpArray[ 26] = 0x244c49c65fffffffffffffffffffffffff;\n    //  maxExpArray[ 27] = 0x22ce03cd5fffffffffffffffffffffffff;\n    //  maxExpArray[ 28] = 0x215f77c047ffffffffffffffffffffffff;\n    //  maxExpArray[ 29] = 0x1fffffffffffffffffffffffffffffffff;\n    //  maxExpArray[ 30] = 0x1eaefdbdabffffffffffffffffffffffff;\n    //  maxExpArray[ 31] = 0x1d6bd8b2ebffffffffffffffffffffffff;\n        maxExpArray[ 32] = 0x1c35fedd14ffffffffffffffffffffffff;\n        maxExpArray[ 33] = 0x1b0ce43b323fffffffffffffffffffffff;\n        maxExpArray[ 34] = 0x19f0028ec1ffffffffffffffffffffffff;\n        maxExpArray[ 35] = 0x18ded91f0e7fffffffffffffffffffffff;\n        maxExpArray[ 36] = 0x17d8ec7f0417ffffffffffffffffffffff;\n        maxExpArray[ 37] = 0x16ddc6556cdbffffffffffffffffffffff;\n        maxExpArray[ 38] = 0x15ecf52776a1ffffffffffffffffffffff;\n        maxExpArray[ 39] = 0x15060c256cb2ffffffffffffffffffffff;\n        maxExpArray[ 40] = 0x1428a2f98d72ffffffffffffffffffffff;\n        maxExpArray[ 41] = 0x13545598e5c23fffffffffffffffffffff;\n        maxExpArray[ 42] = 0x1288c4161ce1dfffffffffffffffffffff;\n        maxExpArray[ 43] = 0x11c592761c666fffffffffffffffffffff;\n        maxExpArray[ 44] = 0x110a688680a757ffffffffffffffffffff;\n        maxExpArray[ 45] = 0x1056f1b5bedf77ffffffffffffffffffff;\n        maxExpArray[ 46] = 0x0faadceceeff8bffffffffffffffffffff;\n        maxExpArray[ 47] = 0x0f05dc6b27edadffffffffffffffffffff;\n        maxExpArray[ 48] = 0x0e67a5a25da4107fffffffffffffffffff;\n        maxExpArray[ 49] = 0x0dcff115b14eedffffffffffffffffffff;\n        maxExpArray[ 50] = 0x0d3e7a392431239fffffffffffffffffff;\n        maxExpArray[ 51] = 0x0cb2ff529eb71e4fffffffffffffffffff;\n        maxExpArray[ 52] = 0x0c2d415c3db974afffffffffffffffffff;\n        maxExpArray[ 53] = 0x0bad03e7d883f69bffffffffffffffffff;\n        maxExpArray[ 54] = 0x0b320d03b2c343d5ffffffffffffffffff;\n        maxExpArray[ 55] = 0x0abc25204e02828dffffffffffffffffff;\n        maxExpArray[ 56] = 0x0a4b16f74ee4bb207fffffffffffffffff;\n        maxExpArray[ 57] = 0x09deaf736ac1f569ffffffffffffffffff;\n        maxExpArray[ 58] = 0x0976bd9952c7aa957fffffffffffffffff;\n        maxExpArray[ 59] = 0x09131271922eaa606fffffffffffffffff;\n        maxExpArray[ 60] = 0x08b380f3558668c46fffffffffffffffff;\n        maxExpArray[ 61] = 0x0857ddf0117efa215bffffffffffffffff;\n        maxExpArray[ 62] = 0x07ffffffffffffffffffffffffffffffff;\n        maxExpArray[ 63] = 0x07abbf6f6abb9d087fffffffffffffffff;\n        maxExpArray[ 64] = 0x075af62cbac95f7dfa7fffffffffffffff;\n        maxExpArray[ 65] = 0x070d7fb7452e187ac13fffffffffffffff;\n        maxExpArray[ 66] = 0x06c3390ecc8af379295fffffffffffffff;\n        maxExpArray[ 67] = 0x067c00a3b07ffc01fd6fffffffffffffff;\n        maxExpArray[ 68] = 0x0637b647c39cbb9d3d27ffffffffffffff;\n        maxExpArray[ 69] = 0x05f63b1fc104dbd39587ffffffffffffff;\n        maxExpArray[ 70] = 0x05b771955b36e12f7235ffffffffffffff;\n        maxExpArray[ 71] = 0x057b3d49dda84556d6f6ffffffffffffff;\n        maxExpArray[ 72] = 0x054183095b2c8ececf30ffffffffffffff;\n        maxExpArray[ 73] = 0x050a28be635ca2b888f77fffffffffffff;\n        maxExpArray[ 74] = 0x04d5156639708c9db33c3fffffffffffff;\n        maxExpArray[ 75] = 0x04a23105873875bd52dfdfffffffffffff;\n        maxExpArray[ 76] = 0x0471649d87199aa990756fffffffffffff;\n        maxExpArray[ 77] = 0x04429a21a029d4c1457cfbffffffffffff;\n        maxExpArray[ 78] = 0x0415bc6d6fb7dd71af2cb3ffffffffffff;\n        maxExpArray[ 79] = 0x03eab73b3bbfe282243ce1ffffffffffff;\n        maxExpArray[ 80] = 0x03c1771ac9fb6b4c18e229ffffffffffff;\n        maxExpArray[ 81] = 0x0399e96897690418f785257fffffffffff;\n        maxExpArray[ 82] = 0x0373fc456c53bb779bf0ea9fffffffffff;\n        maxExpArray[ 83] = 0x034f9e8e490c48e67e6ab8bfffffffffff;\n        maxExpArray[ 84] = 0x032cbfd4a7adc790560b3337ffffffffff;\n        maxExpArray[ 85] = 0x030b50570f6e5d2acca94613ffffffffff;\n        maxExpArray[ 86] = 0x02eb40f9f620fda6b56c2861ffffffffff;\n        maxExpArray[ 87] = 0x02cc8340ecb0d0f520a6af58ffffffffff;\n        maxExpArray[ 88] = 0x02af09481380a0a35cf1ba02ffffffffff;\n        maxExpArray[ 89] = 0x0292c5bdd3b92ec810287b1b3fffffffff;\n        maxExpArray[ 90] = 0x0277abdcdab07d5a77ac6d6b9fffffffff;\n        maxExpArray[ 91] = 0x025daf6654b1eaa55fd64df5efffffffff;\n        maxExpArray[ 92] = 0x0244c49c648baa98192dce88b7ffffffff;\n        maxExpArray[ 93] = 0x022ce03cd5619a311b2471268bffffffff;\n        maxExpArray[ 94] = 0x0215f77c045fbe885654a44a0fffffffff;\n        maxExpArray[ 95] = 0x01ffffffffffffffffffffffffffffffff;\n        maxExpArray[ 96] = 0x01eaefdbdaaee7421fc4d3ede5ffffffff;\n        maxExpArray[ 97] = 0x01d6bd8b2eb257df7e8ca57b09bfffffff;\n        maxExpArray[ 98] = 0x01c35fedd14b861eb0443f7f133fffffff;\n        maxExpArray[ 99] = 0x01b0ce43b322bcde4a56e8ada5afffffff;\n        maxExpArray[100] = 0x019f0028ec1fff007f5a195a39dfffffff;\n        maxExpArray[101] = 0x018ded91f0e72ee74f49b15ba527ffffff;\n        maxExpArray[102] = 0x017d8ec7f04136f4e5615fd41a63ffffff;\n        maxExpArray[103] = 0x016ddc6556cdb84bdc8d12d22e6fffffff;\n        maxExpArray[104] = 0x015ecf52776a1155b5bd8395814f7fffff;\n        maxExpArray[105] = 0x015060c256cb23b3b3cc3754cf40ffffff;\n        maxExpArray[106] = 0x01428a2f98d728ae223ddab715be3fffff;\n        maxExpArray[107] = 0x013545598e5c23276ccf0ede68034fffff;\n        maxExpArray[108] = 0x01288c4161ce1d6f54b7f61081194fffff;\n        maxExpArray[109] = 0x011c592761c666aa641d5a01a40f17ffff;\n        maxExpArray[110] = 0x0110a688680a7530515f3e6e6cfdcdffff;\n        maxExpArray[111] = 0x01056f1b5bedf75c6bcb2ce8aed428ffff;\n        maxExpArray[112] = 0x00faadceceeff8a0890f3875f008277fff;\n        maxExpArray[113] = 0x00f05dc6b27edad306388a600f6ba0bfff;\n        maxExpArray[114] = 0x00e67a5a25da41063de1495d5b18cdbfff;\n        maxExpArray[115] = 0x00dcff115b14eedde6fc3aa5353f2e4fff;\n        maxExpArray[116] = 0x00d3e7a3924312399f9aae2e0f868f8fff;\n        maxExpArray[117] = 0x00cb2ff529eb71e41582cccd5a1ee26fff;\n        maxExpArray[118] = 0x00c2d415c3db974ab32a51840c0b67edff;\n        maxExpArray[119] = 0x00bad03e7d883f69ad5b0a186184e06bff;\n        maxExpArray[120] = 0x00b320d03b2c343d4829abd6075f0cc5ff;\n        maxExpArray[121] = 0x00abc25204e02828d73c6e80bcdb1a95bf;\n        maxExpArray[122] = 0x00a4b16f74ee4bb2040a1ec6c15fbbf2df;\n        maxExpArray[123] = 0x009deaf736ac1f569deb1b5ae3f36c130f;\n        maxExpArray[124] = 0x00976bd9952c7aa957f5937d790ef65037;\n        maxExpArray[125] = 0x009131271922eaa6064b73a22d0bd4f2bf;\n        maxExpArray[126] = 0x008b380f3558668c46c91c49a2f8e967b9;\n        maxExpArray[127] = 0x00857ddf0117efa215952912839f6473e6;\n    }\n\n    // auxiliary function\n    function mulDivLog(uint256 x, uint256 y, uint256 z) private pure returns (uint256) {\n        return fixedLog(IntegralMath.mulDivF(x, y, z));\n    }\n\n    // auxiliary function\n    function mulDivExp(uint256 x, uint256 y, uint256 z) private view returns (uint256, uint256) {\n        return fixedExp(IntegralMath.mulDivF(x, y, z));\n    }\n}\n"
    },
    "contracts/libraries/FullMath.sol": {
      "content": "// SPDX-License-Identifier: MIT\npragma solidity ^0.8.0;\n\n/// @title Contains 512-bit math functions\n/// @notice Facilitates multiplication and division that can have overflow of an intermediate value without any loss of precision\n/// @dev Handles \"phantom overflow\" i.e., allows multiplication and division where an intermediate value overflows 256 bits\nlibrary FullMath {\n    /// @notice Calculates floor(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0\n    /// @param a The multiplicand\n    /// @param b The multiplier\n    /// @param denominator The divisor\n    /// @return result The 256-bit result\n    /// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv\n    function mulDiv(\n        uint256 a,\n        uint256 b,\n        uint256 denominator\n    ) internal pure returns (uint256 result) {\n        // 512-bit multiply [prod1 prod0] = a * b\n        // Compute the product mod 2**256 and mod 2**256 - 1\n        // then use the Chinese Remainder Theorem to reconstruct\n        // the 512 bit result. The result is stored in two 256\n        // variables such that product = prod1 * 2**256 + prod0\n        uint256 prod0; // Least significant 256 bits of the product\n        uint256 prod1; // Most significant 256 bits of the product\n\n        // todo unchecked\n        unchecked {\n            assembly {\n                let mm := mulmod(a, b, not(0))\n                prod0 := mul(a, b)\n                prod1 := sub(sub(mm, prod0), lt(mm, prod0))\n            }\n\n            // Handle non-overflow cases, 256 by 256 division\n            if (prod1 == 0) {\n                require(denominator > 0);\n                assembly {\n                    result := div(prod0, denominator)\n                }\n                return result;\n            }\n\n            // Make sure the result is less than 2**256.\n            // Also prevents denominator == 0\n            require(denominator > prod1);\n\n            ///////////////////////////////////////////////\n            // 512 by 256 division.\n            ///////////////////////////////////////////////\n\n            // Make division exact by subtracting the remainder from [prod1 prod0]\n            // Compute remainder using mulmod\n            uint256 remainder;\n            assembly {\n                remainder := mulmod(a, b, denominator)\n            }\n            // Subtract 256 bit number from 512 bit number\n            assembly {\n                prod1 := sub(prod1, gt(remainder, prod0))\n                prod0 := sub(prod0, remainder)\n            }\n\n            // Factor powers of two out of denominator\n            // Compute largest power of two divisor of denominator.\n            // Always >= 1.\n            uint256 twos = (~denominator + 1) & denominator;\n            // Divide denominator by power of two\n            assembly {\n                denominator := div(denominator, twos)\n            }\n\n            // Divide [prod1 prod0] by the factors of two\n            assembly {\n                prod0 := div(prod0, twos)\n            }\n            // Shift in bits from prod1 into prod0. For this we need\n            // to flip `twos` such that it is 2**256 / twos.\n            // If twos is zero, then it becomes one\n            assembly {\n                twos := add(div(sub(0, twos), twos), 1)\n            }\n\n            prod0 |= prod1 * twos;\n\n            // Invert denominator mod 2**256\n            // Now that denominator is an odd number, it has an inverse\n            // modulo 2**256 such that denominator * inv = 1 mod 2**256.\n            // Compute the inverse by starting with a seed that is correct\n            // correct for four bits. That is, denominator * inv = 1 mod 2**4\n            uint256 inv = (3 * denominator) ^ 2;\n            // Now use Newton-Raphson iteration to improve the precision.\n            // Thanks to Hensel's lifting lemma, this also works in modular\n            // arithmetic, doubling the correct bits in each step.\n\n            inv *= 2 - denominator * inv; // inverse mod 2**8\n            inv *= 2 - denominator * inv; // inverse mod 2**16\n            inv *= 2 - denominator * inv; // inverse mod 2**32\n            inv *= 2 - denominator * inv; // inverse mod 2**64\n            inv *= 2 - denominator * inv; // inverse mod 2**128\n            inv *= 2 - denominator * inv; // inverse mod 2**256\n\n            // Because the division is now exact we can divide by multiplying\n            // with the modular inverse of denominator. This will give us the\n            // correct result modulo 2**256. Since the precoditions guarantee\n            // that the outcome is less than 2**256, this is the final result.\n            // We don't need to compute the high bits of the result and prod1\n            // is no longer required.\n            result = prod0 * inv;\n            return result;\n        }\n    }\n\n    /// @notice Calculates ceil(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0\n    /// @param a The multiplicand\n    /// @param b The multiplier\n    /// @param denominator The divisor\n    /// @return result The 256-bit result\n    function mulDivRoundingUp(\n        uint256 a,\n        uint256 b,\n        uint256 denominator\n    ) internal pure returns (uint256 result) {\n        result = mulDiv(a, b, denominator);\n        if (mulmod(a, b, denominator) > 0) {\n            require(result < type(uint256).max);\n            result++;\n        }\n    }\n}\n"
    },
    "contracts/libraries/TransferHelper.sol": {
      "content": "// SPDX-License-Identifier: MIT\npragma solidity ^0.8.0;\n\n// helper methods for interacting with ERC20 tokens and sending ETH that do not consistently return true/false\nlibrary TransferHelper {\n    function safeApprove(\n        address token,\n        address to,\n        uint256 value\n    ) internal {\n        // bytes4(keccak256(bytes('approve(address,uint256)')));\n        (bool success, bytes memory data) = token.call(abi.encodeWithSelector(0x095ea7b3, to, value));\n        require(\n            success && (data.length == 0 || abi.decode(data, (bool))),\n            \"TransferHelper::safeApprove: approve failed\"\n        );\n    }\n\n    function safeTransfer(\n        address token,\n        address to,\n        uint256 value\n    ) internal {\n        // bytes4(keccak256(bytes('transfer(address,uint256)')));\n        (bool success, bytes memory data) = token.call(abi.encodeWithSelector(0xa9059cbb, to, value));\n        require(\n            success && (data.length == 0 || abi.decode(data, (bool))),\n            \"TransferHelper::safeTransfer: transfer failed\"\n        );\n    }\n\n    function safeTransferFrom(\n        address token,\n        address from,\n        address to,\n        uint256 value\n    ) internal {\n        // bytes4(keccak256(bytes('transferFrom(address,address,uint256)')));\n        (bool success, bytes memory data) = token.call(abi.encodeWithSelector(0x23b872dd, from, to, value));\n        require(\n            success && (data.length == 0 || abi.decode(data, (bool))),\n            \"TransferHelper::transferFrom: transferFrom failed\"\n        );\n    }\n\n    function safeTransferETH(address to, uint256 value) internal {\n        (bool success, ) = to.call{value: value}(new bytes(0));\n        require(success, \"TransferHelper::safeTransferETH: ETH transfer failed\");\n    }\n}\n"
    },
    "contracts/libraries/IntegralMath.sol": {
      "content": "// SPDX-License-Identifier: GPL-2.0-or-later\npragma solidity ^0.8.0;\n\nlibrary IntegralMath {\n    uint256 constant MAX_VAL = type(uint256).max;\n\n    // reverts on overflow\n    function safeAdd(uint256 x, uint256 y) internal pure returns (uint256) {\n        return x + y;\n    }\n\n    // does not revert on overflow\n    function unsafeAdd(uint256 x, uint256 y) internal pure returns (uint256) { unchecked {\n        return x + y;\n    }}\n\n    // does not revert on overflow\n    function unsafeSub(uint256 x, uint256 y) internal pure returns (uint256) { unchecked {\n        return x - y;\n    }}\n\n    // does not revert on overflow\n    function unsafeMul(uint256 x, uint256 y) internal pure returns (uint256) { unchecked {\n        return x * y;\n    }}\n\n    // does not overflow\n    function mulModMax(uint256 x, uint256 y) internal pure returns (uint256) { unchecked {\n        return mulmod(x, y, MAX_VAL);\n    }}\n\n    // does not overflow\n    function mulMod(uint256 x, uint256 y, uint256 z) internal pure returns (uint256) { unchecked {\n        return mulmod(x, y, z);\n    }}\n    /**\n      * @dev Compute the largest integer smaller than or equal to the binary logarithm of `n`\n    */\n    function floorLog2(uint256 n) internal pure returns (uint8) { unchecked {\n        uint8 res = 0;\n\n        if (n < 256) {\n            // at most 8 iterations\n            while (n > 1) {\n                n >>= 1;\n                res += 1;\n            }\n        }\n        else {\n            // exactly 8 iterations\n            for (uint8 s = 128; s > 0; s >>= 1) {\n                if (n >= 1 << s) {\n                    n >>= s;\n                    res |= s;\n                }\n            }\n        }\n\n        return res;\n    }}\n\n    /**\n      * @dev Compute the largest integer smaller than or equal to the square root of `n`\n    */\n    function floorSqrt(uint256 n) internal pure returns (uint256) { unchecked {\n        if (n > 0) {\n            uint256 x = n / 2 + 1;\n            uint256 y = (x + n / x) / 2;\n            while (x > y) {\n                x = y;\n                y = (x + n / x) / 2;\n            }\n            return x;\n        }\n        return 0;\n    }}\n\n    /**\n      * @dev Compute the smallest integer larger than or equal to the square root of `n`\n    */\n    function ceilSqrt(uint256 n) internal pure returns (uint256) { unchecked {\n        uint256 x = floorSqrt(n);\n        return x ** 2 == n ? x : x + 1;\n    }}\n\n    /**\n      * @dev Compute the largest integer smaller than or equal to the cubic root of `n`\n    */\n    function floorCbrt(uint256 n) internal pure returns (uint256) { unchecked {\n        uint256 x = 0;\n        for (uint256 y = 1 << 255; y > 0; y >>= 3) {\n            x <<= 1;\n            uint256 z = 3 * x * (x + 1) + 1;\n            if (n / y >= z) {\n                n -= y * z;\n                x += 1;\n            }\n        }\n        return x;\n    }}\n\n    /**\n      * @dev Compute the smallest integer larger than or equal to the cubic root of `n`\n    */\n    function ceilCbrt(uint256 n) internal pure returns (uint256) { unchecked {\n        uint256 x = floorCbrt(n);\n        return x ** 3 == n ? x : x + 1;\n    }}\n\n    /**\n      * @dev Compute the nearest integer to the quotient of `n` and `d` (or `n / d`)\n    */\n    function roundDiv(uint256 n, uint256 d) internal pure returns (uint256) { unchecked {\n        return n / d + (n % d) / (d - d / 2);\n    }}\n\n    /**\n      * @dev Compute the largest integer smaller than or equal to `x * y / z`\n    */\n    function mulDivF(uint256 x, uint256 y, uint256 z) internal pure returns (uint256) { unchecked {\n        (uint256 xyh, uint256 xyl) = mul512(x, y);\n        if (xyh == 0) { // `x * y < 2 ^ 256`\n            return xyl / z;\n        }\n        if (xyh < z) { // `x * y / z < 2 ^ 256`\n            uint256 m = mulMod(x, y, z);                    // `m = x * y % z`\n            (uint256 nh, uint256 nl) = sub512(xyh, xyl, m); // `n = x * y - m` hence `n / z = floor(x * y / z)`\n            if (nh == 0) { // `n < 2 ^ 256`\n                return nl / z;\n            }\n            uint256 p = unsafeSub(0, z) & z; // `p` is the largest power of 2 which `z` is divisible by\n            uint256 q = div512(nh, nl, p);   // `n` is divisible by `p` because `n` is divisible by `z` and `z` is divisible by `p`\n            uint256 r = inv256(z / p);       // `z / p = 1 mod 2` hence `inverse(z / p) = 1 mod 2 ^ 256`\n            return unsafeMul(q, r);          // `q * r = (n / p) * inverse(z / p) = n / z`\n        }\n        revert(); // `x * y / z >= 2 ^ 256`\n    }}\n\n    /**\n      * @dev Compute the smallest integer larger than or equal to `x * y / z`\n    */\n    function mulDivC(uint256 x, uint256 y, uint256 z) internal pure returns (uint256) { unchecked {\n        uint256 w = mulDivF(x, y, z);\n        if (mulMod(x, y, z) > 0)\n            return safeAdd(w, 1);\n        return w;\n    }}\n\n    /**\n      * @dev Compute the value of `x * y`\n    */\n    function mul512(uint256 x, uint256 y) private pure returns (uint256, uint256) { unchecked {\n        uint256 p = mulModMax(x, y);\n        uint256 q = unsafeMul(x, y);\n        if (p >= q)\n            return (p - q, q);\n        return (unsafeSub(p, q) - 1, q);\n    }}\n\n    /**\n      * @dev Compute the value of `2 ^ 256 * xh + xl - y`, where `2 ^ 256 * xh + xl >= y`\n    */\n    function sub512(uint256 xh, uint256 xl, uint256 y) private pure returns (uint256, uint256) { unchecked {\n        if (xl >= y)\n            return (xh, xl - y);\n        return (xh - 1, unsafeSub(xl, y));\n    }}\n\n    /**\n      * @dev Compute the value of `(2 ^ 256 * xh + xl) / pow2n`, where `xl` is divisible by `pow2n`\n    */\n    function div512(uint256 xh, uint256 xl, uint256 pow2n) private pure returns (uint256) { unchecked {\n        uint256 pow2nInv = unsafeAdd(unsafeSub(0, pow2n) / pow2n, 1); // `1 << (256 - n)`\n        return unsafeMul(xh, pow2nInv) | (xl / pow2n); // `(xh << (256 - n)) | (xl >> n)`\n    }}\n\n    /**\n      * @dev Compute the inverse of `d` modulo `2 ^ 256`, where `d` is congruent to `1` modulo `2`\n    */\n    function inv256(uint256 d) private pure returns (uint256) { unchecked {\n        // approximate the root of `f(x) = 1 / x - d` using the newton–raphson convergence method\n        uint256 x = 1;\n        for (uint256 i = 0; i < 8; ++i)\n            x = unsafeMul(x, unsafeSub(2, unsafeMul(x, d))); // `x = x * (2 - x * d) mod 2 ^ 256`\n        return x;\n    }}\n}\n"
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