// File: lib/delphi/src/factory/NotionalFinanceFactory.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

import {NotionalFinanceValueProvider} from "../oracle_implementations/discount_rate/NotionalFinance/NotionalFinanceValueProvider.sol";
import {Relayer} from "../relayer/Relayer.sol";
import {IRelayer} from "../relayer/IRelayer.sol";

contract NotionalFinanceFactory {
    event NotionalFinanceDeployed(
        address relayerAddress,
        address oracleAddress
    );

    /// @param collybus_ Address of the collybus
    /// @param tokenId_ Token Id that will be used to push values to Collybus
    /// @param minimumPercentageDeltaValue_ Minimum delta value used to determine when to
    /// push data to Collybus
    /// @param timeUpdateWindow_ Minimum time between updates of the value
    /// @param notional_ The address of the deployed notional contract
    /// @param currencyId_ Currency ID(eth = 1, dai = 2, usdc = 3, wbtc = 4)
    /// @param oracleRateDecimals_ Precision of the Notional oracle rate
    /// @param maturity_ Maturity date.
    /// @return The address of the Relayer
    function create(
        // Relayer parameters
        address collybus_,
        uint256 tokenId_,
        uint256 minimumPercentageDeltaValue_,
        // Oracle parameters
        uint256 timeUpdateWindow_,
        // Notional specific parameters, see NotionalFinanceValueProvider for more info
        address notional_,
        uint256 currencyId_,
        uint256 oracleRateDecimals_,
        uint256 maturity_
    ) external returns (address) {
        // Create the oracle that will fetch data from the NotionalFinance contract
        NotionalFinanceValueProvider notionalFinanceValueProvider = new NotionalFinanceValueProvider(
                timeUpdateWindow_,
                notional_,
                currencyId_,
                oracleRateDecimals_,
                maturity_
            );

        // Create the relayer that manages the oracle and pushes data to Collybus
        Relayer relayer = new Relayer(
            collybus_,
            IRelayer.RelayerType.DiscountRate,
            address(notionalFinanceValueProvider),
            bytes32(tokenId_),
            minimumPercentageDeltaValue_
        );

        // Whitelist the Relayer in the Oracle so it can trigger updates
        notionalFinanceValueProvider.allowCaller(
            notionalFinanceValueProvider.ANY_SIG(),
            address(relayer)
        );
        // Whitelist the deployer
        notionalFinanceValueProvider.allowCaller(
            notionalFinanceValueProvider.ANY_SIG(),
            msg.sender
        );
        // Whitelist the deployer
        relayer.allowCaller(relayer.ANY_SIG(), msg.sender);

        // Renounce permissions
        relayer.blockCaller(relayer.ANY_SIG(), address(this));
        notionalFinanceValueProvider.blockCaller(
            notionalFinanceValueProvider.ANY_SIG(),
            address(this)
        );

        emit NotionalFinanceDeployed(
            address(relayer),
            address(notionalFinanceValueProvider)
        );

        // Return the address of the relayer
        return address(relayer);
    }
}

// File: lib/delphi/src/guarded/Guarded.sol



// SPDX-License-Identifier: Unlicensed
pragma solidity ^0.8.0;

/// @title Guarded
/// @notice Mixin implementing an authentication scheme on a method level
abstract contract Guarded {
    /// ======== Custom Errors ======== ///

    error Guarded__notRoot();
    error Guarded__notGranted();

    /// ======== Storage ======== ///

    /// @notice Wildcard for granting a caller to call every guarded method
    bytes32 public constant ANY_SIG = keccak256("ANY_SIG");
    /// @notice Wildcard for granting a caller to call every guarded method
    address public constant ANY_CALLER =
        address(uint160(uint256(bytes32(keccak256("ANY_CALLER")))));

    /// @notice Mapping storing who is granted to which method
    /// @dev Method Signature => Caller => Bool
    mapping(bytes32 => mapping(address => bool)) private _canCall;

    /// ======== Events ======== ///

    event AllowCaller(bytes32 sig, address who);
    event BlockCaller(bytes32 sig, address who);

    constructor() {
        // set root
        _setRoot(msg.sender);
    }

    /// ======== Auth ======== ///

    modifier callerIsRoot() {
        if (_canCall[ANY_SIG][msg.sender]) {
            _;
        } else revert Guarded__notRoot();
    }

    modifier checkCaller() {
        if (canCall(msg.sig, msg.sender)) {
            _;
        } else revert Guarded__notGranted();
    }

    /// @notice Grant the right to call method `sig` to `who`
    /// @dev Only the root user (granted `ANY_SIG`) is able to call this method
    /// @param sig_ Method signature (4Byte)
    /// @param who_ Address of who should be able to call `sig`
    function allowCaller(bytes32 sig_, address who_) public callerIsRoot {
        _canCall[sig_][who_] = true;
        emit AllowCaller(sig_, who_);
    }

    /// @notice Revoke the right to call method `sig` from `who`
    /// @dev Only the root user (granted `ANY_SIG`) is able to call this method
    /// @param sig_ Method signature (4Byte)
    /// @param who_ Address of who should not be able to call `sig` anymore
    function blockCaller(bytes32 sig_, address who_) public callerIsRoot {
        _canCall[sig_][who_] = false;
        emit BlockCaller(sig_, who_);
    }

    /// @notice Returns if `who` can call `sig`
    /// @param sig_ Method signature (4Byte)
    /// @param who_ Address of who should be able to call `sig`
    function canCall(bytes32 sig_, address who_) public view returns (bool) {
        return (_canCall[sig_][who_] ||
            _canCall[ANY_SIG][who_] ||
            _canCall[sig_][ANY_CALLER]);
    }

    /// @notice Sets the root user (granted `ANY_SIG`)
    /// @param root_ Address of who should be set as root
    function _setRoot(address root_) internal {
        _canCall[ANY_SIG][root_] = true;
        emit AllowCaller(ANY_SIG, root_);
    }
}

// File: lib/delphi/src/oracle/IOracle.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

interface IOracle {
    function value() external view returns (int256, bool);

    function nextValue() external view returns (int256);

    function update() external returns (bool);
}

// File: lib/delphi/src/oracle/Oracle.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

import {IOracle} from "./IOracle.sol";

import {Pausable} from "../pausable/Pausable.sol";

abstract contract Oracle is Pausable, IOracle {
    /// @notice Emitted when a method is reentered
    error Oracle__nonReentrant();

    /// ======== Events ======== ///

    event ValueInvalid();
    event ValueUpdated(int256 currentValue, int256 nextValue);
    event OracleReset();

    /// ======== Storage ======== ///
    // Time interval between the value updates
    uint256 public immutable timeUpdateWindow;

    // Timestamp of the current value
    uint256 public lastTimestamp;

    // The next value that will replace the current value once the timeUpdateWindow has passed
    int256 public override(IOracle) nextValue;

    // Current value that will be returned by the Oracle
    int256 private _currentValue;

    // Flag that tells if the value provider returned successfully
    bool private _validReturnedValue;

    // Reentrancy constants
    uint256 private constant _NOT_ENTERED = 1;
    uint256 private constant _ENTERED = 2;

    // Reentrancy guard flag
    uint256 private _reentrantGuard = _NOT_ENTERED;

    /// ======== Modifiers ======== ///

    modifier nonReentrant() {
        // Check if the guard is set
        if (_reentrantGuard != _NOT_ENTERED) {
            revert Oracle__nonReentrant();
        }

        // Set the guard
        _reentrantGuard = _ENTERED;

        // Allow execution
        _;

        // Reset the guard
        _reentrantGuard = _NOT_ENTERED;
    }

    constructor(uint256 timeUpdateWindow_) {
        timeUpdateWindow = timeUpdateWindow_;
        _validReturnedValue = false;
    }

    /// @notice Get the current value of the oracle
    /// @return The current value of the oracle
    /// @return Whether the value is valid
    function value()
        public
        view
        override(IOracle)
        whenNotPaused
        returns (int256, bool)
    {
        // Value is considered valid if the value provider successfully returned a value
        return (_currentValue, _validReturnedValue);
    }

    function getValue() external virtual returns (int256);

    function update()
        public
        override(IOracle)
        checkCaller
        nonReentrant
        returns (bool)
    {
        // Not enough time has passed since the last update
        if (lastTimestamp + timeUpdateWindow > block.timestamp) {
            // Exit early if no update is needed
            return false;
        }

        // Oracle update should not fail even if the value provider fails to return a value
        try this.getValue() returns (int256 returnedValue) {
            // Update the value using an exponential moving average
            if (_currentValue == 0) {
                // First update takes the current value
                nextValue = returnedValue;
                _currentValue = nextValue;
            } else {
                // Update the current value with the next value
                _currentValue = nextValue;
                // Set the returnedValue as the next value
                nextValue = returnedValue;
            }

            // Save when the value was last updated
            lastTimestamp = block.timestamp;
            _validReturnedValue = true;

            emit ValueUpdated(_currentValue, nextValue);

            return true;
        } catch {
            // When a value provider fails, we update the valid flag which will
            // invalidate the value instantly
            _validReturnedValue = false;
            emit ValueInvalid();
        }

        return false;
    }

    function pause() public checkCaller {
        _pause();
    }

    function unpause() public checkCaller {
        _unpause();
    }

    function reset() public whenPaused checkCaller {
        _currentValue = 0;
        nextValue = 0;
        lastTimestamp = 0;
        _validReturnedValue = false;

        emit OracleReset();
    }
}

// File: lib/delphi/src/oracle_implementations/discount_rate/NotionalFinance/INotionalView.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

/// Imported from:
/// https://github.com/notional-finance/contracts-v2/blob/23a3d5fcdba8a2e2ae6b0730f73eed810484e4cc/contracts/global/Types.sol
/// @dev Holds information about a market, total storage is 42 bytes so this spans
/// two storage words
struct MarketStorage {
    // Total fCash in the market
    uint80 totalfCash;
    // Total asset cash in the market
    uint80 totalAssetCash;
    // Last annualized interest rate the market traded at
    uint32 lastImpliedRate;
    // Last recorded oracle rate for the market
    uint32 oracleRate;
    // Last time a trade was made
    uint32 previousTradeTime;
    // This is stored in slot + 1
    uint80 totalLiquidity;
}

/// Imported from:
/// https://github.com/notional-finance/contracts-v2/blob/23a3d5fcdba8a2e2ae6b0730f73eed810484e4cc/contracts/global/Types.sol
/// @dev Market object as represented in memory
struct MarketParameters {
    bytes32 storageSlot;
    uint256 maturity;
    // Total amount of fCash available for purchase in the market.
    int256 totalfCash;
    // Total amount of cash available for purchase in the market.
    int256 totalAssetCash;
    // Total amount of liquidity tokens (representing a claim on liquidity) in the market.
    int256 totalLiquidity;
    // This is the previous annualized interest rate in RATE_PRECISION that the market traded
    // at. This is used to calculate the rate anchor to smooth interest rates over time.
    // RATE_PRECISION is defined as 1e9 in the constants contract deployed here:
    // https://github.com/notional-finance/contracts-v2/blob/23a3d5fcdba8a2e2ae6b0730f73eed810484e4cc/contracts/global/Constants.sol
    uint256 lastImpliedRate;
    // Time lagged version of lastImpliedRate, used to value fCash assets at market rates while
    // remaining resistent to flash loan attacks.
    uint256 oracleRate;
    // This is the timestamp of the previous trade
    uint256 previousTradeTime;
}

interface INotionalView {
    /// @notice Returns a single market
    function getMarket(
        uint16 currencyId_,
        uint256 maturity_,
        uint256 settlementDate_
    ) external view returns (MarketParameters memory);

    function getActiveMarkets(uint16 currencyId)
        external
        view
        returns (MarketParameters[] memory);
}

// File: lib/delphi/src/oracle_implementations/discount_rate/NotionalFinance/NotionalFinanceValueProvider.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

import {Convert} from "../utils/Convert.sol";
import {Oracle} from "../../../oracle/Oracle.sol";
import "prb-math/contracts/PRBMathSD59x18.sol";
import {INotionalView} from "./INotionalView.sol";
import {MarketParameters} from "./INotionalView.sol";

contract NotionalFinanceValueProvider is Oracle, Convert {
    /// @notice Emitted when trying to add pull a value for an expired pool
    error NotionalFinanceValueProvider__getValue_maturityLessThanBlocktime(
        uint256 maturity
    );

    /// @notice Emitted when an invalid currencyId is used to deploy the contract
    error NotionalFinanceValueProvider__constructor_invalidCurrencyId(
        uint256 currencyId
    );

    /// @notice Emitted when the parameters do not map to an active / initialized Notional Market
    error NotionalFinanceValueProvider__getValue_invalidMarketParameters(
        uint256 currencyId,
        uint256 maturityDate,
        uint256 settlementDate
    );

    // Seconds in a 360 days year as used by Notional in 18 digits precision
    int256 internal constant SECONDS_PER_YEAR = 31104000 * 1e18;

    // Quarter computed as it's defined in the Notional protocol
    // 3 months , 5 weeks per month, 6 days per week computed in seconds
    // Reference: https://github.com/notional-finance/contracts-v2/blob/master/contracts/global/Constants.sol#L56
    uint256 internal constant QUARTER = 3 * 5 * 6 * 86400;

    address public immutable notional;
    uint256 public immutable currencyId;
    uint256 public immutable maturityDate;
    uint256 public immutable oracleRateDecimals;

    /// @notice Constructs the Value provider contracts with the needed Notional contract data in order to
    /// calculate the per-second rate.
    /// @param timeUpdateWindow_ Minimum time between updates of the value
    /// @param notional_ The address of the deployed notional contract
    /// @param currencyId_ Currency ID(eth = 1, dai = 2, usdc = 3, wbtc = 4)
    /// @param oracleRateDecimals_ Precision of the Notional oracle rate
    /// @param maturityDate_ Maturity date.
    /// @dev reverts if the CurrencyId is bigger than uint16 max value
    constructor(
        // Oracle parameters
        uint256 timeUpdateWindow_,
        // Notional specific parameters
        address notional_,
        uint256 currencyId_,
        uint256 oracleRateDecimals_,
        uint256 maturityDate_
    ) Oracle(timeUpdateWindow_) {
        if (currencyId_ > type(uint16).max) {
            revert NotionalFinanceValueProvider__constructor_invalidCurrencyId(
                currencyId_
            );
        }

        oracleRateDecimals = oracleRateDecimals_;
        notional = notional_;
        currencyId = currencyId_;
        maturityDate = maturityDate_;
    }

    /// @notice Calculates the annual rate used by the FIAT DAO contracts
    /// the rate is precomputed by the notional contract and scaled to 1e18 precision
    /// @dev For more details regarding the computed rate in the Notional contracts:
    /// https://github.com/notional-finance/contracts-v2/blob/b8e3792e39486b2719c6153acc270199377cc6b9/contracts/internal/markets/Market.sol#L495
    /// @return result The result as an signed 59.18-decimal fixed-point number
    function getValue() external view override(Oracle) returns (int256) {
        // No values for matured pools
        if (block.timestamp >= maturityDate) {
            revert NotionalFinanceValueProvider__getValue_maturityLessThanBlocktime(
                maturityDate
            );
        }

        // Retrieve the oracle rate from Notional
        uint256 oracleRate = getOracleRate();

        // Convert rate per annum to 18 digits precision.
        uint256 ratePerAnnum = uconvert(oracleRate, oracleRateDecimals, 18);

        // Convert per annum to per second rate
        int256 ratePerSecondD59x18 = PRBMathSD59x18.div(
            int256(ratePerAnnum),
            SECONDS_PER_YEAR
        );

        // Convert continuous compounding to discrete compounding rate
        int256 discreteRateD59x18 = PRBMathSD59x18.exp(ratePerSecondD59x18) -
            PRBMathSD59x18.SCALE;

        // The result is a 59.18 fixed-point number.
        return discreteRateD59x18;
    }

    /// @notice Retrieve the oracle rate from the NotionalFinance Market
    function getOracleRate() internal view returns (uint256) {
        uint256 settlementDate = getSettlementDate();
        // Attempt to retrieve the oracle rate directly by using the maturity and settlement date
        MarketParameters memory marketParams = INotionalView(notional)
            .getMarket(uint16(currencyId), maturityDate, settlementDate);

        // If we find an active market we can return the oracle rate otherwise we revert
        if (marketParams.oracleRate <= 0) {
            revert NotionalFinanceValueProvider__getValue_invalidMarketParameters(
                currencyId,
                maturityDate,
                settlementDate
            );
        }

        return marketParams.oracleRate;
    }

    /// @notice Computes the settlement date as is done in the NotionalFinance contracts
    /// Reference 1:
    /// https://github.com/notional-finance/contracts-v2/blob/d89be9474e181b322480830501728ea625e853d0/contracts/internal/markets/DateTime.sol#L14
    /// Reference 2:
    /// https://github.com/notional-finance/contracts-v2/blob/d89be9474e181b322480830501728ea625e853d0/contracts/internal/markets/Market.sol#L536
    function getSettlementDate() public view returns (uint256) {
        return block.timestamp - (block.timestamp % QUARTER) + QUARTER;
    }
}

// File: lib/delphi/src/oracle_implementations/discount_rate/utils/Convert.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

contract Convert {
    function convert(
        int256 x_,
        uint256 currentPrecision_,
        uint256 targetPrecision_
    ) internal pure returns (int256) {
        if (targetPrecision_ > currentPrecision_)
            return x_ * int256(10**(targetPrecision_ - currentPrecision_));

        return x_ / int256(10**(currentPrecision_ - targetPrecision_));
    }

    function uconvert(
        uint256 x_,
        uint256 currentPrecision_,
        uint256 targetPrecision_
    ) internal pure returns (uint256) {
        if (targetPrecision_ > currentPrecision_)
            return x_ * 10**(targetPrecision_ - currentPrecision_);

        return x_ / 10**(currentPrecision_ - targetPrecision_);
    }
}

// File: lib/delphi/src/pausable/Pausable.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

/// @notice Emitted when paused
error Pausable__whenNotPaused_paused();

/// @notice Emitted when not paused
error Pausable__whenPaused_notPaused();

import {Guarded} from "../guarded/Guarded.sol";

contract Pausable is Guarded {
    event Paused(address who);
    event Unpaused(address who);

    bool private _paused;

    function paused() public view virtual returns (bool) {
        return _paused;
    }

    modifier whenNotPaused() {
        // If the contract is paused, throw an error
        if (_paused) {
            revert Pausable__whenNotPaused_paused();
        }
        _;
    }

    modifier whenPaused() {
        // If the contract is not paused, throw an error
        if (_paused == false) {
            revert Pausable__whenPaused_notPaused();
        }
        _;
    }

    function _pause() internal whenNotPaused {
        _paused = true;
        emit Paused(msg.sender);
    }

    function _unpause() internal whenPaused {
        _paused = false;
        emit Unpaused(msg.sender);
    }
}

// File: lib/delphi/src/relayer/ICollybus.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

// Lightweight interface for Collybus
// Source: https://github.com/fiatdao/fiat-lux/blob/f49a9457fbcbdac1969c35b4714722f00caa462c/src/interfaces/ICollybus.sol
interface ICollybus {
    function updateDiscountRate(uint256 tokenId_, uint256 rate_) external;

    function updateSpot(address token_, uint256 spot_) external;
}

// File: lib/delphi/src/relayer/IRelayer.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

interface IRelayer {
    enum RelayerType {
        DiscountRate,
        SpotPrice,
        COUNT
    }

    function execute() external returns (bool);

    function executeWithRevert() external;
}

// File: lib/delphi/src/relayer/Relayer.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.0;

import {IRelayer} from "./IRelayer.sol";
import {IOracle} from "../oracle/IOracle.sol";
import {ICollybus} from "./ICollybus.sol";
import {Guarded} from "../guarded/Guarded.sol";

/// @notice The Relayer contract manages the relationship between an oracle and Collybus.
/// The Relayer manages an Oracle for which it controls the update flow and via execute() calls
/// pushes data to Collybus when it's needed
/// @dev The Relayer should be the single entity that updates the oracle so that the Relayer and the Oracle
/// are value synched. The same is true for the Relayer-Collybus relationship as we do not interrogate the Collybus
/// for the current value and use a storage cached last updated value.
contract Relayer is Guarded, IRelayer {
    /// @notice Emitter during executeWithRevert() if the oracle is not updated successfully
    error Relayer__executeWithRevert_noUpdate(RelayerType relayerType);

    /// @notice Emitted when trying to set a parameter that does not exist
    error Relayer__setParam_unrecognizedParam(bytes32 param);

    event SetParam(bytes32 param, uint256 value);
    event UpdateOracle(address oracle, int256 value, bool valid);
    event UpdatedCollybus(bytes32 tokenId, uint256 rate, RelayerType);

    /// ======== Storage ======== ///

    address public immutable collybus;
    RelayerType public immutable relayerType;
    address public immutable oracle;
    bytes32 public immutable encodedTokenId;

    uint256 public minimumPercentageDeltaValue;
    int256 private _lastUpdateValue;

    /// @param collybusAddress_ Address of the collybus
    /// @param type_ Relayer type, DiscountRate or SpotPrice
    /// @param oracleAddress_ The address of the oracle used by the Relayer
    /// @param encodedTokenId_ Encoded token Id that will be used to push values to Collybus
    /// uint256 for discount rate, address for spot price
    /// @param minimumPercentageDeltaValue_ Minimum delta value used to determine when to
    /// push data to Collybus
    constructor(
        address collybusAddress_,
        RelayerType type_,
        address oracleAddress_,
        bytes32 encodedTokenId_,
        uint256 minimumPercentageDeltaValue_
    ) {
        collybus = collybusAddress_;
        relayerType = type_;
        oracle = oracleAddress_;
        encodedTokenId = encodedTokenId_;
        minimumPercentageDeltaValue = minimumPercentageDeltaValue_;
        _lastUpdateValue = 0;
    }

    /// @notice Sets a Relayer parameter
    /// Supported parameters are:
    /// - minimumPercentageDeltaValue
    /// @param param_ The identifier of the parameter that should be updated
    /// @param value_ The new value
    /// @dev Reverts if parameter is not found
    function setParam(bytes32 param_, uint256 value_) public checkCaller {
        if (param_ == "minimumPercentageDeltaValue") {
            minimumPercentageDeltaValue = value_;
        } else revert Relayer__setParam_unrecognizedParam(param_);

        emit SetParam(param_, value_);
    }

    /// @notice Updates the oracle and pushes the updated data to Collybus if the
    /// delta change in value is larger than the minimum threshold value
    /// @return Whether the Collybus was or is about to be updated
    /// @dev Return value is mainly meant to be used by Keepers in order to optimize costs
    function execute() public override(IRelayer) returns (bool) {
        // We always update the oracles before retrieving the rates
        bool oracleUpdated = IOracle(oracle).update();
        (int256 oracleValue, ) = IOracle(oracle).value();

        // If the oracle was not updated we exit early because no value was updated
        if (!oracleUpdated) return false;

        // We calculate whether the returned value is over the minimumPercentageDeltaValue
        // If the deviation is large enough we push the value to Collybus and return true
        bool currentValueThresholdPassed = checkDeviation(
            _lastUpdateValue,
            oracleValue,
            minimumPercentageDeltaValue
        );
        if (currentValueThresholdPassed) {
            updateCollybus(oracleValue);
            return true;
        }

        // If the oracle received a new value (nextValue != oracleValue), but wasn't updated yet,
        // we return true to indicate that the Collybus is about to be updated
        bool nextValueThresholdPassed = checkDeviation(
            _lastUpdateValue,
            IOracle(oracle).nextValue(),
            minimumPercentageDeltaValue
        );

        return nextValueThresholdPassed;
    }

    /// @notice The function will call execute() and will revert if false
    /// @dev This method is needed for services that run on each block and only call the method if it doesn't fail
    /// For extra information on the revert conditions check the execute() function
    function executeWithRevert() public override(IRelayer) {
        if (!execute()) {
            revert Relayer__executeWithRevert_noUpdate(relayerType);
        }
    }

    /// @notice Updates Collybus with the new value
    /// @param oracleValue_ the new value pushed into Collybus
    function updateCollybus(int256 oracleValue_) internal {
        // Save the new value to be able to check if the next value is over the threshold
        _lastUpdateValue = oracleValue_;

        if (relayerType == RelayerType.DiscountRate) {
            ICollybus(collybus).updateDiscountRate(
                uint256(encodedTokenId),
                uint256(oracleValue_)
            );
        } else if (relayerType == RelayerType.SpotPrice) {
            ICollybus(collybus).updateSpot(
                address(uint160(uint256(encodedTokenId))),
                uint256(oracleValue_)
            );
        }

        emit UpdatedCollybus(
            encodedTokenId,
            uint256(oracleValue_),
            relayerType
        );
    }

    /// @notice Returns true if the percentage difference between the two values is larger than the percentage
    /// @param baseValue_ The value that the percentage is based on
    /// @param newValue_ The new value
    /// @param percentage_ The percentage threshold value (100% = 100_00, 50% = 50_00, etc)
    function checkDeviation(
        int256 baseValue_,
        int256 newValue_,
        uint256 percentage_
    ) public pure returns (bool) {
        int256 deviation = (baseValue_ * int256(percentage_)) / 100_00;

        if (
            baseValue_ + deviation <= newValue_ ||
            baseValue_ - deviation >= newValue_
        ) return true;

        return false;
    }
}

// File: lib/fiat/lib/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 least power of two that is greater than or equal to sqrt(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;
        }
    }
}

// File: lib/fiat/lib/prb-math/contracts/PRBMathSD59x18.sol



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

import "./PRBMath.sol";

/// @title PRBMathSD59x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math that works with int256 numbers considered to have 18
/// trailing decimals. We call this number representation signed 59.18-decimal fixed-point, since the numbers can have
/// a sign and there can be up to 59 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 int256.
library PRBMathSD59x18 {
    /// @dev log2(e) as a signed 59.18-decimal fixed-point number.
    int256 internal constant LOG2_E = 1_442695040888963407;

    /// @dev Half the SCALE number.
    int256 internal constant HALF_SCALE = 5e17;

    /// @dev The maximum value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MAX_SD59x18 =
        57896044618658097711785492504343953926634992332820282019728_792003956564819967;

    /// @dev The maximum whole value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MAX_WHOLE_SD59x18 =
        57896044618658097711785492504343953926634992332820282019728_000000000000000000;

    /// @dev The minimum value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MIN_SD59x18 =
        -57896044618658097711785492504343953926634992332820282019728_792003956564819968;

    /// @dev The minimum whole value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MIN_WHOLE_SD59x18 =
        -57896044618658097711785492504343953926634992332820282019728_000000000000000000;

    /// @dev How many trailing decimals can be represented.
    int256 internal constant SCALE = 1e18;

    /// INTERNAL FUNCTIONS ///

    /// @notice Calculate the absolute value of x.
    ///
    /// @dev Requirements:
    /// - x must be greater than MIN_SD59x18.
    ///
    /// @param x The number to calculate the absolute value for.
    /// @param result The absolute value of x.
    function abs(int256 x) internal pure returns (int256 result) {
        unchecked {
            if (x == MIN_SD59x18) {
                revert PRBMathSD59x18__AbsInputTooSmall();
            }
            result = x < 0 ? -x : x;
        }
    }

    /// @notice Calculates the arithmetic average of x and y, rounding down.
    /// @param x The first operand as a signed 59.18-decimal fixed-point number.
    /// @param y The second operand as a signed 59.18-decimal fixed-point number.
    /// @return result The arithmetic average as a signed 59.18-decimal fixed-point number.
    function avg(int256 x, int256 y) internal pure returns (int256 result) {
        // The operations can never overflow.
        unchecked {
            int256 sum = (x >> 1) + (y >> 1);
            if (sum < 0) {
                // If at least one of x and y is odd, we add 1 to the result. This is because shifting negative numbers to the
                // right rounds down to infinity.
                assembly {
                    result := add(sum, and(or(x, y), 1))
                }
            } else {
                // If both x and y are odd, we add 1 to the result. This is because if both numbers are odd, the 0.5
                // remainder gets truncated twice.
                result = sum + (x & y & 1);
            }
        }
    }

    /// @notice Yields the least greatest signed 59.18 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_SD59x18.
    ///
    /// @param x The signed 59.18-decimal fixed-point number to ceil.
    /// @param result The least integer greater than or equal to x, as a signed 58.18-decimal fixed-point number.
    function ceil(int256 x) internal pure returns (int256 result) {
        if (x > MAX_WHOLE_SD59x18) {
            revert PRBMathSD59x18__CeilOverflow(x);
        }
        unchecked {
            int256 remainder = x % SCALE;
            if (remainder == 0) {
                result = x;
            } else {
                // Solidity uses C fmod style, which returns a modulus with the same sign as x.
                result = x - remainder;
                if (x > 0) {
                    result += SCALE;
                }
            }
        }
    }

    /// @notice Divides two signed 59.18-decimal fixed-point numbers, returning a new signed 59.18-decimal fixed-point number.
    ///
    /// @dev Variant of "mulDiv" that works with signed numbers. Works by computing the signs and the absolute values separately.
    ///
    /// Requirements:
    /// - All from "PRBMath.mulDiv".
    /// - None of the inputs can be MIN_SD59x18.
    /// - The denominator cannot be zero.
    /// - The result must fit within int256.
    ///
    /// Caveats:
    /// - All from "PRBMath.mulDiv".
    ///
    /// @param x The numerator as a signed 59.18-decimal fixed-point number.
    /// @param y The denominator as a signed 59.18-decimal fixed-point number.
    /// @param result The quotient as a signed 59.18-decimal fixed-point number.
    function div(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == MIN_SD59x18 || y == MIN_SD59x18) {
            revert PRBMathSD59x18__DivInputTooSmall();
        }

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

        // Compute the absolute value of (x*SCALE)÷y. The result must fit within int256.
        uint256 rAbs = PRBMath.mulDiv(ax, uint256(SCALE), ay);
        if (rAbs > uint256(MAX_SD59x18)) {
            revert PRBMathSD59x18__DivOverflow(rAbs);
        }

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

        // XOR over sx and sy. This is basically checking whether the inputs have the same sign. If yes, the result
        // should be positive. Otherwise, it should be negative.
        result = sx ^ sy == 1 ? -int256(rAbs) : int256(rAbs);
    }

    /// @notice Returns Euler's number as a signed 59.18-decimal fixed-point number.
    /// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
    function e() internal pure returns (int256 result) {
        result = 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 133.084258667509499441.
    ///
    /// Caveats:
    /// - All from "exp2".
    /// - For any x less than -41.446531673892822322, the result is zero.
    ///
    /// @param x The exponent as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function exp(int256 x) internal pure returns (int256 result) {
        // Without this check, the value passed to "exp2" would be less than -59.794705707972522261.
        if (x < -41_446531673892822322) {
            return 0;
        }

        // Without this check, the value passed to "exp2" would be greater than 192.
        if (x >= 133_084258667509499441) {
            revert PRBMathSD59x18__ExpInputTooBig(x);
        }

        // Do the fixed-point multiplication inline to save gas.
        unchecked {
            int256 doubleScaleProduct = x * LOG2_E;
            result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE);
        }
    }

    /// @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_SD59x18.
    ///
    /// Caveats:
    /// - For any x less than -59.794705707972522261, the result is zero.
    ///
    /// @param x The exponent as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function exp2(int256 x) internal pure returns (int256 result) {
        // This works because 2^(-x) = 1/2^x.
        if (x < 0) {
            // 2^59.794705707972522262 is the maximum number whose inverse does not truncate down to zero.
            if (x < -59_794705707972522261) {
                return 0;
            }

            // Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
            unchecked {
                result = 1e36 / exp2(-x);
            }
        } else {
            // 2^192 doesn't fit within the 192.64-bit format used internally in this function.
            if (x >= 192e18) {
                revert PRBMathSD59x18__Exp2InputTooBig(x);
            }

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

                // Safe to convert the result to int256 directly because the maximum input allowed is 192.
                result = int256(PRBMath.exp2(x192x64));
            }
        }
    }

    /// @notice Yields the greatest signed 59.18 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.
    ///
    /// Requirements:
    /// - x must be greater than or equal to MIN_WHOLE_SD59x18.
    ///
    /// @param x The signed 59.18-decimal fixed-point number to floor.
    /// @param result The greatest integer less than or equal to x, as a signed 58.18-decimal fixed-point number.
    function floor(int256 x) internal pure returns (int256 result) {
        if (x < MIN_WHOLE_SD59x18) {
            revert PRBMathSD59x18__FloorUnderflow(x);
        }
        unchecked {
            int256 remainder = x % SCALE;
            if (remainder == 0) {
                result = x;
            } else {
                // Solidity uses C fmod style, which returns a modulus with the same sign as x.
                result = x - remainder;
                if (x < 0) {
                    result -= SCALE;
                }
            }
        }
    }

    /// @notice Yields the excess beyond the floor of x for positive numbers and the part of the number to the right
    /// of the radix point for negative numbers.
    /// @dev Based on the odd function definition. https://en.wikipedia.org/wiki/Fractional_part
    /// @param x The signed 59.18-decimal fixed-point number to get the fractional part of.
    /// @param result The fractional part of x as a signed 59.18-decimal fixed-point number.
    function frac(int256 x) internal pure returns (int256 result) {
        unchecked {
            result = x % SCALE;
        }
    }

    /// @notice Converts a number from basic integer form to signed 59.18-decimal fixed-point representation.
    ///
    /// @dev Requirements:
    /// - x must be greater than or equal to MIN_SD59x18 divided by SCALE.
    /// - x must be less than or equal to MAX_SD59x18 divided by SCALE.
    ///
    /// @param x The basic integer to convert.
    /// @param result The same number in signed 59.18-decimal fixed-point representation.
    function fromInt(int256 x) internal pure returns (int256 result) {
        unchecked {
            if (x < MIN_SD59x18 / SCALE) {
                revert PRBMathSD59x18__FromIntUnderflow(x);
            }
            if (x > MAX_SD59x18 / SCALE) {
                revert PRBMathSD59x18__FromIntOverflow(x);
            }
            result = 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_SD59x18, lest it overflows.
    /// - x * y cannot be negative.
    ///
    /// @param x The first operand as a signed 59.18-decimal fixed-point number.
    /// @param y The second operand as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function gm(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == 0) {
            return 0;
        }

        unchecked {
            // Checking for overflow this way is faster than letting Solidity do it.
            int256 xy = x * y;
            if (xy / x != y) {
                revert PRBMathSD59x18__GmOverflow(x, y);
            }

            // The product cannot be negative.
            if (xy < 0) {
                revert PRBMathSD59x18__GmNegativeProduct(x, y);
            }

            // 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 = int256(PRBMath.sqrt(uint256(xy)));
        }
    }

    /// @notice Calculates 1 / x, rounding toward zero.
    ///
    /// @dev Requirements:
    /// - x cannot be zero.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the inverse.
    /// @return result The inverse as a signed 59.18-decimal fixed-point number.
    function inv(int256 x) internal pure returns (int256 result) {
        unchecked {
            // 1e36 is SCALE * SCALE.
            result = 1e36 / x;
        }
    }

    /// @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 2718281828459045235, for that we would need more fine-grained precision.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the natural logarithm.
    /// @return result The natural logarithm as a signed 59.18-decimal fixed-point number.
    function ln(int256 x) internal pure returns (int256 result) {
        // Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
        // can return is 195205294292027477728.
        unchecked {
            result = (log2(x) * SCALE) / LOG2_E;
        }
    }

    /// @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 signed 59.18-decimal fixed-point number for which to calculate the common logarithm.
    /// @return result The common logarithm as a signed 59.18-decimal fixed-point number.
    function log10(int256 x) internal pure returns (int256 result) {
        if (x <= 0) {
            revert PRBMathSD59x18__LogInputTooSmall(x);
        }

        // Note that the "mul" in this block is the assembly mul operation, not the "mul" function defined in this contract.
        // prettier-ignore
        assembly {
            switch x
            case 1 { result := mul(SCALE, sub(0, 18)) }
            case 10 { result := mul(SCALE, sub(1, 18)) }
            case 100 { result := mul(SCALE, sub(2, 18)) }
            case 1000 { result := mul(SCALE, sub(3, 18)) }
            case 10000 { result := mul(SCALE, sub(4, 18)) }
            case 100000 { result := mul(SCALE, sub(5, 18)) }
            case 1000000 { result := mul(SCALE, sub(6, 18)) }
            case 10000000 { result := mul(SCALE, sub(7, 18)) }
            case 100000000 { result := mul(SCALE, sub(8, 18)) }
            case 1000000000 { result := mul(SCALE, sub(9, 18)) }
            case 10000000000 { result := mul(SCALE, sub(10, 18)) }
            case 100000000000 { result := mul(SCALE, sub(11, 18)) }
            case 1000000000000 { result := mul(SCALE, sub(12, 18)) }
            case 10000000000000 { result := mul(SCALE, sub(13, 18)) }
            case 100000000000000 { result := mul(SCALE, sub(14, 18)) }
            case 1000000000000000 { result := mul(SCALE, sub(15, 18)) }
            case 10000000000000000 { result := mul(SCALE, sub(16, 18)) }
            case 100000000000000000 { result := mul(SCALE, sub(17, 18)) }
            case 1000000000000000000 { result := 0 }
            case 10000000000000000000 { result := SCALE }
            case 100000000000000000000 { result := mul(SCALE, 2) }
            case 1000000000000000000000 { result := mul(SCALE, 3) }
            case 10000000000000000000000 { result := mul(SCALE, 4) }
            case 100000000000000000000000 { result := mul(SCALE, 5) }
            case 1000000000000000000000000 { result := mul(SCALE, 6) }
            case 10000000000000000000000000 { result := mul(SCALE, 7) }
            case 100000000000000000000000000 { result := mul(SCALE, 8) }
            case 1000000000000000000000000000 { result := mul(SCALE, 9) }
            case 10000000000000000000000000000 { result := mul(SCALE, 10) }
            case 100000000000000000000000000000 { result := mul(SCALE, 11) }
            case 1000000000000000000000000000000 { result := mul(SCALE, 12) }
            case 10000000000000000000000000000000 { result := mul(SCALE, 13) }
            case 100000000000000000000000000000000 { result := mul(SCALE, 14) }
            case 1000000000000000000000000000000000 { result := mul(SCALE, 15) }
            case 10000000000000000000000000000000000 { result := mul(SCALE, 16) }
            case 100000000000000000000000000000000000 { result := mul(SCALE, 17) }
            case 1000000000000000000000000000000000000 { result := mul(SCALE, 18) }
            case 10000000000000000000000000000000000000 { result := mul(SCALE, 19) }
            case 100000000000000000000000000000000000000 { result := mul(SCALE, 20) }
            case 1000000000000000000000000000000000000000 { result := mul(SCALE, 21) }
            case 10000000000000000000000000000000000000000 { result := mul(SCALE, 22) }
            case 100000000000000000000000000000000000000000 { result := mul(SCALE, 23) }
            case 1000000000000000000000000000000000000000000 { result := mul(SCALE, 24) }
            case 10000000000000000000000000000000000000000000 { result := mul(SCALE, 25) }
            case 100000000000000000000000000000000000000000000 { result := mul(SCALE, 26) }
            case 1000000000000000000000000000000000000000000000 { result := mul(SCALE, 27) }
            case 10000000000000000000000000000000000000000000000 { result := mul(SCALE, 28) }
            case 100000000000000000000000000000000000000000000000 { result := mul(SCALE, 29) }
            case 1000000000000000000000000000000000000000000000000 { result := mul(SCALE, 30) }
            case 10000000000000000000000000000000000000000000000000 { result := mul(SCALE, 31) }
            case 100000000000000000000000000000000000000000000000000 { result := mul(SCALE, 32) }
            case 1000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 33) }
            case 10000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 34) }
            case 100000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 35) }
            case 1000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 36) }
            case 10000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 37) }
            case 100000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 38) }
            case 1000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 39) }
            case 10000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 40) }
            case 100000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 41) }
            case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 42) }
            case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 43) }
            case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 44) }
            case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 45) }
            case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 46) }
            case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 47) }
            case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 48) }
            case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 49) }
            case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 50) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 51) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 52) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 53) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 54) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 55) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 56) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 57) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 58) }
            default {
                result := MAX_SD59x18
            }
        }

        if (result == MAX_SD59x18) {
            // Do the fixed-point division inline to save gas. The denominator is log2(10).
            unchecked {
                result = (log2(x) * SCALE) / 3_321928094887362347;
            }
        }
    }

    /// @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 zero.
    ///
    /// Caveats:
    /// - The results are not perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the binary logarithm.
    /// @return result The binary logarithm as a signed 59.18-decimal fixed-point number.
    function log2(int256 x) internal pure returns (int256 result) {
        if (x <= 0) {
            revert PRBMathSD59x18__LogInputTooSmall(x);
        }
        unchecked {
            // This works because log2(x) = -log2(1/x).
            int256 sign;
            if (x >= SCALE) {
                sign = 1;
            } else {
                sign = -1;
                // Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
                assembly {
                    x := div(1000000000000000000000000000000000000, x)
                }
            }

            // Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
            uint256 n = PRBMath.mostSignificantBit(uint256(x / SCALE));

            // The integer part of the logarithm as a signed 59.18-decimal fixed-point number. The operation can't overflow
            // because n is maximum 255, SCALE is 1e18 and sign is either 1 or -1.
            result = int256(n) * SCALE;

            // This is y = x * 2^(-n).
            int256 y = x >> n;

            // If y = 1, the fractional part is zero.
            if (y == SCALE) {
                return result * sign;
            }

            // Calculate the fractional part via the iterative approximation.
            // The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster.
            for (int256 delta = int256(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.
                    result += delta;

                    // Corresponds to z/2 on Wikipedia.
                    y >>= 1;
                }
            }
            result *= sign;
        }
    }

    /// @notice Multiplies two signed 59.18-decimal fixed-point numbers together, returning a new signed 59.18-decimal
    /// fixed-point number.
    ///
    /// @dev Variant of "mulDiv" that works with signed numbers and employs constant folding, i.e. the denominator is
    /// always 1e18.
    ///
    /// Requirements:
    /// - All from "PRBMath.mulDivFixedPoint".
    /// - None of the inputs can be MIN_SD59x18
    /// - The result must fit within MAX_SD59x18.
    ///
    /// Caveats:
    /// - The body is purposely left uncommented; see the NatSpec comments in "PRBMath.mulDiv" to understand how this works.
    ///
    /// @param x The multiplicand as a signed 59.18-decimal fixed-point number.
    /// @param y The multiplier as a signed 59.18-decimal fixed-point number.
    /// @return result The product as a signed 59.18-decimal fixed-point number.
    function mul(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == MIN_SD59x18 || y == MIN_SD59x18) {
            revert PRBMathSD59x18__MulInputTooSmall();
        }

        unchecked {
            uint256 ax;
            uint256 ay;
            ax = x < 0 ? uint256(-x) : uint256(x);
            ay = y < 0 ? uint256(-y) : uint256(y);

            uint256 rAbs = PRBMath.mulDivFixedPoint(ax, ay);
            if (rAbs > uint256(MAX_SD59x18)) {
                revert PRBMathSD59x18__MulOverflow(rAbs);
            }

            uint256 sx;
            uint256 sy;
            assembly {
                sx := sgt(x, sub(0, 1))
                sy := sgt(y, sub(0, 1))
            }
            result = sx ^ sy == 1 ? -int256(rAbs) : int256(rAbs);
        }
    }

    /// @notice Returns PI as a signed 59.18-decimal fixed-point number.
    function pi() internal pure returns (int256 result) {
        result = 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".
    /// - z cannot be zero.
    ///
    /// Caveats:
    /// - All from "exp2", "log2" and "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x Number to raise to given power y, as a signed 59.18-decimal fixed-point number.
    /// @param y Exponent to raise x to, as a signed 59.18-decimal fixed-point number.
    /// @return result x raised to power y, as a signed 59.18-decimal fixed-point number.
    function pow(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == 0) {
            result = y == 0 ? SCALE : int256(0);
        } else {
            result = exp2(mul(log2(x), y));
        }
    }

    /// @notice Raises x (signed 59.18-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:
    /// - All from "abs" and "PRBMath.mulDivFixedPoint".
    /// - The result must fit within MAX_SD59x18.
    ///
    /// Caveats:
    /// - All from "PRBMath.mulDivFixedPoint".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x The base as a signed 59.18-decimal fixed-point number.
    /// @param y The exponent as an uint256.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function powu(int256 x, uint256 y) internal pure returns (int256 result) {
        uint256 xAbs = uint256(abs(x));

        // Calculate the first iteration of the loop in advance.
        uint256 rAbs = y & 1 > 0 ? xAbs : uint256(SCALE);

        // Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster.
        uint256 yAux = y;
        for (yAux >>= 1; yAux > 0; yAux >>= 1) {
            xAbs = PRBMath.mulDivFixedPoint(xAbs, xAbs);

            // Equivalent to "y % 2 == 1" but faster.
            if (yAux & 1 > 0) {
                rAbs = PRBMath.mulDivFixedPoint(rAbs, xAbs);
            }
        }

        // The result must fit within the 59.18-decimal fixed-point representation.
        if (rAbs > uint256(MAX_SD59x18)) {
            revert PRBMathSD59x18__PowuOverflow(rAbs);
        }

        // Is the base negative and the exponent an odd number?
        bool isNegative = x < 0 && y & 1 == 1;
        result = isNegative ? -int256(rAbs) : int256(rAbs);
    }

    /// @notice Returns 1 as a signed 59.18-decimal fixed-point number.
    function scale() internal pure returns (int256 result) {
        result = 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 cannot be negative.
    /// - x must be less than MAX_SD59x18 / SCALE.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the square root.
    /// @return result The result as a signed 59.18-decimal fixed-point .
    function sqrt(int256 x) internal pure returns (int256 result) {
        unchecked {
            if (x < 0) {
                revert PRBMathSD59x18__SqrtNegativeInput(x);
            }
            if (x > MAX_SD59x18 / SCALE) {
                revert PRBMathSD59x18__SqrtOverflow(x);
            }
            // Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two signed
            // 59.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
            result = int256(PRBMath.sqrt(uint256(x * SCALE)));
        }
    }

    /// @notice Converts a signed 59.18-decimal fixed-point number to basic integer form, rounding down in the process.
    /// @param x The signed 59.18-decimal fixed-point number to convert.
    /// @return result The same number in basic integer form.
    function toInt(int256 x) internal pure returns (int256 result) {
        unchecked {
            result = x / SCALE;
        }
    }
}