// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.0 (utils/Address.sol)

pragma solidity ^0.8.0;

/**
 * @dev Collection of functions related to the address type
 */
library Address {
    /**
     * @dev Returns true if `account` is a contract.
     *
     * [IMPORTANT]
     * ====
     * It is unsafe to assume that an address for which this function returns
     * false is an externally-owned account (EOA) and not a contract.
     *
     * Among others, `isContract` will return false for the following
     * types of addresses:
     *
     *  - an externally-owned account
     *  - a contract in construction
     *  - an address where a contract will be created
     *  - an address where a contract lived, but was destroyed
     * ====
     */
    function isContract(address account) internal view returns (bool) {
        // This method relies on extcodesize, which returns 0 for contracts in
        // construction, since the code is only stored at the end of the
        // constructor execution.

        uint256 size;
        assembly {
            size := extcodesize(account)
        }
        return size > 0;
    }

    /**
     * @dev Replacement for Solidity's `transfer`: sends `amount` wei to
     * `recipient`, forwarding all available gas and reverting on errors.
     *
     * https://eips.ethereum.org/EIPS/eip-1884[EIP1884] increases the gas cost
     * of certain opcodes, possibly making contracts go over the 2300 gas limit
     * imposed by `transfer`, making them unable to receive funds via
     * `transfer`. {sendValue} removes this limitation.
     *
     * https://diligence.consensys.net/posts/2019/09/stop-using-soliditys-transfer-now/[Learn more].
     *
     * IMPORTANT: because control is transferred to `recipient`, care must be
     * taken to not create reentrancy vulnerabilities. Consider using
     * {ReentrancyGuard} or the
     * https://solidity.readthedocs.io/en/v0.5.11/security-considerations.html#use-the-checks-effects-interactions-pattern[checks-effects-interactions pattern].
     */
    function sendValue(address payable recipient, uint256 amount) internal {
        require(address(this).balance >= amount, "Address: insufficient balance");

        (bool success, ) = recipient.call{value: amount}("");
        require(success, "Address: unable to send value, recipient may have reverted");
    }

    /**
     * @dev Performs a Solidity function call using a low level `call`. A
     * plain `call` is an unsafe replacement for a function call: use this
     * function instead.
     *
     * If `target` reverts with a revert reason, it is bubbled up by this
     * function (like regular Solidity function calls).
     *
     * Returns the raw returned data. To convert to the expected return value,
     * use https://solidity.readthedocs.io/en/latest/units-and-global-variables.html?highlight=abi.decode#abi-encoding-and-decoding-functions[`abi.decode`].
     *
     * Requirements:
     *
     * - `target` must be a contract.
     * - calling `target` with `data` must not revert.
     *
     * _Available since v3.1._
     */
    function functionCall(address target, bytes memory data) internal returns (bytes memory) {
        return functionCall(target, data, "Address: low-level call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`], but with
     * `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal returns (bytes memory) {
        return functionCallWithValue(target, data, 0, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but also transferring `value` wei to `target`.
     *
     * Requirements:
     *
     * - the calling contract must have an ETH balance of at least `value`.
     * - the called Solidity function must be `payable`.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(
        address target,
        bytes memory data,
        uint256 value
    ) internal returns (bytes memory) {
        return functionCallWithValue(target, data, value, "Address: low-level call with value failed");
    }

    /**
     * @dev Same as {xref-Address-functionCallWithValue-address-bytes-uint256-}[`functionCallWithValue`], but
     * with `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(
        address target,
        bytes memory data,
        uint256 value,
        string memory errorMessage
    ) internal returns (bytes memory) {
        require(address(this).balance >= value, "Address: insufficient balance for call");
        require(isContract(target), "Address: call to non-contract");

        (bool success, bytes memory returndata) = target.call{value: value}(data);
        return verifyCallResult(success, returndata, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(address target, bytes memory data) internal view returns (bytes memory) {
        return functionStaticCall(target, data, "Address: low-level static call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal view returns (bytes memory) {
        require(isContract(target), "Address: static call to non-contract");

        (bool success, bytes memory returndata) = target.staticcall(data);
        return verifyCallResult(success, returndata, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(address target, bytes memory data) internal returns (bytes memory) {
        return functionDelegateCall(target, data, "Address: low-level delegate call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal returns (bytes memory) {
        require(isContract(target), "Address: delegate call to non-contract");

        (bool success, bytes memory returndata) = target.delegatecall(data);
        return verifyCallResult(success, returndata, errorMessage);
    }

    /**
     * @dev Tool to verifies that a low level call was successful, and revert if it wasn't, either by bubbling the
     * revert reason using the provided one.
     *
     * _Available since v4.3._
     */
    function verifyCallResult(
        bool success,
        bytes memory returndata,
        string memory errorMessage
    ) internal pure returns (bytes memory) {
        if (success) {
            return returndata;
        } else {
            // Look for revert reason and bubble it up if present
            if (returndata.length > 0) {
                // The easiest way to bubble the revert reason is using memory via assembly

                assembly {
                    let returndata_size := mload(returndata)
                    revert(add(32, returndata), returndata_size)
                }
            } else {
                revert(errorMessage);
            }
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.0 (utils/Context.sol)

pragma solidity ^0.8.0;

/**
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
abstract contract Context {
    function _msgSender() internal view virtual returns (address) {
        return msg.sender;
    }

    function _msgData() internal view virtual returns (bytes calldata) {
        return msg.data;
    }
}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;

interface CustomErrors {
    /// @dev No need to unwind a net zero position
    error PositionNetZero();

    error DebugError(uint256 x, uint256 y);

    /// @dev Cannot have less margin than the minimum requirement
    error MarginLessThanMinimum(int256 marginRequirement);

    /// @dev We can't withdraw more margin than we have
    error WithdrawalExceedsCurrentMargin();

    /// @dev Position must be settled after AMM has reached maturity
    error PositionNotSettled();

    /// The resulting margin does not meet minimum requirements
    error MarginRequirementNotMet(
        int256 marginRequirement,
        int24 tick,
        int256 fixedTokenDelta,
        int256 variableTokenDelta,
        uint256 cumulativeFeeIncurred,
        int256 fixedTokenDeltaUnbalanced
    );

    /// The position/trader needs to be below the liquidation threshold to be liquidated
    error CannotLiquidate();

    /// Only the position/trade owner can update the LP/Trader margin
    error OnlyOwnerCanUpdatePosition();

    error OnlyVAMM();

    error OnlyFCM();

    /// Margin delta must not equal zero
    error InvalidMarginDelta();

    /// Positions and Traders cannot be settled before the applicable interest rate swap has matured
    error CannotSettleBeforeMaturity();

    error closeToOrBeyondMaturity();

    /// @dev There are not enough funds available for the requested operation
    error NotEnoughFunds(uint256 requested, uint256 available);

    /// @dev The two values were expected to have oppostite sigs, but do not
    error ExpectedOppositeSigns(int256 amount0, int256 amount1);

    /// @dev Error which is reverted if the sqrt price of the vamm is non-zero before a vamm is initialized
    error ExpectedSqrtPriceZeroBeforeInit(uint160 sqrtPriceX96);

    /// @dev Error which ensures the liquidity delta is positive if a given LP wishes to mint further liquidity in the vamm
    error LiquidityDeltaMustBePositiveInMint(uint128 amount);

    /// @dev Error which ensures the liquidity delta is positive if a given LP wishes to burn liquidity in the vamm
    error LiquidityDeltaMustBePositiveInBurn(uint128 amount);

    /// @dev Error which ensures the amount of notional specified when initiating an IRS contract (via the swap function in the vamm) is non-zero
    error IRSNotionalAmountSpecifiedMustBeNonZero();

    /// @dev Error which ensures the VAMM is unlocked
    error CanOnlyTradeIfUnlocked(bool unlocked);

    /// @dev only the margin engine can run a certain function
    error OnlyMarginEngine();

    /// The resulting margin does not meet minimum requirements
    error MarginRequirementNotMetFCM(int256 marginRequirement);

    /// @dev getReserveNormalizedIncome() returned zero for underlying asset. Oracle only supports active Aave-V2 assets.
    error AavePoolGetReserveNormalizedIncomeReturnedZero();

    /// @dev ctoken.exchangeRateStored() returned zero for a given Compound ctoken. Oracle only supports active Compound assets.
    error CTokenExchangeRateReturnedZero();

    /// @dev currentTime < queriedTime
    error OOO();
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.0 (proxy/ERC1967/ERC1967Proxy.sol)

pragma solidity ^0.8.0;

import "../Proxy.sol";
import "./ERC1967Upgrade.sol";

/**
 * @dev This contract implements an upgradeable proxy. It is upgradeable because calls are delegated to an
 * implementation address that can be changed. This address is stored in storage in the location specified by
 * https://eips.ethereum.org/EIPS/eip-1967[EIP1967], so that it doesn't conflict with the storage layout of the
 * implementation behind the proxy.
 */
contract ERC1967Proxy is Proxy, ERC1967Upgrade {
    /**
     * @dev Initializes the upgradeable proxy with an initial implementation specified by `_logic`.
     *
     * If `_data` is nonempty, it's used as data in a delegate call to `_logic`. This will typically be an encoded
     * function call, and allows initializating the storage of the proxy like a Solidity constructor.
     */
    constructor(address _logic, bytes memory _data) payable {
        assert(_IMPLEMENTATION_SLOT == bytes32(uint256(keccak256("eip1967.proxy.implementation")) - 1));
        _upgradeToAndCall(_logic, _data, false);
    }

    /**
     * @dev Returns the current implementation address.
     */
    function _implementation() internal view virtual override returns (address impl) {
        return ERC1967Upgrade._getImplementation();
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.0 (proxy/ERC1967/ERC1967Upgrade.sol)

pragma solidity ^0.8.2;

import "../beacon/IBeacon.sol";
import "../../utils/Address.sol";
import "../../utils/StorageSlot.sol";

/**
 * @dev This abstract contract provides getters and event emitting update functions for
 * https://eips.ethereum.org/EIPS/eip-1967[EIP1967] slots.
 *
 * _Available since v4.1._
 *
 * @custom:oz-upgrades-unsafe-allow delegatecall
 */
abstract contract ERC1967Upgrade {
    // This is the keccak-256 hash of "eip1967.proxy.rollback" subtracted by 1
    bytes32 private constant _ROLLBACK_SLOT = 0x4910fdfa16fed3260ed0e7147f7cc6da11a60208b5b9406d12a635614ffd9143;

    /**
     * @dev Storage slot with the address of the current implementation.
     * This is the keccak-256 hash of "eip1967.proxy.implementation" subtracted by 1, and is
     * validated in the constructor.
     */
    bytes32 internal constant _IMPLEMENTATION_SLOT = 0x360894a13ba1a3210667c828492db98dca3e2076cc3735a920a3ca505d382bbc;

    /**
     * @dev Emitted when the implementation is upgraded.
     */
    event Upgraded(address indexed implementation);

    /**
     * @dev Returns the current implementation address.
     */
    function _getImplementation() internal view returns (address) {
        return StorageSlot.getAddressSlot(_IMPLEMENTATION_SLOT).value;
    }

    /**
     * @dev Stores a new address in the EIP1967 implementation slot.
     */
    function _setImplementation(address newImplementation) private {
        require(Address.isContract(newImplementation), "ERC1967: new implementation is not a contract");
        StorageSlot.getAddressSlot(_IMPLEMENTATION_SLOT).value = newImplementation;
    }

    /**
     * @dev Perform implementation upgrade
     *
     * Emits an {Upgraded} event.
     */
    function _upgradeTo(address newImplementation) internal {
        _setImplementation(newImplementation);
        emit Upgraded(newImplementation);
    }

    /**
     * @dev Perform implementation upgrade with additional setup call.
     *
     * Emits an {Upgraded} event.
     */
    function _upgradeToAndCall(
        address newImplementation,
        bytes memory data,
        bool forceCall
    ) internal {
        _upgradeTo(newImplementation);
        if (data.length > 0 || forceCall) {
            Address.functionDelegateCall(newImplementation, data);
        }
    }

    /**
     * @dev Perform implementation upgrade with security checks for UUPS proxies, and additional setup call.
     *
     * Emits an {Upgraded} event.
     */
    function _upgradeToAndCallSecure(
        address newImplementation,
        bytes memory data,
        bool forceCall
    ) internal {
        address oldImplementation = _getImplementation();

        // Initial upgrade and setup call
        _setImplementation(newImplementation);
        if (data.length > 0 || forceCall) {
            Address.functionDelegateCall(newImplementation, data);
        }

        // Perform rollback test if not already in progress
        StorageSlot.BooleanSlot storage rollbackTesting = StorageSlot.getBooleanSlot(_ROLLBACK_SLOT);
        if (!rollbackTesting.value) {
            // Trigger rollback using upgradeTo from the new implementation
            rollbackTesting.value = true;
            Address.functionDelegateCall(
                newImplementation,
                abi.encodeWithSignature("upgradeTo(address)", oldImplementation)
            );
            rollbackTesting.value = false;
            // Check rollback was effective
            require(oldImplementation == _getImplementation(), "ERC1967Upgrade: upgrade breaks further upgrades");
            // Finally reset to the new implementation and log the upgrade
            _upgradeTo(newImplementation);
        }
    }

    /**
     * @dev Storage slot with the admin of the contract.
     * This is the keccak-256 hash of "eip1967.proxy.admin" subtracted by 1, and is
     * validated in the constructor.
     */
    bytes32 internal constant _ADMIN_SLOT = 0xb53127684a568b3173ae13b9f8a6016e243e63b6e8ee1178d6a717850b5d6103;

    /**
     * @dev Emitted when the admin account has changed.
     */
    event AdminChanged(address previousAdmin, address newAdmin);

    /**
     * @dev Returns the current admin.
     */
    function _getAdmin() internal view returns (address) {
        return StorageSlot.getAddressSlot(_ADMIN_SLOT).value;
    }

    /**
     * @dev Stores a new address in the EIP1967 admin slot.
     */
    function _setAdmin(address newAdmin) private {
        require(newAdmin != address(0), "ERC1967: new admin is the zero address");
        StorageSlot.getAddressSlot(_ADMIN_SLOT).value = newAdmin;
    }

    /**
     * @dev Changes the admin of the proxy.
     *
     * Emits an {AdminChanged} event.
     */
    function _changeAdmin(address newAdmin) internal {
        emit AdminChanged(_getAdmin(), newAdmin);
        _setAdmin(newAdmin);
    }

    /**
     * @dev The storage slot of the UpgradeableBeacon contract which defines the implementation for this proxy.
     * This is bytes32(uint256(keccak256('eip1967.proxy.beacon')) - 1)) and is validated in the constructor.
     */
    bytes32 internal constant _BEACON_SLOT = 0xa3f0ad74e5423aebfd80d3ef4346578335a9a72aeaee59ff6cb3582b35133d50;

    /**
     * @dev Emitted when the beacon is upgraded.
     */
    event BeaconUpgraded(address indexed beacon);

    /**
     * @dev Returns the current beacon.
     */
    function _getBeacon() internal view returns (address) {
        return StorageSlot.getAddressSlot(_BEACON_SLOT).value;
    }

    /**
     * @dev Stores a new beacon in the EIP1967 beacon slot.
     */
    function _setBeacon(address newBeacon) private {
        require(Address.isContract(newBeacon), "ERC1967: new beacon is not a contract");
        require(
            Address.isContract(IBeacon(newBeacon).implementation()),
            "ERC1967: beacon implementation is not a contract"
        );
        StorageSlot.getAddressSlot(_BEACON_SLOT).value = newBeacon;
    }

    /**
     * @dev Perform beacon upgrade with additional setup call. Note: This upgrades the address of the beacon, it does
     * not upgrade the implementation contained in the beacon (see {UpgradeableBeacon-_setImplementation} for that).
     *
     * Emits a {BeaconUpgraded} event.
     */
    function _upgradeBeaconToAndCall(
        address newBeacon,
        bytes memory data,
        bool forceCall
    ) internal {
        _setBeacon(newBeacon);
        emit BeaconUpgraded(newBeacon);
        if (data.length > 0 || forceCall) {
            Address.functionDelegateCall(IBeacon(newBeacon).implementation(), data);
        }
    }
}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;
import "./interfaces/IFactory.sol";
import "./interfaces/IPeriphery.sol";
import "./interfaces/rate_oracles/IRateOracle.sol";
import "./interfaces/IMarginEngine.sol";
import "./interfaces/IVAMM.sol";
import "./interfaces/fcms/IFCM.sol";
import "@openzeppelin/contracts/access/Ownable.sol";
import "@openzeppelin/contracts/proxy/ERC1967/ERC1967Proxy.sol";
import "contracts/utils/CustomErrors.sol";

contract VoltzERC1967Proxy is ERC1967Proxy, CustomErrors {
  constructor(address _logic, bytes memory _data) payable ERC1967Proxy(_logic, _data) {}
}


/// @title Voltz Factory Contract
/// @notice Deploys Voltz VAMMs and MarginEngines and manages ownership and control over amm protocol fees
// Following this example https://github.com/OriginProtocol/minimal-proxy-example/blob/master/contracts/PairFactory.sol
contract Factory is IFactory, Ownable {

  /// @dev master MarginEngine implementation that MarginEngine proxies can delegate call to
  IMarginEngine public override masterMarginEngine;

  /// @dev master VAMM implementation that VAMM proxies can delegate call to
  IVAMM public override masterVAMM;

  /// @dev yieldBearingProtocolID --> master FCM implementation for the underlying yield bearing protocol with the corresponding id
  mapping(uint8 => IFCM) public override masterFCMs;

  /// @dev owner --> integration contract address --> isApproved
  /// @dev if an owner wishes to allow a given intergration contract to act on thir behalf with Voltz Core
  /// @dev they need to set the approval via the setApproval function
  mapping(address => mapping(address => bool)) private _isApproved;
  
  /// @dev Voltz Periphery
  IPeriphery public override periphery;  

  function setApproval(address intAddress, bool allowIntegration) external override {
    _isApproved[msg.sender][intAddress] = allowIntegration;
    emit Approval(msg.sender, intAddress, allowIntegration);
  }

  function isApproved(address _owner, address _intAddress) override view public returns (bool) {

    require(_owner != address(0), "owner does not exist");
    require(_intAddress != address(0), "int does not exist");

    /// @dev Voltz periphery is always approved to act on behalf of the owner
    if (_intAddress == address(periphery)) {
      return true;
    } else {
      return _isApproved[_owner][_intAddress];
    }

  }

  constructor(IMarginEngine _masterMarginEngine, IVAMM _masterVAMM) {
    require(address(_masterMarginEngine) != address(0), "master me must exist");
    require(address(_masterVAMM) != address(0), "master vamm must exist");

    masterMarginEngine = _masterMarginEngine;
    masterVAMM = _masterVAMM;
  }

  function setMasterFCM(IFCM _masterFCM, uint8 _yieldBearingProtocolID) external override onlyOwner {

    require(address(_masterFCM) != address(0), "master fcm must exist");
    masterFCMs[_yieldBearingProtocolID] = _masterFCM;
    emit MasterFCM(_masterFCM, _yieldBearingProtocolID);
  }

  function setMasterMarginEngine(IMarginEngine _masterMarginEngine) external override onlyOwner {
    require(address(_masterMarginEngine) != address(0), "master me must exist");

    if (address(masterMarginEngine) != address(_masterMarginEngine)) {
      masterMarginEngine = _masterMarginEngine;
    }

  }


  function setMasterVAMM(IVAMM _masterVAMM) external override onlyOwner {

    require(address(_masterVAMM) != address(0), "master vamm must exist");

    if (address(masterVAMM) != address(_masterVAMM)) {
      masterVAMM = _masterVAMM;
    }

  }


  function setPeriphery(IPeriphery _periphery) external override onlyOwner {
    
    require(address(_periphery) != address(0), "periphery must exist");

    if (address(periphery) != address(_periphery)) {
      periphery = _periphery;
      emit PeripheryUpdate(periphery);
    }

  }


  function deployIrsInstance(IERC20Minimal _underlyingToken, IRateOracle _rateOracle, uint256 _termStartTimestampWad, uint256 _termEndTimestampWad, int24 _tickSpacing) external override onlyOwner returns (IMarginEngine marginEngineProxy, IVAMM vammProxy, IFCM fcmProxy) {
    IMarginEngine marginEngine = IMarginEngine(address(new VoltzERC1967Proxy(address(masterMarginEngine), "")));
    IVAMM vamm = IVAMM(address(new VoltzERC1967Proxy(address(masterVAMM), "")));
    marginEngine.initialize(_underlyingToken, _rateOracle, _termStartTimestampWad, _termEndTimestampWad);
    vamm.initialize(marginEngine, _tickSpacing);
    marginEngine.setVAMM(vamm);

    IRateOracle r = IRateOracle(_rateOracle);
    require(r.underlying() == _underlyingToken, "Tokens do not match");
    uint8 yieldBearingProtocolID = r.UNDERLYING_YIELD_BEARING_PROTOCOL_ID();
    IFCM _masterFCM = masterFCMs[yieldBearingProtocolID];
    IFCM fcm;

    if (address(_masterFCM) != address(0)) {
      fcm = IFCM(address(new VoltzERC1967Proxy(address(_masterFCM), "")));
      fcm.initialize(vamm, marginEngine);
      marginEngine.setFCM(fcm);
      Ownable(address(fcm)).transferOwnership(msg.sender);
    }

    uint8 underlyingTokenDecimals = _underlyingToken.decimals();

    emit IrsInstance(_underlyingToken, _rateOracle, _termStartTimestampWad, _termEndTimestampWad, _tickSpacing, marginEngine, vamm, fcm, yieldBearingProtocolID, underlyingTokenDecimals);

    // Transfer ownership of all instances to the factory owner
    Ownable(address(vamm)).transferOwnership(msg.sender);
    Ownable(address(marginEngine)).transferOwnership(msg.sender);

    return(marginEngine, vamm, fcm);
  }



}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;
import "prb-math/contracts/PRBMathSD59x18.sol";
import "prb-math/contracts/PRBMathUD60x18.sol";
import "./Time.sol";

/// @title A utility library for mathematics of fixed and variable token amounts.
library FixedAndVariableMath {
    using PRBMathSD59x18 for int256;
    using PRBMathUD60x18 for uint256;

    /// @notice Number of wei-seconds in a year
    /// @dev Ignoring leap years since we're only using it to calculate the eventual APY rate

    uint256 public constant SECONDS_IN_YEAR_IN_WAD = 31536000e18;
    uint256 public constant ONE_HUNDRED_IN_WAD = 100e18;

    /// @notice Caclulate the remaining cashflow to settle a position
    /// @param fixedTokenBalance The current balance of the fixed side of the position
    /// @param variableTokenBalance The current balance of the variable side of the position
    /// @param termStartTimestampWad When did the position begin, in seconds
    /// @param termEndTimestampWad When does the position reach maturity, in seconds
    /// @param variableFactorToMaturityWad What factor expresses the current remaining variable rate, up to position maturity? (in WAD)
    /// @return cashflow The remaining cashflow of the position
    function calculateSettlementCashflow(
        int256 fixedTokenBalance,
        int256 variableTokenBalance,
        uint256 termStartTimestampWad,
        uint256 termEndTimestampWad,
        uint256 variableFactorToMaturityWad
    ) internal view returns (int256 cashflow) {
        /// @dev convert fixed and variable token balances to their respective fixed token representations

        int256 fixedTokenBalanceWad = fixedTokenBalance.fromInt();
        int256 variableTokenBalanceWad = variableTokenBalance.fromInt();
        int256 fixedCashflowWad = fixedTokenBalanceWad.mul(
            int256(
                fixedFactor(true, termStartTimestampWad, termEndTimestampWad)
            )
        );

        int256 variableCashflowWad = variableTokenBalanceWad.mul(
            int256(variableFactorToMaturityWad)
        );

        int256 cashflowWad = fixedCashflowWad + variableCashflowWad;

        /// @dev convert back to non-fixed point representation
        cashflow = cashflowWad.toInt();
    }

    /// @notice Divide a given time in seconds by the number of seconds in a year
    /// @param timeInSecondsAsWad A time in seconds in Wad (i.e. scaled up by 10^18)
    /// @return timeInYearsWad An annualised factor of timeInSeconds, also in Wad
    function accrualFact(uint256 timeInSecondsAsWad)
        internal
        pure
        returns (uint256 timeInYearsWad)
    {
        timeInYearsWad = timeInSecondsAsWad.div(SECONDS_IN_YEAR_IN_WAD);
    }

    /// @notice Calculate the fixed factor for a position - that is, the percentage earned over
    /// the specified period of time, assuming 1% per year
    /// @param atMaturity Whether to calculate the factor at maturity (true), or now (false)
    /// @param termStartTimestampWad When does the period of time begin, in wei-seconds
    /// @param termEndTimestampWad When does the period of time end, in wei-seconds
    /// @return fixedFactorValueWad The fixed factor for the position (in Wad)
    function fixedFactor(
        bool atMaturity,
        uint256 termStartTimestampWad,
        uint256 termEndTimestampWad
    ) internal view returns (uint256 fixedFactorValueWad) {
        require(termEndTimestampWad > termStartTimestampWad, "E<=S");

        uint256 currentTimestampWad = Time.blockTimestampScaled();

        require(currentTimestampWad >= termStartTimestampWad, "B.T<S");

        uint256 timeInSecondsWad;

        if (atMaturity || (currentTimestampWad >= termEndTimestampWad)) {
            timeInSecondsWad = termEndTimestampWad - termStartTimestampWad;
        } else {
            timeInSecondsWad = currentTimestampWad - termStartTimestampWad;
        }

        uint256 timeInYearsWad = accrualFact(timeInSecondsWad);
        fixedFactorValueWad = timeInYearsWad.div(ONE_HUNDRED_IN_WAD);
    }

    /// @notice Calculate the fixed token balance for a position over a timespan
    /// @param amountFixedWad  A fixed amount
    /// @param excessBalanceWad Cashflows accrued to the fixed and variable token amounts since the inception of the IRS AMM
    /// @param termStartTimestampWad When does the period of time begin, in wei-seconds
    /// @param termEndTimestampWad When does the period of time end, in wei-seconds
    /// @return fixedTokenBalanceWad The fixed token balance for that time period
    function calculateFixedTokenBalance(
        int256 amountFixedWad,
        int256 excessBalanceWad,
        uint256 termStartTimestampWad,
        uint256 termEndTimestampWad
    ) internal view returns (int256 fixedTokenBalanceWad) {
        require(termEndTimestampWad > termStartTimestampWad, "E<=S");

        return
            amountFixedWad -
            excessBalanceWad.div(
                int256(
                    fixedFactor(
                        true,
                        termStartTimestampWad,
                        termEndTimestampWad
                    )
                )
            );
    }

    /// @notice Calculate the excess balance of both sides of a position in Wad
    /// @param amountFixedWad A fixed balance
    /// @param amountVariableWad A variable balance
    /// @param accruedVariableFactorWad An annualised factor in wei-years
    /// @param termStartTimestampWad When does the period of time begin, in wei-seconds
    /// @param termEndTimestampWad When does the period of time end, in wei-seconds
    /// @return excessBalanceWad The excess balance in wad
    function getExcessBalance(
        int256 amountFixedWad,
        int256 amountVariableWad,
        uint256 accruedVariableFactorWad,
        uint256 termStartTimestampWad,
        uint256 termEndTimestampWad
    ) internal view returns (int256) {
        int256 excessFixedAccruedBalanceWad;
        int256 excessVariableAccruedBalanceWad;
        int256 excessBalanceWad;

        excessFixedAccruedBalanceWad = amountFixedWad.mul(
            int256(
                fixedFactor(false, termStartTimestampWad, termEndTimestampWad)
            )
        );

        excessVariableAccruedBalanceWad = amountVariableWad.mul(
            int256(accruedVariableFactorWad)
        );

        /// @dev cashflows accrued since the inception of the IRS AMM

        excessBalanceWad =
            excessFixedAccruedBalanceWad +
            excessVariableAccruedBalanceWad;

        return excessBalanceWad;
    }

    /// @notice Calculate the fixed token balance given both fixed and variable balances
    /// @param amountFixed A fixed balance
    /// @param amountVariable A variable balance
    /// @param accruedVariableFactorWad An annualised factor in wei-years
    /// @param termStartTimestampWad When does the period of time begin, in wei-seconds
    /// @param termEndTimestampWad When does the period of time end, in wei-seconds
    /// @return fixedTokenBalance The fixed token balance for that time period
    function getFixedTokenBalance(
        int256 amountFixed,
        int256 amountVariable,
        uint256 accruedVariableFactorWad,
        uint256 termStartTimestampWad,
        uint256 termEndTimestampWad
    ) internal view returns (int256 fixedTokenBalance) {
        require(termEndTimestampWad > termStartTimestampWad, "E<=S");

        if (amountFixed == 0 && amountVariable == 0) return 0;

        int256 amountFixedWad = amountFixed.fromInt();
        int256 amountVariableWad = amountVariable.fromInt();

        int256 excessBalanceWad = getExcessBalance(
            amountFixedWad,
            amountVariableWad,
            accruedVariableFactorWad,
            termStartTimestampWad,
            termEndTimestampWad
        );

        int256 fixedTokenBalanceWad = calculateFixedTokenBalance(
            amountFixedWad,
            excessBalanceWad,
            termStartTimestampWad,
            termEndTimestampWad
        );

        fixedTokenBalance = fixedTokenBalanceWad.toInt();
    }
}

// SPDX-License-Identifier: GPL-2.0-or-later

pragma solidity =0.8.9;

/// @title FixedPoint128
/// @notice A library for handling binary fixed point numbers, see https://en.wikipedia.org/wiki/Q_(number_format)
library FixedPoint128 {
    uint256 internal constant Q128 = 0x100000000000000000000000000000000;
}

// SPDX-License-Identifier: MIT

// solhint-disable no-inline-assembly

pragma solidity =0.8.9;

/// @title Contains 512-bit math functions
/// @notice Facilitates multiplication and division that can have overflow of an intermediate value without any loss of precision
/// @dev Handles "phantom overflow" i.e., allows multiplication and division where an intermediate value overflows 256 bits
library FullMath {
    /// @notice Calculates floor(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
    /// @param a The multiplicand
    /// @param b The multiplier
    /// @param denominator The divisor
    /// @return result The 256-bit result
    /// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv

    function mulDivSigned(
        int256 a,
        uint256 b,
        uint256 denominator
    ) internal pure returns (int256 result) {
        if (a < 0) return -int256(mulDiv(uint256(-a), b, denominator));
        return int256(mulDiv(uint256(a), b, denominator));
    }

    function mulDiv(
        uint256 a,
        uint256 b,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        // 512-bit multiply [prod1 prod0] = a * b
        // Compute the product mod 2**256 and mod 2**256 - 1
        // then 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

        unchecked {
            assembly {
                let mm := mulmod(a, b, not(0))
                prod0 := mul(a, b)
                prod1 := sub(sub(mm, prod0), lt(mm, prod0))
            }

            // Handle non-overflow cases, 256 by 256 division
            if (prod1 == 0) {
                require(denominator > 0, "Division by zero");
                assembly {
                    result := div(prod0, denominator)
                }
                return result;
            }

            // Make sure the result is less than 2**256.
            // Also prevents denominator == 0
            require(denominator > prod1, "overflow");

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

            // Make division exact by subtracting the remainder from [prod1 prod0]
            // Compute remainder using mulmod
            uint256 remainder;
            assembly {
                remainder := mulmod(a, b, denominator)
            }
            // Subtract 256 bit number from 512 bit number
            assembly {
                prod1 := sub(prod1, gt(remainder, prod0))
                prod0 := sub(prod0, remainder)
            }

            // Factor powers of two out of denominator
            // Compute largest power of two divisor of denominator.
            // Always >= 1.
            // uint256 twos = -denominator & denominator;
            // https://ethereum.stackexchange.com/questions/96642/unary-operator-cannot-be-applied-to-type-uint256
            uint256 twos = (type(uint256).max - denominator + 1) & denominator;
            // Divide denominator by power of two
            assembly {
                denominator := div(denominator, twos)
            }

            // Divide [prod1 prod0] by the factors of two
            assembly {
                prod0 := div(prod0, twos)
            }
            // Shift in bits from prod1 into prod0. For this we need
            // to flip `twos` such that it is 2**256 / twos.
            // If twos is zero, then it becomes one
            assembly {
                twos := add(div(sub(0, twos), twos), 1)
            }
            prod0 |= prod1 * twos;

            // 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
            // correct for four bits. That is, denominator * inv = 1 mod 2**4
            uint256 inv = (3 * denominator) ^ 2;
            // Now use 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.
            inv *= 2 - denominator * inv; // inverse mod 2**8
            inv *= 2 - denominator * inv; // inverse mod 2**16
            inv *= 2 - denominator * inv; // inverse mod 2**32
            inv *= 2 - denominator * inv; // inverse mod 2**64
            inv *= 2 - denominator * inv; // inverse mod 2**128
            inv *= 2 - denominator * inv; // 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 precoditions 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 * inv;
            return result;
        }
    }

    /// @notice Calculates ceil(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
    /// @param a The multiplicand
    /// @param b The multiplier
    /// @param denominator The divisor
    /// @return result The 256-bit result
    function mulDivRoundingUp(
        uint256 a,
        uint256 b,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        result = mulDiv(a, b, denominator);
        if (mulmod(a, b, denominator) > 0) {
            require(result < type(uint256).max, "overflow");
            result++;
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.0 (proxy/beacon/IBeacon.sol)

pragma solidity ^0.8.0;

/**
 * @dev This is the interface that {BeaconProxy} expects of its beacon.
 */
interface IBeacon {
    /**
     * @dev Must return an address that can be used as a delegate call target.
     *
     * {BeaconProxy} will check that this address is a contract.
     */
    function implementation() external view returns (address);
}

// SPDX-License-Identifier: GPL-2.0-or-later

pragma solidity =0.8.9;

/// @title Minimal ERC20 interface for Voltz
/// @notice Contains a subset of the full ERC20 interface that is used in Voltz
interface IERC20Minimal {
    /// @notice Returns the balance of a token
    /// @param account The account for which to look up the number of tokens it has, i.e. its balance
    /// @return The number of tokens held by the account
    function balanceOf(address account) external view returns (uint256);

    /// @notice Transfers the amount of token from the `msg.sender` to the recipient
    /// @param recipient The account that will receive the amount transferred
    /// @param amount The number of tokens to send from the sender to the recipient
    /// @return Returns true for a successful transfer, false for an unsuccessful transfer
    function transfer(address recipient, uint256 amount)
        external
        returns (bool);

    /// @notice Returns the current allowance given to a spender by an owner
    /// @param owner The account of the token owner
    /// @param spender The account of the token spender
    /// @return The current allowance granted by `owner` to `spender`
    function allowance(address owner, address spender)
        external
        view
        returns (uint256);

    /// @notice Sets the allowance of a spender from the `msg.sender` to the value `amount`
    /// @param spender The account which will be allowed to spend a given amount of the owners tokens
    /// @param amount The amount of tokens allowed to be used by `spender`
    /// @return Returns true for a successful approval, false for unsuccessful
    function approve(address spender, uint256 amount) external returns (bool);

    /// @notice Transfers `amount` tokens from `sender` to `recipient` up to the allowance given to the `msg.sender`
    /// @param sender The account from which the transfer will be initiated
    /// @param recipient The recipient of the transfer
    /// @param amount The amount of the transfer
    /// @return Returns true for a successful transfer, false for unsuccessful
    function transferFrom(
        address sender,
        address recipient,
        uint256 amount
    ) external returns (bool);

    /// @dev Returns the number of decimals used to get its user representation.
    // For example, if decimals equals 2, a balance of 505 tokens should be displayed to a user as 5,05 (505 / 10 ** 2).
    // Tokens usually opt for a value of 18, imitating the relationship between Ether and Wei.
    function decimals() external view returns (uint8);

    /// @notice Event emitted when tokens are transferred from one address to another, either via `#transfer` or `#transferFrom`.
    /// @param from The account from which the tokens were sent, i.e. the balance decreased
    /// @param to The account to which the tokens were sent, i.e. the balance increased
    /// @param value The amount of tokens that were transferred
    event Transfer(address indexed from, address indexed to, uint256 value);

    /// @notice Event emitted when the approval amount for the spender of a given owner's tokens changes.
    /// @param owner The account that approved spending of its tokens
    /// @param spender The account for which the spending allowance was modified
    /// @param value The new allowance from the owner to the spender
    event Approval(
        address indexed owner,
        address indexed spender,
        uint256 value
    );
}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;

import "../IMarginEngine.sol";
import "../../utils/CustomErrors.sol";
import "../IERC20Minimal.sol";
import "../../core_libraries/TraderWithYieldBearingAssets.sol";
import "../IVAMM.sol";
import "../rate_oracles/IRateOracle.sol";

interface IFCM is CustomErrors {
    function getTraderWithYieldBearingAssets(address trader)
        external
        view
        returns (TraderWithYieldBearingAssets.Info memory traderInfo);

    /// @notice Initiate a Fully Collateralised Fixed Taker Swap
    /// @param notional amount of notional (in terms of the underlying token) to trade
    /// @param sqrtPriceLimitX96 the sqrtPriceLimit (in binary fixed point math notation) beyond which swaps won't be executed
    /// @dev An example of an initiated fully collateralised fixed taker swap is a scenario where a trader with 100 aTokens wishes to get a fixed return on them
    /// @dev they can choose to deposit their 100aTokens into the FCM (enter into a fixed taker position with a notional of 100) to swap variable cashflows from the aTokens
    /// @dev with the fixed cashflows from the variable takers
    function initiateFullyCollateralisedFixedTakerSwap(
        uint256 notional,
        uint160 sqrtPriceLimitX96
    ) external returns (int256 fixedTokenDelta, int256 variableTokenDelta, uint256 cumulativeFeeIncurred, int256 fixedTokenDeltaUnbalanced);

    /// @notice Unwind a Fully Collateralised Fixed Taker Swap
    /// @param notionalToUnwind The amount of notional of the original Fully Collateralised Fixed Taker swap to be unwound at the current VAMM fixed rates
    /// @param sqrtPriceLimitX96 the sqrtPriceLimit (in binary fixed point math notation) beyond which the unwind swaps won't be executed
    /// @dev The purpose of this function is to let fully collateralised fixed takers to exist their swaps by entering into variable taker positions against the VAMM
    /// @dev thus effectively releasing the margin in yield bearing tokens from the fixed swap contract
    function unwindFullyCollateralisedFixedTakerSwap(
        uint256 notionalToUnwind,
        uint160 sqrtPriceLimitX96
    ) external returns (int256 fixedTokenDelta, int256 variableTokenDelta, uint256 cumulativeFeeIncurred, int256 fixedTokenDeltaUnbalanced);

    /// @notice Settle Trader
    /// @dev this function in the fcm let's traders settle with the MarginEngine based on their settlement cashflows which is a functon of their fixed and variable token balances
    function settleTrader() external returns (int256);

    /// @notice
    /// @param account address of the position owner from the MarginEngine who wishes to settle with the FCM in underlying tokens
    /// @param marginDeltaInUnderlyingTokens amount in terms of underlying tokens that needs to be settled with the trader from the MarginEngine
    function transferMarginToMarginEngineTrader(
        address account,
        uint256 marginDeltaInUnderlyingTokens
    ) external;

    /// @notice initialize is the constructor for the proxy instances of the FCM
    /// @dev "constructor" for proxy instances
    /// @dev in the initialize function we set the vamm and the margiEngine associated with the fcm
    /// @dev different FCM implementations are free to have different implementations for the initialisation logic
    function initialize(IVAMM __vamm, IMarginEngine __marginEngine)
        external;

    /// @notice Margine Engine linked to the Full Collateralisation Module
    /// @return marginEngine Margine Engine linked to the Full Collateralisation Module
    function marginEngine() external view returns (IMarginEngine);

    /// @notice VAMM linked to the Full Collateralisation Module
    /// @return VAMM linked to the Full Collateralisation Module
    function vamm() external view returns (IVAMM);

    /// @notice Rate Oracle linked to the Full Collateralisation Module
    /// @return Rate Oracle linked to the Full Collateralisation Module
    function rateOracle() external view returns (IRateOracle);

    event FullyCollateralisedSwap(
        address indexed trader,
        uint256 desiredNotional,
        uint160 sqrtPriceLimitX96,
        uint256 cumulativeFeeIncurred,
        int256 fixedTokenDelta,
        int256 variableTokenDelta,
        int256 fixedTokenDeltaUnbalanced
    );

    event FullyCollateralisedUnwind(
        address indexed trader,
        uint256 desiredNotional,
        uint160 sqrtPriceLimitX96,
        uint256 cumulativeFeeIncurred,
        int256 fixedTokenDelta,
        int256 variableTokenDelta,
        int256 fixedTokenDeltaUnbalanced
    );

    event fcmPositionSettlement(
        address indexed trader,
        int256 settlementCashflow
    );

    event FCMTraderUpdate(
        address indexed trader,
        uint256 marginInScaledYieldBearingTokens,
        int256 fixedTokenBalance,
        int256 variableTokenBalance
    );
}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;

import "contracts/utils/CustomErrors.sol";
import "./rate_oracles/IRateOracle.sol";
import "./IMarginEngine.sol";
import "./IVAMM.sol";
import "./fcms/IFCM.sol";
import "./IERC20Minimal.sol";
import "./IPeriphery.sol";

/// @title The interface for the Voltz AMM Factory
/// @notice The AMM Factory facilitates creation of Voltz AMMs
interface IFactory is CustomErrors {
    event IrsInstance(
        IERC20Minimal indexed underlyingToken,
        IRateOracle indexed rateOracle,
        uint256 termStartTimestampWad,
        uint256 termEndTimestampWad,
        int24 tickSpacing,
        IMarginEngine marginEngine,
        IVAMM vamm,
        IFCM fcm,
        uint8 yieldBearingProtocolID,
        uint8 underlyingTokenDecimals
    );

    event MasterFCM(IFCM masterFCMAddress, uint8 yieldBearingProtocolID);

    event Approval(
        address indexed owner,
        address indexed intAddress,
        bool indexed isApproved
    );

    event PeripheryUpdate(IPeriphery periphery);

    // view functions

    function isApproved(address _owner, address intAddress)
        external
        view
        returns (bool);

    function masterVAMM() external view returns (IVAMM);

    function masterMarginEngine() external view returns (IMarginEngine);

    function periphery() external view returns (IPeriphery);

    // settters

    function setApproval(address intAddress, bool allowIntegration) external;

    function setMasterFCM(IFCM masterFCM, uint8 yieldBearingProtocolID)
        external;

    function setMasterVAMM(IVAMM _masterVAMM) external;

    function setMasterMarginEngine(IMarginEngine _masterMarginEngine) external;

    function setPeriphery(IPeriphery _periphery) external;

    /// @notice Deploys the contracts required for a new Interest Rate Swap instance
    function deployIrsInstance(
        IERC20Minimal _underlyingToken,
        IRateOracle _rateOracle,
        uint256 _termStartTimestampWad,
        uint256 _termEndTimestampWad,
        int24 _tickSpacing
    )
        external
        returns (
            IMarginEngine marginEngineProxy,
            IVAMM vammProxy,
            IFCM fcmProxy
        );

    function masterFCMs(uint8 yieldBearingProtocolID)
        external
        view
        returns (IFCM masterFCM);
}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;
import "./IVAMM.sol";
import "./IPositionStructs.sol";
import "../core_libraries/Position.sol";
import "./rate_oracles/IRateOracle.sol";
import "./fcms/IFCM.sol";
import "./IFactory.sol";
import "./IERC20Minimal.sol";
import "contracts/utils/CustomErrors.sol";

interface IMarginEngine is IPositionStructs, CustomErrors {
    // structs

    struct MarginCalculatorParameters {
        /// @dev Upper bound of the underlying pool (e.g. Aave v2 USDC lending pool) APY from the initiation of the IRS AMM and until its maturity (18 decimals fixed point number)
        uint256 apyUpperMultiplierWad;
        /// @dev Lower bound of the underlying pool (e.g. Aave v2 USDC lending pool) APY from the initiation of the IRS AMM and until its maturity (18 decimals)
        uint256 apyLowerMultiplierWad;
        /// @dev The volatility of the underlying pool APY (settable by the owner of the Margin Engine) (18 decimals)
        int256 sigmaSquaredWad;
        /// @dev Margin Engine Parameter estimated via CIR model calibration (for details refer to litepaper) (18 decimals)
        int256 alphaWad;
        /// @dev Margin Engine Parameter estimated via CIR model calibration (for details refer to litepaper) (18 decimals)
        int256 betaWad;
        /// @dev Standard normal critical value used in the computation of the Upper APY Bound of the underlying pool
        int256 xiUpperWad;
        /// @dev Standard normal critical value used in the computation of the Lower APY Bound of the underlying pool
        int256 xiLowerWad;
        /// @dev Max term possible for a Voltz IRS AMM in seconds (18 decimals)
        int256 tMaxWad;
        /// @dev multiplier of the starting fixed rate (refer to the litepaper) if simulating a counterfactual fixed taker unwind (moving to the left along the VAMM) for purposes of calculating liquidation margin requirement
        uint256 devMulLeftUnwindLMWad;
        /// @dev multiplier of the starting fixed rate (refer to the litepaper) if simulating a counterfactual variable taker unwind (moving to the right along the VAMM) for purposes of calculating liquidation margin requirement
        uint256 devMulRightUnwindLMWad;
        /// @dev same as devMulLeftUnwindLMWad but for purposes of calculating the initial margin requirement
        uint256 devMulLeftUnwindIMWad;
        /// @dev same as devMulRightUnwindLMWad but for purposes of calculating the initial margin requirement
        uint256 devMulRightUnwindIMWad;
        /// @dev r_min from the litepaper eq. 11 for a scenario where counterfactual is a simulated fixed taker unwind (left unwind along the VAMM), used for liquidation margin calculation
        uint256 fixedRateDeviationMinLeftUnwindLMWad;
        /// @dev r_min from the litepaper eq. 11 for a scenario where counterfactual is a simulated variable taker unwind (right unwind along the VAMM), used for liquidation margin calculation
        uint256 fixedRateDeviationMinRightUnwindLMWad;
        /// @dev same as fixedRateDeviationMinLeftUnwindLMWad but for Initial Margin Requirement
        uint256 fixedRateDeviationMinLeftUnwindIMWad;
        /// @dev same as fixedRateDeviationMinRightUnwindLMWad but for Initial Margin Requirement
        uint256 fixedRateDeviationMinRightUnwindIMWad;
        /// @dev gamma from eqn. 12 [append this logic to the litepaper] from the litepaper, gamma is an adjustable parameter necessary to calculate scaled deviations to the fixed rate in counterfactual unwinds for minimum margin requirement calculations
        uint256 gammaWad;
        /// @dev settable parameter that ensures that minimumMarginRequirement is always above or equal to the minMarginToIncentiviseLiquidators which ensures there is always sufficient incentive for liquidators to liquidate positions given the fact their income is a proportion of position margin
        uint256 minMarginToIncentiviseLiquidators;
    }

    // Events
    event HistoricalApyWindowSetting(uint256 secondsAgo);
    event CacheMaxAgeSetting(uint256 cacheMaxAgeInSeconds);
    event RateOracle(uint256 cacheMaxAgeInSeconds);

    event ProtocolCollection(
        address sender,
        address indexed recipient,
        uint256 amount
    );
    event LiquidatorRewardSetting(uint256 liquidatorRewardWad);

    event VAMMSetting(IVAMM indexed vamm);

    event RateOracleSetting(IRateOracle indexed rateOracle);

    event FCMSetting(IFCM indexed fcm);

    event MarginCalculatorParametersSetting(
        MarginCalculatorParameters marginCalculatorParameters
    );

    event PositionMarginUpdate(
        address sender,
        address indexed owner,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        int256 marginDelta
    );

    event HistoricalApy(uint256 value);

    event PositionSettlement(
        address indexed owner,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        int256 settlementCashflow
    );

    event PositionLiquidation(
        address indexed owner,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        address liquidator,
        int256 notionalUnwound,
        uint256 liquidatorReward
    );

    event PositionUpdate(
        address indexed owner,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        uint128 _liquidity,
        int256 margin,
        int256 fixedTokenBalance,
        int256 variableTokenBalance,
        uint256 accumulatedFees
    );

    // immutables

    /// @notice The Full Collateralisation Module (FCM)
    /// @dev The FCM is a smart contract that acts as an intermediary Position between the Voltz Core and traders who wish to take fully collateralised fixed taker positions
    /// @dev An example FCM is the AaveFCM.sol module which inherits from the IFCM interface, it lets fixed takers deposit underlying yield bearing tokens (e.g.) aUSDC as margin to enter into a fixed taker swap without the need to worry about liquidations
    /// @dev since the MarginEngine is confident the FCM is always fully collateralised, it does not let liquidators liquidate the FCM Position
    /// @return The Full Collateralisation Module linked to the MarginEngine
    function fcm() external view returns (IFCM);

    /// @notice The Factory
    /// @dev the factory that deployed the master Margin Engine
    function factory() external view returns (IFactory);

    /// @notice The address of the underlying (non-yield bearing) token - e.g. USDC
    /// @return The underlying ERC20 token (e.g. USDC)
    function underlyingToken() external view returns (IERC20Minimal);

    /// @notice The rateOracle contract which lets the protocol access historical apys in the yield bearing pools it is built on top of
    /// @return The underlying ERC20 token (e.g. USDC)
    function rateOracle() external view returns (IRateOracle);

    /// @notice The unix termStartTimestamp of the MarginEngine in Wad
    /// @return Term Start Timestamp in Wad
    function termStartTimestampWad() external view returns (uint256);

    /// @notice The unix termEndTimestamp of the MarginEngine in Wad
    /// @return Term End Timestamp in Wad
    function termEndTimestampWad() external view returns (uint256);

    /// @dev "constructor" for proxy instances
    function initialize(
        IERC20Minimal __underlyingToken,
        IRateOracle __rateOracle,
        uint256 __termStartTimestampWad,
        uint256 __termEndTimestampWad
    ) external;

    // view functions

    /// @notice The liquidator Reward Percentage (in Wad)
    /// @dev liquidatorReward (in wad) is the percentage of the margin (of a liquidated position) that is sent to the liquidator
    /// @dev following a successful liquidation that results in a trader/position unwind; example value:  2 * 10**16 => 2% of position margin is used to cover liquidator reward
    /// @return Liquidator Reward in Wad
    function liquidatorRewardWad() external view returns (uint256);

    /// @notice VAMM (Virtual Automated Market Maker) linked to the MarginEngine
    /// @dev The VAMM is responsible for pricing only (determining the effective fixed rate at which a given Interest Rate Swap notional will be executed)
    /// @return The VAMM
    function vamm() external view returns (IVAMM);

    /// @notice Returns the information about a position by the position's key
    /// @param _owner The address of the position owner
    /// @param _tickLower The lower tick boundary of the position
    /// @param _tickUpper The upper tick boundary of the position
    /// Returns position The Position.Info corresponding to the requested position
    function getPosition(
        address _owner,
        int24 _tickLower,
        int24 _tickUpper
    ) external returns (Position.Info memory position);

    /// @notice Gets the look-back window size that's used to request the historical APY from the rate Oracle
    /// @dev The historical APY of the Rate Oracle is necessary for MarginEngine computations
    /// @dev The look-back window is seconds from the current timestamp
    /// @dev This value is only settable by the the Factory owner and may be unique for each MarginEngine
    /// @dev When setting secondAgo, the setter needs to take into consideration the underlying volatility of the APYs in the reference yield-bearing pool (e.g. Aave v2 USDC)
    function lookbackWindowInSeconds() external view returns (uint256);

    // non-view functions

    /// @notice Sets secondsAgo: The look-back window size used to calculate the historical APY for margin purposes
    /// @param _secondsAgo the duration of the lookback window in seconds
    /// @dev Can only be set by the Factory Owner
    function setLookbackWindowInSeconds(uint256 _secondsAgo) external;

    /// @notice Set the MarginCalculatorParameters (each margin engine can have its own custom set of margin calculator parameters)
    /// @param _marginCalculatorParameters the MarginCalculatorParameters to set
    /// @dev marginCalculatorParameteres is of type MarginCalculatorParameters (refer to the definition of the struct for elaboration on what each parameter means)
    function setMarginCalculatorParameters(
        MarginCalculatorParameters memory _marginCalculatorParameters
    ) external;

    /// @notice Sets the liquidator reward: proportion of liquidated position's margin paid as a reward to the liquidator
    function setLiquidatorReward(uint256 _liquidatorRewardWad) external;

    /// @notice updates the margin account of a position which can be uniquily identified with its _owner, tickLower, tickUpper
    /// @dev if the position has positive liquidity then before the margin update, we call the updatePositionTokenBalancesAndAccountForFees functon that calculates up to date
    /// @dev margin, fixed and variable token balances by taking into account the fee income from their tick range and fixed and variable deltas settled along their tick range
    /// @dev marginDelta is the delta applied to the current margin of a position, if the marginDelta is negative, the position is withdrawing margin, if the marginDelta is positive, the position is depositing funds in terms of the underlying tokens
    /// @dev if marginDelta is negative, we need to check if the msg.sender is either the _owner of the position or the msg.sender is apporved by the _owner to act on their behalf in Voltz Protocol
    /// @dev the approval logic is implemented in the Factory.sol
    /// @dev if marginDelta is negative, we additionally need to check if post the initial margin requirement is still satisfied post withdrawal
    /// @dev if marginDelta is positive, the depositor of the margin is either the msg.sender or the owner who interacted through an approved peripheral contract
    function updatePositionMargin(
        address _owner,
        int24 _tickLower,
        int24 _tickUpper,
        int256 marginDelta
    ) external;

    /// @notice Settles a Position
    /// @dev Can be called by anyone
    /// @dev A position cannot be settled before maturity
    /// @dev Steps to settle a position:
    /// @dev 1. Retrieve the current fixed and variable token growth inside the tick range of a position
    /// @dev 2. Calculate accumulated fixed and variable balances of the position since the last mint/poke/burn
    /// @dev 3. Update the postion's fixed and variable token balances
    /// @dev 4. Update the postion's fixed and varaible token growth inside last to enable future updates
    /// @dev 5. Calculates the settlement cashflow from all of the IRS contracts the position has entered since entering the AMM
    /// @dev 6. Updates the fixed and variable token balances of the position to be zero, adds the settlement cashflow to the position's current margin
    function settlePosition(
        address _owner,
        int24 _tickLower,
        int24 _tickUpper
    ) external;

    /// @notice Liquidate a Position
    /// @dev Steps to liquidate: update position's fixed and variable token balances to account for balances accumulated throughout the trades made since the last mint/burn/poke,
    /// @dev Check if the position is liquidatable by calling the isLiquidatablePosition function of the calculator, revert if that is not the case,
    /// @dev Calculate the liquidation reward = current margin of the position * liquidatorReward, subtract the liquidator reward from the position margin,
    /// @dev Burn the position's liquidity, unwind unnetted fixed and variable balances of a position, transfer the reward to the liquidator
    function liquidatePosition(
        address _owner,
        int24 _tickLower,
        int24 _tickUpper
    ) external returns (uint256);

    /// @notice Update a Position post VAMM induced mint or burn
    /// @dev Steps taken:
    /// @dev 1. Update position liquidity based on params.liquidityDelta
    /// @dev 2. Update fixed and variable token balances of the position based on how much has been accumulated since the last mint/burn/poke
    /// @dev 3. Update position's margin by taking into account the position accumulated fees since the last mint/burn/poke
    /// @dev 4. Update fixed and variable token growth + fee growth in the position info struct for future interactions with the position
    /// @param _params necessary for the purposes of referencing the position being updated (owner, tickLower, tickUpper, _) and the liquidity delta that needs to be applied to position._liquidity
    function updatePositionPostVAMMInducedMintBurn(
        IPositionStructs.ModifyPositionParams memory _params
    ) external returns (int256 _positionMarginRequirement);

    // @notive Update a position post VAMM induced swap
    /// @dev Since every position can also engage in swaps with the VAMM, this function needs to be invoked after non-external calls are made to the VAMM's swap function
    /// @dev This purpose of this function is to:
    /// @dev 1. updatePositionTokenBalancesAndAccountForFees
    /// @dev 2. update position margin to account for fees paid to execute the swap
    /// @dev 3. calculate the position margin requrement given the swap, check if the position marigin satisfies the most up to date requirement
    /// @dev 4. if all the requirements are satisfied then position gets updated to take into account the swap that it just entered, if the minimum margin requirement is not satisfied then the transaction will revert
    function updatePositionPostVAMMInducedSwap(
        address _owner,
        int24 _tickLower,
        int24 _tickUpper,
        int256 _fixedTokenDelta,
        int256 _variableTokenDelta,
        uint256 _cumulativeFeeIncurred,
        int256 _fixedTokenDeltaUnbalanced
    ) external returns (int256 _positionMarginRequirement);

    /// @notice function that can only be called by the owner enables collection of protocol generated fees from any give margin engine
    /// @param _recipient the address which collects the protocol generated fees
    /// @param _amount the amount in terms of underlying tokens collected from the protocol's earnings
    function collectProtocol(address _recipient, uint256 _amount) external;

    /// @notice sets the Virtual Automated Market Maker (VAMM) attached to the MarginEngine
    /// @dev the VAMM is responsible for price discovery, whereas the management of the underlying collateral and liquidations are handled by the Margin Engine
    function setVAMM(IVAMM _vAMM) external;

    /// @notice sets the Virtual Automated Market Maker (VAMM) attached to the MarginEngine
    /// @dev the VAMM is responsible for price discovery, whereas the management of the underlying collateral and liquidations are handled by the Margin Engine
    function setRateOracle(IRateOracle __rateOracle) external;

    /// @notice sets the Full Collateralisation Module
    function setFCM(IFCM _newFCM) external;

    /// @notice transfers margin in terms of underlying tokens to a trader from the Full Collateralisation Module
    /// @dev post maturity date of the MarginEngine, the traders from the Full Collateralisation module will be able to settle with the MarginEngine
    /// @dev to ensure their fixed yield is guaranteed, in order to collect the funds from the MarginEngine, the FCM needs to invoke the transferMarginToFCMTrader function whcih is only callable by the FCM attached to a particular Margin Engine
    function transferMarginToFCMTrader(address _account, uint256 _marginDelta)
        external;

    /// @notice Gets the maximum age of the cached historical APY value can be without being refreshed
    function cacheMaxAgeInSeconds() external view returns (uint256);

    /// @notice Sets the maximum age that the cached historical APY value
    /// @param _cacheMaxAgeInSeconds The new maximum age that the historical APY cache can be before being considered stale
    function setCacheMaxAgeInSeconds(uint256 _cacheMaxAgeInSeconds) external;

    /// @notice Get Historical APY
    /// @dev The lookback window used by this function is determined by the secondsAgo state variable
    /// @dev refresh the historical apy cache if necessary
    /// @return historicalAPY (Wad)
    function getHistoricalApy() external returns (uint256);

    function getPositionMarginRequirement(
        address _recipient,
        int24 _tickLower,
        int24 _tickUpper,
        bool _isLM
    ) external returns (uint256);
}

// SPDX-License-Identifier: Apache-2.0

pragma solidity =0.8.9;

import "../interfaces/IMarginEngine.sol";
import "../interfaces/IVAMM.sol";
import "contracts/utils/CustomErrors.sol";

interface IPeriphery is CustomErrors {
    // events

    /// @dev emitted after new lp notional cap is set
    event NotionalCap(IVAMM _vamm, uint256 _lpNotionalCapNew);

    // structs

    struct MintOrBurnParams {
        IMarginEngine marginEngine;
        int24 tickLower;
        int24 tickUpper;
        uint256 notional;
        bool isMint;
        uint256 marginDelta;
    }

    struct SwapPeripheryParams {
        IMarginEngine marginEngine;
        bool isFT;
        uint256 notional;
        uint160 sqrtPriceLimitX96;
        int24 tickLower;
        int24 tickUpper;
        uint256 marginDelta;
    }

    // view functions

    function getCurrentTick(IMarginEngine marginEngine)
        external
        view
        returns (int24 currentTick);

    /// @param _vamm VAMM for which to get the lp cap in underlying tokens
    /// @return Notional Cap for liquidity providers that mint or burn via periphery (enforced in the core if isAlpha is set to true)
    function lpNotionalCaps(IVAMM _vamm) external returns (uint256);

    /// @param _vamm VAMM for which to get the lp notional cumulative in underlying tokens
    /// @return Total amount of notional supplied by the LPs to a given VAMM via the periphery
    function lpNotionalCumulatives(IVAMM _vamm) external returns (uint256);

    // non-view functions

    function mintOrBurn(MintOrBurnParams memory params)
        external
        returns (int256 positionMarginRequirement);

    function swap(SwapPeripheryParams memory params)
        external
        returns (
            int256 _fixedTokenDelta,
            int256 _variableTokenDelta,
            uint256 _cumulativeFeeIncurred,
            int256 _fixedTokenDeltaUnbalanced,
            int256 _marginRequirement,
            int24 _tickAfter
        );
}

// SPDX-License-Identifier: BUSL-1.1

pragma solidity =0.8.9;

interface IPositionStructs {
    struct ModifyPositionParams {
        // the address that owns the position
        address owner;
        // the lower and upper tick of the position
        int24 tickLower;
        int24 tickUpper;
        // any change in liquidity
        int128 liquidityDelta;
    }
}

// SPDX-License-Identifier: Apache-2.0

import "contracts/utils/CustomErrors.sol";

pragma solidity =0.8.9;

import "contracts/utils/CustomErrors.sol";
import "../IERC20Minimal.sol";

/// @dev The RateOracle is used for two purposes on the Voltz Protocol
/// @dev Settlement: in order to be able to settle IRS positions after the termEndTimestamp of a given AMM
/// @dev Margin Engine Computations: getApyFromTo is used by the MarginCalculator and MarginEngine
/// @dev It is necessary to produce margin requirements for Trader and Liquidity Providers
interface IRateOracle is CustomErrors {

    // events
    event MinSecondsSinceLastUpdate(uint256 _minSecondsSinceLastUpdate);
    event OracleBufferUpdate(
        uint256 blockTimestampScaled,
        address source,
        uint16 index,
        uint32 blockTimestamp,
        uint256 observedValue,
        uint16 cardinality,
        uint16 cardinalityNext
    );

    /// @notice Emitted by the rate oracle for increases to the number of observations that can be stored
    /// @param observationCardinalityNextNew The updated value of the next observation cardinality
    event RateCardinalityNext(
        uint16 observationCardinalityNextNew
    );

    // view functions

    /// @notice Gets minimum number of seconds that need to pass since the last update to the rates array
    /// @dev This is a throttling mechanic that needs to ensure we don't run out of space in the rates array
    /// @dev The maximum size of the rates array is 65535 entries
    // AB: as long as this doesn't affect the termEndTimestamp rateValue too much
    // AB: can have a different minSecondsSinceLastUpdate close to termEndTimestamp to have more granularity for settlement purposes
    /// @return minSecondsSinceLastUpdate in seconds
    function minSecondsSinceLastUpdate() external view returns (uint256);

    /// @notice Gets the address of the underlying token of the RateOracle
    /// @return underlying The address of the underlying token
    function underlying() external view returns (IERC20Minimal);

    /// @notice Gets the variable factor between termStartTimestamp and termEndTimestamp
    /// @return result The variable factor
    /// @dev If the current block timestamp is beyond the maturity of the AMM, then the variableFactor is getRateFromTo(termStartTimestamp, termEndTimestamp). Term end timestamps are cached for quick retrieval later.
    /// @dev If the current block timestamp is before the maturity of the AMM, then the variableFactor is getRateFromTo(termStartTimestamp,Time.blockTimestampScaled());
    /// @dev if queried before maturity then returns the rate of return between pool initiation and current timestamp (in wad)
    /// @dev if queried after maturity then returns the rate of return between pool initiation and maturity timestamp (in wad)
    function variableFactor(uint256 termStartTimestamp, uint256 termEndTimestamp) external returns(uint256 result);

    /// @notice Gets the variable factor between termStartTimestamp and termEndTimestamp
    /// @return result The variable factor
    /// @dev If the current block timestamp is beyond the maturity of the AMM, then the variableFactor is getRateFromTo(termStartTimestamp, termEndTimestamp). No caching takes place.
    /// @dev If the current block timestamp is before the maturity of the AMM, then the variableFactor is getRateFromTo(termStartTimestamp,Time.blockTimestampScaled());
    function variableFactorNoCache(uint256 termStartTimestamp, uint256 termEndTimestamp) external view returns(uint256 result);

    
    /// @notice Calculates the observed interest returned by the underlying in a given period
    /// @dev Reverts if we have no data point for either timestamp
    /// @param from The timestamp of the start of the period, in seconds
    /// @param to The timestamp of the end of the period, in seconds
    /// @return The "floating rate" expressed in Wad, e.g. 4% is encoded as 0.04*10**18 = 4*10*16
    function getRateFromTo(uint256 from, uint256 to)
        external
        view
        returns (uint256);

    /// @notice Calculates the observed APY returned by the rate oracle in a given period
    /// @param from The timestamp of the start of the period, in seconds
    /// @param to The timestamp of the end of the period, in seconds
    /// @dev Reverts if we have no data point for either timestamp
    //  how is the returned rate encoded? Floating rate?
    function getApyFromTo(uint256 from, uint256 to)
        external
        view
        returns (uint256 apyFromTo);

    // non-view functions

    /// @notice Sets minSecondsSinceLastUpdate: The minimum number of seconds that need to pass since the last update to the rates array
    /// @dev Can only be set by the Factory Owner
    function setMinSecondsSinceLastUpdate(uint256 _minSecondsSinceLastUpdate) external;

    /// @notice Increase the maximum number of rates observations that this RateOracle will store
    /// @dev This method is no-op if the RateOracle already has an observationCardinalityNext greater than or equal to
    /// the input observationCardinalityNext.
    /// @param rateCardinalityNext The desired minimum number of observations for the pool to store
    function increaseObservationCardinalityNext(uint16 rateCardinalityNext) external;

    /// @notice Writes a rate observation to the rates array given the current rate cardinality, rate index and rate cardinality next
    /// Write oracle entry is called whenever a new position is minted via the vamm or when a swap is initiated via the vamm
    /// That way the gas costs of Rate Oracle updates can be distributed across organic interactions with the protocol
    function writeOracleEntry() external;

    /// @notice unique ID of the underlying yield bearing protocol (e.g. Aave v2 has id 1)
    /// @return yieldBearingProtocolID unique id of the underlying yield bearing protocol
    function UNDERLYING_YIELD_BEARING_PROTOCOL_ID() external view returns(uint8 yieldBearingProtocolID);
}

// SPDX-License-Identifier: BUSL-1.1

pragma solidity =0.8.9;
import "./IMarginEngine.sol";
import "./IFactory.sol";
import "./IPositionStructs.sol";
import "../core_libraries/Tick.sol";
import "contracts/utils/CustomErrors.sol";

interface IVAMM is IPositionStructs, CustomErrors {
    // events
    event Swap(
        address sender,
        address indexed recipient,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        int256 desiredNotional,
        uint160 sqrtPriceLimitX96,
        uint256 cumulativeFeeIncurred,
        int256 fixedTokenDelta,
        int256 variableTokenDelta,
        int256 fixedTokenDeltaUnbalanced
    );

    /// @dev emitted after a given vamm is successfully initialized
    event VAMMInitialization(uint160 sqrtPriceX96, int24 tick);

    /// @dev emitted after a successful minting of a given LP position
    event Mint(
        address sender,
        address indexed owner,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        uint128 amount
    );

    /// @dev emitted after a successful burning of a given LP position
    event Burn(
        address sender,
        address indexed owner,
        int24 indexed tickLower,
        int24 indexed tickUpper,
        uint128 amount
    );

    /// @dev emitted after setting feeProtocol
    event FeeProtocol(uint8 feeProtocol);

    /// @dev emitted after fee is set
    event Fee(uint256 feeWad);

    /// @dev emitted after the _isAlpha boolean is updated by the owner of the VAMM
    /// @dev _isAlpha boolean dictates whether the Margin Engine is in the Alpha State, i.e. mints can only be done via the periphery
    /// @dev additionally, the periphery has the logic to take care of lp notional caps in the Alpha State phase of VAMM
    /// @dev __isAlpha is the newly set value for the _isAlpha boolean
    event IsAlpha(bool __isAlpha);

    event VAMMPriceChange(int24 tick);

    // structs

    struct VAMMVars {
        /// @dev The current price of the pool as a sqrt(variableToken/fixedToken) Q64.96 value
        uint160 sqrtPriceX96;
        /// @dev The current tick of the vamm, i.e. according to the last tick transition that was run.
        int24 tick;
        // the current protocol fee as a percentage of the swap fee taken on withdrawal
        // represented as an integer denominator (1/x)
        uint8 feeProtocol;
    }

    struct SwapParams {
        /// @dev Address of the trader initiating the swap
        address recipient;
        /// @dev The amount of the swap, which implicitly configures the swap as exact input (positive), or exact output (negative)
        int256 amountSpecified;
        /// @dev The Q64.96 sqrt price limit. If !isFT, the price cannot be less than this
        uint160 sqrtPriceLimitX96;
        /// @dev lower tick of the position
        int24 tickLower;
        /// @dev upper tick of the position
        int24 tickUpper;
    }

    struct SwapCache {
        /// @dev liquidity at the beginning of the swap
        uint128 liquidityStart;
        // the current protocol fee as a percentage of the swap fee taken on withdrawal
        // represented as an integer denominator (1/x)%
        uint8 feeProtocol;
    }

    /// @dev the top level state of the swap, the results of which are recorded in storage at the end
    struct SwapState {
        /// @dev the amount remaining to be swapped in/out of the input/output asset
        int256 amountSpecifiedRemaining;
        /// @dev the amount already swapped out/in of the output/input asset
        int256 amountCalculated;
        /// @dev current sqrt(price)
        uint160 sqrtPriceX96;
        /// @dev the tick associated with the current price
        int24 tick;
        /// @dev the global fixed token growth
        int256 fixedTokenGrowthGlobalX128;
        /// @dev the global variable token growth
        int256 variableTokenGrowthGlobalX128;
        /// @dev the current liquidity in range
        uint128 liquidity;
        /// @dev the global fee growth of the underlying token
        uint256 feeGrowthGlobalX128;
        /// @dev amount of underlying token paid as protocol fee
        uint256 protocolFee;
        /// @dev cumulative fee incurred while initiating a swap
        uint256 cumulativeFeeIncurred;
        /// @dev fixedTokenDelta that will be applied to the fixed token balance of the position executing the swap (recipient)
        int256 fixedTokenDeltaCumulative;
        /// @dev variableTokenDelta that will be applied to the variable token balance of the position executing the swap (recipient)
        int256 variableTokenDeltaCumulative;
        /// @dev fixed token delta cumulative but without rebalancings applied
        int256 fixedTokenDeltaUnbalancedCumulative;
    }

    struct StepComputations {
        /// @dev the price at the beginning of the step
        uint160 sqrtPriceStartX96;
        /// @dev the next tick to swap to from the current tick in the swap direction
        int24 tickNext;
        /// @dev whether tickNext is initialized or not
        bool initialized;
        /// @dev sqrt(price) for the next tick (1/0)
        uint160 sqrtPriceNextX96;
        /// @dev how much is being swapped in in this step
        uint256 amountIn;
        /// @dev how much is being swapped out
        uint256 amountOut;
        /// @dev how much fee is being paid in (underlying token)
        uint256 feeAmount;
        /// @dev ...
        uint256 feeProtocolDelta;
        /// @dev ...
        int256 fixedTokenDeltaUnbalanced; // for LP
        /// @dev ...
        int256 fixedTokenDelta; // for LP
        /// @dev ...
        int256 variableTokenDelta; // for LP
    }

    /// @dev "constructor" for proxy instances
    function initialize(IMarginEngine __marginEngine, int24 __tickSpacing)
        external;

    // immutables

    /// @notice The vamm's fee (proportion) in wad
    /// @return The fee in wad
    function feeWad() external view returns (uint256);

    /// @notice The vamm tick spacing
    /// @dev Ticks can only be used at multiples of this value, minimum of 1 and always positive
    /// e.g.: a tickSpacing of 3 means ticks can be initialized every 3rd tick, i.e., ..., -6, -3, 0, 3, 6, ...
    /// This value is an int24 to avoid casting even though it is always positive.
    /// @return The tick spacing
    function tickSpacing() external view returns (int24);

    /// @notice The maximum amount of position liquidity that can use any tick in the range
    /// @dev This parameter should be enforced per tick (when setting) to prevent liquidity from overflowing a uint128 at any point, and
    /// also prevents out-of-range liquidity from being used to prevent adding in-range liquidity to the vamm
    /// @return The max amount of liquidity per tick
    function maxLiquidityPerTick() external view returns (uint128);

    // state variables

    /// @return The current VAMM Vars (see struct definition for semantics)
    function vammVars() external view returns (VAMMVars memory);

    /// @return If true, the VAMM Proxy is currently in alpha state, hence minting can only be done via the periphery. If false, minting can be done directly via VAMM.
    function isAlpha() external view returns (bool);

    /// @notice The fixed token growth accumulated per unit of liquidity for the entire life of the vamm
    /// @dev This value can overflow the uint256
    function fixedTokenGrowthGlobalX128() external view returns (int256);

    /// @notice The variable token growth accumulated per unit of liquidity for the entire life of the vamm
    /// @dev This value can overflow the uint256
    function variableTokenGrowthGlobalX128() external view returns (int256);

    /// @notice The fee growth collected per unit of liquidity for the entire life of the vamm
    /// @dev This value can overflow the uint256
    function feeGrowthGlobalX128() external view returns (uint256);

    /// @notice The currently in range liquidity available to the vamm
    function liquidity() external view returns (uint128);

    /// @notice The amount underlying token that are owed to the protocol
    /// @dev Protocol fees will never exceed uint256
    function protocolFees() external view returns (uint256);

    function marginEngine() external view returns (IMarginEngine);

    function factory() external view returns (IFactory);

    /// @notice Function that sets the feeProtocol of the vamm
    /// @dev the current protocol fee as a percentage of the swap fee taken on withdrawal
    // represented as an integer denominator (1/x)
    function setFeeProtocol(uint8 feeProtocol) external;

    /// @notice Function that sets the _isAlpha state variable, if it is set to true the protocol is in the Alpha State
    /// @dev if the VAMM is at the alpha state, mints can only be done via the periphery which in turn takes care of notional caps for the LPs
    /// @dev this function can only be called by the owner of the VAMM
    function setIsAlpha(bool __isAlpha) external;

    /// @notice Function that sets fee of the vamm
    /// @dev The vamm's fee (proportion) in wad
    function setFee(uint256 _fee) external;

    /// @notice Updates internal accounting to reflect a collection of protocol fees. The actual transfer of fees must happen separately in the AMM
    /// @dev can only be done via the collectProtocol function of the parent AMM of the vamm
    function updateProtocolFees(uint256 protocolFeesCollected) external;

    /// @notice Sets the initial price for the vamm
    /// @dev Price is represented as a sqrt(amountVariableToken/amountFixedToken) Q64.96 value
    /// @param sqrtPriceX96 the initial sqrt price of the vamm as a Q64.96
    function initializeVAMM(uint160 sqrtPriceX96) external;

    /// @notice removes liquidity given recipient/tickLower/tickUpper of the position
    /// @param recipient The address for which the liquidity will be removed
    /// @param tickLower The lower tick of the position in which to remove liquidity
    /// @param tickUpper The upper tick of the position in which to remove liqudiity
    /// @param amount The amount of liquidity to burn
    function burn(
        address recipient,
        int24 tickLower,
        int24 tickUpper,
        uint128 amount
    ) external returns (int256 positionMarginRequirement);

    /// @notice Adds liquidity for the given recipient/tickLower/tickUpper position
    /// @param recipient The address for which the liquidity will be created
    /// @param tickLower The lower tick of the position in which to add liquidity
    /// @param tickUpper The upper tick of the position in which to add liquidity
    /// @param amount The amount of liquidity to mint
    function mint(
        address recipient,
        int24 tickLower,
        int24 tickUpper,
        uint128 amount
    ) external returns (int256 positionMarginRequirement);

    /// @notice Initiate an Interest Rate Swap
    /// @param params SwapParams necessary to initiate an Interest Rate Swap
    /// @return fixedTokenDelta Fixed Token Delta
    /// @return variableTokenDelta Variable Token Delta
    /// @return cumulativeFeeIncurred Cumulative Fee Incurred
    function swap(SwapParams memory params)
        external
        returns (
            int256 fixedTokenDelta,
            int256 variableTokenDelta,
            uint256 cumulativeFeeIncurred,
            int256 fixedTokenDeltaUnbalanced,
            int256 marginRequirement
        );

    /// @notice Look up information about a specific tick in the amm
    /// @param tick The tick to look up
    /// @return liquidityGross: the total amount of position liquidity that uses the vamm either as tick lower or tick upper,
    /// liquidityNet: how much liquidity changes when the vamm price crosses the tick,
    /// feeGrowthOutsideX128: the fee growth on the other side of the tick from the current tick in underlying token. i.e. if liquidityGross is greater than 0. In addition, these values are only relative.
    function ticks(int24 tick) external view returns (Tick.Info memory);

    /// @notice Returns 256 packed tick initialized boolean values. See TickBitmap for more information
    function tickBitmap(int16 wordPosition) external view returns (uint256);

    /// @notice Computes the current fixed and variable token growth inside a given tick range given the current tick in the vamm
    /// @param tickLower The lower tick of the position
    /// @param tickUpper The upper tick of the position
    /// @return fixedTokenGrowthInsideX128 Fixed Token Growth inside the given tick range
    /// @return variableTokenGrowthInsideX128 Variable Token Growth inside the given tick range
    /// @return feeGrowthInsideX128 Fee Growth Inside given tick range
    function computeGrowthInside(int24 tickLower, int24 tickUpper)
        external
        view
        returns (
            int256 fixedTokenGrowthInsideX128,
            int256 variableTokenGrowthInsideX128,
            uint256 feeGrowthInsideX128
        );
}

// SPDX-License-Identifier: GPL-2.0-or-later

pragma solidity =0.8.9;

/// @title Math library for liquidity
library LiquidityMath {
    /// @notice Add a signed liquidity delta to liquidity and revert if it overflows or underflows
    /// @param x The liquidity before change
    /// @param y The delta by which liquidity should be changed
    /// @return z The liquidity delta
    function addDelta(uint128 x, int128 y) internal pure returns (uint128 z) {
        if (y < 0) {
            uint128 yAbsolute;

            unchecked {
                yAbsolute = uint128(-y);
            }

            z = x - yAbsolute;
        } else {
            z = x + uint128(y);
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.0 (access/Ownable.sol)

pragma solidity ^0.8.0;

import "../utils/Context.sol";

/**
 * @dev Contract module which provides a basic access control mechanism, where
 * there is an account (an owner) that can be granted exclusive access to
 * specific functions.
 *
 * By default, the owner account will be the one that deploys the contract. This
 * can later be changed with {transferOwnership}.
 *
 * This module is used through inheritance. It will make available the modifier
 * `onlyOwner`, which can be applied to your functions to restrict their use to
 * the owner.
 */
abstract contract Ownable is Context {
    address private _owner;

    event OwnershipTransferred(address indexed previousOwner, address indexed newOwner);

    /**
     * @dev Initializes the contract setting the deployer as the initial owner.
     */
    constructor() {
        _transferOwnership(_msgSender());
    }

    /**
     * @dev Returns the address of the current owner.
     */
    function owner() public view virtual returns (address) {
        return _owner;
    }

    /**
     * @dev Throws if called by any account other than the owner.
     */
    modifier onlyOwner() {
        require(owner() == _msgSender(), "Ownable: caller is not the owner");
        _;
    }

    /**
     * @dev Leaves the contract without owner. It will not be possible to call
     * `onlyOwner` functions anymore. Can only be called by the current owner.
     *
     * NOTE: Renouncing ownership will leave the contract without an owner,
     * thereby removing any functionality that is only available to the owner.
     */
    function renounceOwnership() public virtual onlyOwner {
        _transferOwnership(address(0));
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Can only be called by the current owner.
     */
    function transferOwnership(address newOwner) public virtual onlyOwner {
        require(newOwner != address(0), "Ownable: new owner is the zero address");
        _transferOwnership(newOwner);
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Internal function without access restriction.
     */
    function _transferOwnership(address newOwner) internal virtual {
        address oldOwner = _owner;
        _owner = newOwner;
        emit OwnershipTransferred(oldOwner, newOwner);
    }
}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    struct SD59x18 {
        int256 value;
    }

    struct UD60x18 {
        uint256 value;
    }

    /// STORAGE ///

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

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

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

    /// FUNCTIONS ///

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

// 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;
        }
    }
}

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

import "./PRBMath.sol";

/// @title PRBMathUD60x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math that works with uint256 numbers considered to have 18
/// trailing decimals. We call this number representation unsigned 60.18-decimal fixed-point, since there can be up to 60
/// digits in the integer part and up to 18 decimals in the fractional part. The numbers are bound by the minimum and the
/// maximum values permitted by the Solidity type uint256.
library PRBMathUD60x18 {
    /// @dev Half the SCALE number.
    uint256 internal constant HALF_SCALE = 5e17;

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

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

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

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

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

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

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

            // Equivalent to "x + delta * (remainder > 0 ? 1 : 0)" but faster.
            result := add(x, mul(delta, gt(remainder, 0)))
        }
    }

    /// @notice Divides two unsigned 60.18-decimal fixed-point numbers, returning a new unsigned 60.18-decimal fixed-point number.
    ///
    /// @dev Uses mulDiv to enable overflow-safe multiplication and division.
    ///
    /// Requirements:
    /// - The denominator cannot be zero.
    ///
    /// @param x The numerator as an unsigned 60.18-decimal fixed-point number.
    /// @param y The denominator as an unsigned 60.18-decimal fixed-point number.
    /// @param result The quotient as an unsigned 60.18-decimal fixed-point number.
    function div(uint256 x, uint256 y) internal pure returns (uint256 result) {
        result = PRBMath.mulDiv(x, SCALE, y);
    }

    /// @notice Returns Euler's number as an unsigned 60.18-decimal fixed-point number.
    /// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
    function e() internal pure returns (uint256 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.
    ///
    /// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp(uint256 x) internal pure returns (uint256 result) {
        // Without this check, the value passed to "exp2" would be greater than 192.
        if (x >= 133_084258667509499441) {
            revert PRBMathUD60x18__ExpInputTooBig(x);
        }

        // Do the fixed-point multiplication inline to save gas.
        unchecked {
            uint256 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_UD60x18.
    ///
    /// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp2(uint256 x) internal pure returns (uint256 result) {
        // 2^192 doesn't fit within the 192.64-bit format used internally in this function.
        if (x >= 192e18) {
            revert PRBMathUD60x18__Exp2InputTooBig(x);
        }

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

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

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

            // Equivalent to "x - remainder * (remainder > 0 ? 1 : 0)" but faster.
            result := sub(x, mul(remainder, gt(remainder, 0)))
        }
    }

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

    /// @notice Converts a number from basic integer form to unsigned 60.18-decimal fixed-point representation.
    ///
    /// @dev Requirements:
    /// - x must be less than or equal to MAX_UD60x18 divided by SCALE.
    ///
    /// @param x The basic integer to convert.
    /// @param result The same number in unsigned 60.18-decimal fixed-point representation.
    function fromUint(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            if (x > MAX_UD60x18 / SCALE) {
                revert PRBMathUD60x18__FromUintOverflow(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_UD60x18, lest it overflows.
    ///
    /// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function gm(uint256 x, uint256 y) internal pure returns (uint256 result) {
        if (x == 0) {
            return 0;
        }

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

    /// @notice Calculates 1 / x, rounding toward zero.
    ///
    /// @dev Requirements:
    /// - x cannot be zero.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the inverse.
    /// @return result The inverse as an unsigned 60.18-decimal fixed-point number.
    function inv(uint256 x) internal pure returns (uint256 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 2.718281828459045235, for that we would need more fine-grained precision.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the natural logarithm.
    /// @return result The natural logarithm as an unsigned 60.18-decimal fixed-point number.
    function ln(uint256 x) internal pure returns (uint256 result) {
        // Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
        // can return is 196205294292027477728.
        unchecked {
            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 unsigned 60.18-decimal fixed-point number for which to calculate the common logarithm.
    /// @return result The common logarithm as an unsigned 60.18-decimal fixed-point number.
    function log10(uint256 x) internal pure returns (uint256 result) {
        if (x < SCALE) {
            revert PRBMathUD60x18__LogInputTooSmall(x);
        }

        // Note that the "mul" in this block is the assembly multiplication 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) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 59) }
            default {
                result := MAX_UD60x18
            }
        }

        if (result == MAX_UD60x18) {
            // 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 or equal to SCALE, otherwise the result would be negative.
    ///
    /// Caveats:
    /// - The results are nor perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the binary logarithm.
    /// @return result The binary logarithm as an unsigned 60.18-decimal fixed-point number.
    function log2(uint256 x) internal pure returns (uint256 result) {
        if (x < SCALE) {
            revert PRBMathUD60x18__LogInputTooSmall(x);
        }
        unchecked {
            // Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
            uint256 n = PRBMath.mostSignificantBit(x / SCALE);

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

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

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

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

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

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

    /// @notice Multiplies two unsigned 60.18-decimal fixed-point numbers together, returning a new unsigned 60.18-decimal
    /// fixed-point number.
    /// @dev See the documentation for the "PRBMath.mulDivFixedPoint" function.
    /// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
    /// @return result The product as an unsigned 60.18-decimal fixed-point number.
    function mul(uint256 x, uint256 y) internal pure returns (uint256 result) {
        result = PRBMath.mulDivFixedPoint(x, y);
    }

    /// @notice Returns PI as an unsigned 60.18-decimal fixed-point number.
    function pi() internal pure returns (uint256 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".
    ///
    /// Caveats:
    /// - All from "exp2", "log2" and "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x Number to raise to given power y, as an unsigned 60.18-decimal fixed-point number.
    /// @param y Exponent to raise x to, as an unsigned 60.18-decimal fixed-point number.
    /// @return result x raised to power y, as an unsigned 60.18-decimal fixed-point number.
    function pow(uint256 x, uint256 y) internal pure returns (uint256 result) {
        if (x == 0) {
            result = y == 0 ? SCALE : uint256(0);
        } else {
            result = exp2(mul(log2(x), y));
        }
    }

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

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

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

    /// @notice Returns 1 as an unsigned 60.18-decimal fixed-point number.
    function scale() internal pure returns (uint256 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 must be less than MAX_UD60x18 / SCALE.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the square root.
    /// @return result The result as an unsigned 60.18-decimal fixed-point .
    function sqrt(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            if (x > MAX_UD60x18 / SCALE) {
                revert PRBMathUD60x18__SqrtOverflow(x);
            }
            // Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two unsigned
            // 60.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
            result = PRBMath.sqrt(x * SCALE);
        }
    }

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