Contract Name:
DataStorageOperator
Contract Source Code:
// SPDX-License-Identifier: BUSL-1.1
pragma solidity =0.7.6;
pragma abicoder v2;
import './interfaces/IAlgebraFactory.sol';
import './interfaces/IDataStorageOperator.sol';
import './libraries/DataStorage.sol';
import './libraries/Sqrt.sol';
import './libraries/AdaptiveFee.sol';
import './libraries/Constants.sol';
/// @title Algebra timepoints data operator
/// @notice This contract stores timepoints and calculates adaptive fee and statistical averages
contract DataStorageOperator is IDataStorageOperator {
uint256 constant UINT16_MODULO = 65536;
uint128 constant MAX_VOLUME_PER_LIQUIDITY = 100000 << 64; // maximum meaningful ratio of volume to liquidity
using DataStorage for DataStorage.Timepoint[UINT16_MODULO];
DataStorage.Timepoint[UINT16_MODULO] public override timepoints;
AdaptiveFee.Configuration public feeConfigZto;
AdaptiveFee.Configuration public feeConfigOtz;
address private immutable pool;
address private immutable factory;
modifier onlyPool() {
require(msg.sender == pool, 'only pool can call this');
_;
}
constructor(address _pool) {
factory = msg.sender;
pool = _pool;
}
/// @inheritdoc IDataStorageOperator
function initialize(uint32 time, int24 tick) external override onlyPool {
return timepoints.initialize(time, tick);
}
/// @inheritdoc IDataStorageOperator
function changeFeeConfiguration(bool zto, AdaptiveFee.Configuration calldata _feeConfig) external override {
require(msg.sender == factory || msg.sender == IAlgebraFactory(factory).owner());
require(uint256(_feeConfig.alpha1) + uint256(_feeConfig.alpha2) + uint256(_feeConfig.baseFee) <= type(uint16).max, 'Max fee exceeded');
require(_feeConfig.gamma1 != 0 && _feeConfig.gamma2 != 0 && _feeConfig.volumeGamma != 0, 'Gammas must be > 0');
if (zto) feeConfigZto = _feeConfig;
else feeConfigOtz = _feeConfig;
emit FeeConfiguration(zto, _feeConfig);
}
/// @inheritdoc IDataStorageOperator
function getSingleTimepoint(
uint32 time,
uint32 secondsAgo,
int24 tick,
uint16 index,
uint128 liquidity
)
external
view
override
onlyPool
returns (int56 tickCumulative, uint160 secondsPerLiquidityCumulative, uint112 volatilityCumulative, uint256 volumePerAvgLiquidity)
{
uint16 oldestIndex;
// check if we have overflow in the past
uint16 nextIndex = index + 1; // considering overflow
if (timepoints[nextIndex].initialized) {
oldestIndex = nextIndex;
}
DataStorage.Timepoint memory result = timepoints.getSingleTimepoint(time, secondsAgo, tick, index, oldestIndex, liquidity);
(tickCumulative, secondsPerLiquidityCumulative, volatilityCumulative, volumePerAvgLiquidity) = (
result.tickCumulative,
result.secondsPerLiquidityCumulative,
result.volatilityCumulative,
result.volumePerLiquidityCumulative
);
}
/// @inheritdoc IDataStorageOperator
function getTimepoints(
uint32 time,
uint32[] memory secondsAgos,
int24 tick,
uint16 index,
uint128 liquidity
)
external
view
override
onlyPool
returns (
int56[] memory tickCumulatives,
uint160[] memory secondsPerLiquidityCumulatives,
uint112[] memory volatilityCumulatives,
uint256[] memory volumePerAvgLiquiditys
)
{
return timepoints.getTimepoints(time, secondsAgos, tick, index, liquidity);
}
/// @inheritdoc IDataStorageOperator
function getAverages(
uint32 time,
int24 tick,
uint16 index,
uint128 liquidity
) external view override onlyPool returns (uint112 TWVolatilityAverage, uint256 TWVolumePerLiqAverage) {
return timepoints.getAverages(time, tick, index, liquidity);
}
/// @inheritdoc IDataStorageOperator
function write(
uint16 index,
uint32 blockTimestamp,
int24 tick,
uint128 liquidity,
uint128 volumePerLiquidity
) external override onlyPool returns (uint16 indexUpdated) {
return timepoints.write(index, blockTimestamp, tick, liquidity, volumePerLiquidity);
}
/// @inheritdoc IDataStorageOperator
function calculateVolumePerLiquidity(
uint128 liquidity,
int256 amount0,
int256 amount1
) external pure override returns (uint128 volumePerLiquidity) {
uint256 volume = Sqrt.sqrtAbs(amount0) * Sqrt.sqrtAbs(amount1);
uint256 volumeShifted;
if (volume >= 2 ** 192) volumeShifted = (type(uint256).max) / (liquidity > 0 ? liquidity : 1);
else volumeShifted = (volume << 64) / (liquidity > 0 ? liquidity : 1);
if (volumeShifted >= MAX_VOLUME_PER_LIQUIDITY) return MAX_VOLUME_PER_LIQUIDITY;
else return uint128(volumeShifted);
}
/// @inheritdoc IDataStorageOperator
function window() external pure override returns (uint32) {
return DataStorage.WINDOW;
}
/// @inheritdoc IDataStorageOperator
function getFees(
uint32 _time,
int24 _tick,
uint16 _index,
uint128 _liquidity
) external view override onlyPool returns (uint16 feeZto, uint16 feeOtz) {
(uint88 volatilityAverage, uint256 volumePerLiqAverage) = timepoints.getAverages(_time, _tick, _index, _liquidity);
feeZto = AdaptiveFee.getFee(volatilityAverage / 15, volumePerLiqAverage, feeConfigZto);
feeOtz = AdaptiveFee.getFee(volatilityAverage / 15, volumePerLiqAverage, feeConfigOtz);
}
}
// SPDX-License-Identifier: GPL-2.0-or-later
pragma solidity >=0.5.0;
/**
* @title The interface for the Algebra Factory
* @dev Credit to Uniswap Labs under GPL-2.0-or-later license:
* https://github.com/Uniswap/v3-core/tree/main/contracts/interfaces
*/
interface IAlgebraFactory {
/**
* @notice Emitted when the owner of the factory is changed
* @param newOwner The owner after the owner was changed
*/
event Owner(address indexed newOwner);
/**
* @notice Emitted when the vault address is changed
* @param newVaultAddress The vault address after the address was changed
*/
event VaultAddress(address indexed newVaultAddress);
/**
* @notice Emitted when a pool is created
* @param token0 The first token of the pool by address sort order
* @param token1 The second token of the pool by address sort order
* @param pool The address of the created pool
*/
event Pool(address indexed token0, address indexed token1, address pool);
/**
* @notice Emitted when the farming address is changed
* @param newFarmingAddress The farming address after the address was changed
*/
event FarmingAddress(address indexed newFarmingAddress);
/**
* @notice Emitted when the default community fee is changed
* @param newDefaultCommunityFee The new default community fee value
*/
event DefaultCommunityFee(uint8 newDefaultCommunityFee);
event FeeConfiguration(
uint16 alpha1,
uint16 alpha2,
uint32 beta1,
uint32 beta2,
uint16 gamma1,
uint16 gamma2,
uint32 volumeBeta,
uint16 volumeGamma,
uint16 baseFee
);
/**
* @notice Returns the current owner of the factory
* @dev Can be changed by the current owner via setOwner
* @return The address of the factory owner
*/
function owner() external view returns (address);
/**
* @notice Returns the current poolDeployerAddress
* @return The address of the poolDeployer
*/
function poolDeployer() external view returns (address);
/**
* @dev Is retrieved from the pools to restrict calling
* certain functions not by a tokenomics contract
* @return The tokenomics contract address
*/
function farmingAddress() external view returns (address);
/**
* @notice Returns the default community fee
* @return Fee which will be set at the creation of the pool
*/
function defaultCommunityFee() external view returns (uint8);
function vaultAddress() external view returns (address);
/**
* @notice Returns the pool address for a given pair of tokens and a fee, or address 0 if it does not exist
* @dev tokenA and tokenB may be passed in either token0/token1 or token1/token0 order
* @param tokenA The contract address of either token0 or token1
* @param tokenB The contract address of the other token
* @return pool The pool address
*/
function poolByPair(address tokenA, address tokenB) external view returns (address pool);
/**
* @notice Creates a pool for the given two tokens and fee
* @param tokenA One of the two tokens in the desired pool
* @param tokenB The other of the two tokens in the desired pool
* @dev tokenA and tokenB may be passed in either order: token0/token1 or token1/token0. tickSpacing is retrieved
* from the fee. The call will revert if the pool already exists, the fee is invalid, or the token arguments
* are invalid.
* @return pool The address of the newly created pool
*/
function createPool(address tokenA, address tokenB) external returns (address pool);
/**
* @notice Updates the owner of the factory
* @dev Must be called by the current owner
* @param _owner The new owner of the factory
*/
function setOwner(address _owner) external;
/**
* @dev updates tokenomics address on the factory
* @param _farmingAddress The new tokenomics contract address
*/
function setFarmingAddress(address _farmingAddress) external;
/**
* @dev updates default community fee for new pools
* @param newDefaultCommunityFee The new community fee, _must_ be <= MAX_COMMUNITY_FEE
*/
function setDefaultCommunityFee(uint8 newDefaultCommunityFee) external;
/**
* @dev updates vault address on the factory
* @param _vaultAddress The new vault contract address
*/
function setVaultAddress(address _vaultAddress) external;
/**
* @notice Changes initial fee configuration for new pools
* @dev changes coefficients for sigmoids: α / (1 + e^( (β-x) / γ))
* alpha1 + alpha2 + baseFee (max possible fee) must be <= type(uint16).max
* gammas must be > 0
* @param alpha1 max value of the first sigmoid
* @param alpha2 max value of the second sigmoid
* @param beta1 shift along the x-axis for the first sigmoid
* @param beta2 shift along the x-axis for the second sigmoid
* @param gamma1 horizontal stretch factor for the first sigmoid
* @param gamma2 horizontal stretch factor for the second sigmoid
* @param volumeBeta shift along the x-axis for the outer volume-sigmoid
* @param volumeGamma horizontal stretch factor the outer volume-sigmoid
* @param baseFee minimum possible fee
*/
function setBaseFeeConfiguration(
uint16 alpha1,
uint16 alpha2,
uint32 beta1,
uint32 beta2,
uint16 gamma1,
uint16 gamma2,
uint32 volumeBeta,
uint16 volumeGamma,
uint16 baseFee
) external;
}
// SPDX-License-Identifier: GPL-2.0-or-later
pragma solidity >=0.5.0;
pragma abicoder v2;
import '../libraries/AdaptiveFee.sol';
interface IDataStorageOperator {
event FeeConfiguration(bool zto, AdaptiveFee.Configuration feeConfig);
/**
* @notice Returns data belonging to a certain timepoint
* @param index The index of timepoint in the array
* @dev There is more convenient function to fetch a timepoint: getTimepoints(). Which requires not an index but seconds
* @return initialized Whether the timepoint has been initialized and the values are safe to use,
* blockTimestamp The timestamp of the observation,
* tickCumulative The tick multiplied by seconds elapsed for the life of the pool as of the timepoint timestamp,
* secondsPerLiquidityCumulative The seconds per in range liquidity for the life of the pool as of the timepoint timestamp,
* volatilityCumulative Cumulative standard deviation for the life of the pool as of the timepoint timestamp,
* averageTick Time-weighted average tick,
* volumePerLiquidityCumulative Cumulative swap volume per liquidity for the life of the pool as of the timepoint timestamp
*/
function timepoints(uint256 index)
external
view
returns (
bool initialized,
uint32 blockTimestamp,
int56 tickCumulative,
uint160 secondsPerLiquidityCumulative,
uint88 volatilityCumulative,
int24 averageTick,
uint144 volumePerLiquidityCumulative
);
/// @notice Initialize the dataStorage array by writing the first slot. Called once for the lifecycle of the timepoints array
/// @param time The time of the dataStorage initialization, via block.timestamp truncated to uint32
/// @param tick Initial tick
function initialize(uint32 time, int24 tick) external;
/// @dev Reverts if an timepoint at or before the desired timepoint timestamp does not exist.
/// 0 may be passed as `secondsAgo' to return the current cumulative values.
/// If called with a timestamp falling between two timepoints, returns the counterfactual accumulator values
/// at exactly the timestamp between the two timepoints.
/// @param time The current block timestamp
/// @param secondsAgo The amount of time to look back, in seconds, at which point to return an timepoint
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param liquidity The current in-range pool liquidity
/// @return tickCumulative The cumulative tick since the pool was first initialized, as of `secondsAgo`
/// @return secondsPerLiquidityCumulative The cumulative seconds / max(1, liquidity) since the pool was first initialized, as of `secondsAgo`
/// @return volatilityCumulative The cumulative volatility value since the pool was first initialized, as of `secondsAgo`
/// @return volumePerAvgLiquidity The cumulative volume per liquidity value since the pool was first initialized, as of `secondsAgo`
function getSingleTimepoint(
uint32 time,
uint32 secondsAgo,
int24 tick,
uint16 index,
uint128 liquidity
)
external
view
returns (
int56 tickCumulative,
uint160 secondsPerLiquidityCumulative,
uint112 volatilityCumulative,
uint256 volumePerAvgLiquidity
);
/// @notice Returns the accumulator values as of each time seconds ago from the given time in the array of `secondsAgos`
/// @dev Reverts if `secondsAgos` > oldest timepoint
/// @param time The current block.timestamp
/// @param secondsAgos Each amount of time to look back, in seconds, at which point to return an timepoint
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param liquidity The current in-range pool liquidity
/// @return tickCumulatives The cumulative tick since the pool was first initialized, as of each `secondsAgo`
/// @return secondsPerLiquidityCumulatives The cumulative seconds / max(1, liquidity) since the pool was first initialized, as of each `secondsAgo`
/// @return volatilityCumulatives The cumulative volatility values since the pool was first initialized, as of each `secondsAgo`
/// @return volumePerAvgLiquiditys The cumulative volume per liquidity values since the pool was first initialized, as of each `secondsAgo`
function getTimepoints(
uint32 time,
uint32[] memory secondsAgos,
int24 tick,
uint16 index,
uint128 liquidity
)
external
view
returns (
int56[] memory tickCumulatives,
uint160[] memory secondsPerLiquidityCumulatives,
uint112[] memory volatilityCumulatives,
uint256[] memory volumePerAvgLiquiditys
);
/// @notice Returns average volatility in the range from time-WINDOW to time
/// @param time The current block.timestamp
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param liquidity The current in-range pool liquidity
/// @return TWVolatilityAverage The average volatility in the recent range
/// @return TWVolumePerLiqAverage The average volume per liquidity in the recent range
function getAverages(
uint32 time,
int24 tick,
uint16 index,
uint128 liquidity
) external view returns (uint112 TWVolatilityAverage, uint256 TWVolumePerLiqAverage);
/// @notice Writes an dataStorage timepoint to the array
/// @dev Writable at most once per block. Index represents the most recently written element. index must be tracked externally.
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param blockTimestamp The timestamp of the new timepoint
/// @param tick The active tick at the time of the new timepoint
/// @param liquidity The total in-range liquidity at the time of the new timepoint
/// @param volumePerLiquidity The gmean(volumes)/liquidity at the time of the new timepoint
/// @return indexUpdated The new index of the most recently written element in the dataStorage array
function write(
uint16 index,
uint32 blockTimestamp,
int24 tick,
uint128 liquidity,
uint128 volumePerLiquidity
) external returns (uint16 indexUpdated);
/// @notice Changes fee configuration for the pool
function changeFeeConfiguration(bool zto, AdaptiveFee.Configuration calldata feeConfig) external;
/// @notice Calculates gmean(volume/liquidity) for block
/// @param liquidity The current in-range pool liquidity
/// @param amount0 Total amount of swapped token0
/// @param amount1 Total amount of swapped token1
/// @return volumePerLiquidity gmean(volume/liquidity) capped by 100000 << 64
function calculateVolumePerLiquidity(
uint128 liquidity,
int256 amount0,
int256 amount1
) external pure returns (uint128 volumePerLiquidity);
/// @return windowLength Length of window used to calculate averages
function window() external view returns (uint32 windowLength);
/// @notice Calculates fee based on combination of sigmoids
/// @param time The current block.timestamp
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param liquidity The current in-range pool liquidity
/// @return feeZto The fee for ZtO swaps in hundredths of a bip, i.e. 1e-6
/// @return feeOtz The fee for OtZ swaps in hundredths of a bip, i.e. 1e-6
function getFees(
uint32 time,
int24 tick,
uint16 index,
uint128 liquidity
) external view returns (uint16 feeZto, uint16 feeOtz);
}
// SPDX-License-Identifier: BUSL-1.1
pragma solidity =0.7.6;
import './Constants.sol';
/// @title AdaptiveFee
/// @notice Calculates fee based on combination of sigmoids
library AdaptiveFee {
// alpha1 + alpha2 + baseFee must be <= type(uint16).max
struct Configuration {
uint16 alpha1; // max value of the first sigmoid
uint16 alpha2; // max value of the second sigmoid
uint32 beta1; // shift along the x-axis for the first sigmoid
uint32 beta2; // shift along the x-axis for the second sigmoid
uint16 gamma1; // horizontal stretch factor for the first sigmoid
uint16 gamma2; // horizontal stretch factor for the second sigmoid
uint32 volumeBeta; // shift along the x-axis for the outer volume-sigmoid
uint16 volumeGamma; // horizontal stretch factor the outer volume-sigmoid
uint16 baseFee; // minimum possible fee
}
/// @notice Calculates fee based on formula:
/// baseFee + sigmoidVolume(sigmoid1(volatility, volumePerLiquidity) + sigmoid2(volatility, volumePerLiquidity))
/// maximum value capped by baseFee + alpha1 + alpha2
function getFee(
uint88 volatility,
uint256 volumePerLiquidity,
Configuration memory config
) internal pure returns (uint16 fee) {
uint256 sumOfSigmoids = sigmoid(volatility, config.gamma1, config.alpha1, config.beta1) +
sigmoid(volatility, config.gamma2, config.alpha2, config.beta2);
if (sumOfSigmoids > type(uint16).max) {
// should be impossible, just in case
sumOfSigmoids = type(uint16).max;
}
return uint16(config.baseFee + sigmoid(volumePerLiquidity, config.volumeGamma, uint16(sumOfSigmoids), config.volumeBeta)); // safe since alpha1 + alpha2 + baseFee _must_ be <= type(uint16).max
}
/// @notice calculates α / (1 + e^( (β-x) / γ))
/// that is a sigmoid with a maximum value of α, x-shifted by β, and stretched by γ
/// @dev returns uint256 for fuzzy testing. Guaranteed that the result is not greater than alpha
function sigmoid(
uint256 x,
uint16 g,
uint16 alpha,
uint256 beta
) internal pure returns (uint256 res) {
if (x > beta) {
x = x - beta;
if (x >= 6 * uint256(g)) return alpha; // so x < 19 bits
uint256 g8 = uint256(g)**8; // < 128 bits (8*16)
uint256 ex = exp(x, g, g8); // < 155 bits
res = (alpha * ex) / (g8 + ex); // in worst case: (16 + 155 bits) / 155 bits
// so res <= alpha
} else {
x = beta - x;
if (x >= 6 * uint256(g)) return 0; // so x < 19 bits
uint256 g8 = uint256(g)**8; // < 128 bits (8*16)
uint256 ex = g8 + exp(x, g, g8); // < 156 bits
res = (alpha * g8) / ex; // in worst case: (16 + 128 bits) / 156 bits
// g8 <= ex, so res <= alpha
}
}
/// @notice calculates e^(x/g) * g^8 in a series, since (around zero):
/// e^x = 1 + x + x^2/2 + ... + x^n/n! + ...
/// e^(x/g) = 1 + x/g + x^2/(2*g^2) + ... + x^(n)/(g^n * n!) + ...
function exp(
uint256 x,
uint16 g,
uint256 gHighestDegree
) internal pure returns (uint256 res) {
// calculating:
// g**8 + x * g**7 + (x**2 * g**6) / 2 + (x**3 * g**5) / 6 + (x**4 * g**4) / 24 + (x**5 * g**3) / 120 + (x**6 * g^2) / 720 + x**7 * g / 5040 + x**8 / 40320
// x**8 < 152 bits (19*8) and g**8 < 128 bits (8*16)
// so each summand < 152 bits and res < 155 bits
uint256 xLowestDegree = x;
res = gHighestDegree; // g**8
gHighestDegree /= g; // g**7
res += xLowestDegree * gHighestDegree;
gHighestDegree /= g; // g**6
xLowestDegree *= x; // x**2
res += (xLowestDegree * gHighestDegree) / 2;
gHighestDegree /= g; // g**5
xLowestDegree *= x; // x**3
res += (xLowestDegree * gHighestDegree) / 6;
gHighestDegree /= g; // g**4
xLowestDegree *= x; // x**4
res += (xLowestDegree * gHighestDegree) / 24;
gHighestDegree /= g; // g**3
xLowestDegree *= x; // x**5
res += (xLowestDegree * gHighestDegree) / 120;
gHighestDegree /= g; // g**2
xLowestDegree *= x; // x**6
res += (xLowestDegree * gHighestDegree) / 720;
xLowestDegree *= x; // x**7
res += (xLowestDegree * g) / 5040 + (xLowestDegree * x) / (40320);
}
}
// SPDX-License-Identifier: GPL-2.0-or-later
pragma solidity =0.7.6;
library Constants {
uint8 internal constant RESOLUTION = 96;
uint256 internal constant Q96 = 0x1000000000000000000000000;
uint256 internal constant Q128 = 0x100000000000000000000000000000000;
// fee value in hundredths of a bip, i.e. 1e-6
uint16 internal constant BASE_FEE = 100;
int24 internal constant MAX_TICK_SPACING = 500;
// max(uint128) / (MAX_TICK - MIN_TICK)
uint128 internal constant MAX_LIQUIDITY_PER_TICK = 191757638537527648490752896198553;
uint32 internal constant MAX_LIQUIDITY_COOLDOWN = 1 days;
uint8 internal constant MAX_COMMUNITY_FEE = 250;
uint256 internal constant COMMUNITY_FEE_DENOMINATOR = 1000;
}
// SPDX-License-Identifier: BUSL-1.1
pragma solidity =0.7.6;
import './FullMath.sol';
/// @title DataStorage
/// @notice Provides price, liquidity, volatility data useful for a wide variety of system designs
/// @dev Instances of stored dataStorage data, "timepoints", are collected in the dataStorage array
/// Timepoints are overwritten when the full length of the dataStorage array is populated.
/// The most recent timepoint is available by passing 0 to getSingleTimepoint()
library DataStorage {
uint32 public constant WINDOW = 1 days;
uint256 private constant UINT16_MODULO = 65536;
struct Timepoint {
bool initialized; // whether or not the timepoint is initialized
uint32 blockTimestamp; // the block timestamp of the timepoint
int56 tickCumulative; // the tick accumulator, i.e. tick * time elapsed since the pool was first initialized
uint160 secondsPerLiquidityCumulative; // the seconds per liquidity since the pool was first initialized
uint88 volatilityCumulative; // the volatility accumulator; overflow after ~34800 years is desired :)
int24 averageTick; // average tick at this blockTimestamp
uint144 volumePerLiquidityCumulative; // the gmean(volumes)/liquidity accumulator
}
/// @notice Calculates volatility between two sequential timepoints with resampling to 1 sec frequency
/// @param dt Timedelta between timepoints, must be within uint32 range
/// @param tick0 The tick at the left timepoint, must be within int24 range
/// @param tick1 The tick at the right timepoint, must be within int24 range
/// @param avgTick0 The average tick at the left timepoint, must be within int24 range
/// @param avgTick1 The average tick at the right timepoint, must be within int24 range
/// @return volatility The volatility between two sequential timepoints
/// If the requirements for the parameters are met, it always fits 88 bits
function _volatilityOnRange(
int256 dt,
int256 tick0,
int256 tick1,
int256 avgTick0,
int256 avgTick1
) internal pure returns (uint256 volatility) {
// On the time interval from the previous timepoint to the current
// we can represent tick and average tick change as two straight lines:
// tick = k*t + b, where k and b are some constants
// avgTick = p*t + q, where p and q are some constants
// we want to get sum of (tick(t) - avgTick(t))^2 for every t in the interval (0; dt]
// so: (tick(t) - avgTick(t))^2 = ((k*t + b) - (p*t + q))^2 = (k-p)^2 * t^2 + 2(k-p)(b-q)t + (b-q)^2
// since everything except t is a constant, we need to use progressions for t and t^2:
// sum(t) for t from 1 to dt = dt*(dt + 1)/2 = sumOfSequence
// sum(t^2) for t from 1 to dt = dt*(dt+1)*(2dt + 1)/6 = sumOfSquares
// so result will be: (k-p)^2 * sumOfSquares + 2(k-p)(b-q)*sumOfSequence + dt*(b-q)^2
int256 K = (tick1 - tick0) - (avgTick1 - avgTick0); // (k - p)*dt
int256 B = (tick0 - avgTick0) * dt; // (b - q)*dt
int256 sumOfSquares = (dt * (dt + 1) * (2 * dt + 1)); // sumOfSquares * 6
int256 sumOfSequence = (dt * (dt + 1)); // sumOfSequence * 2
volatility = uint256((K**2 * sumOfSquares + 6 * B * K * sumOfSequence + 6 * dt * B**2) / (6 * dt**2));
}
/// @notice Transforms a previous timepoint into a new timepoint, given the passage of time and the current tick and liquidity values
/// @dev blockTimestamp _must_ be chronologically equal to or greater than last.blockTimestamp, safe for 0 or 1 overflows
/// @param last The specified timepoint to be used in creation of new timepoint
/// @param blockTimestamp The timestamp of the new timepoint
/// @param tick The active tick at the time of the new timepoint
/// @param prevTick The active tick at the time of the last timepoint
/// @param liquidity The total in-range liquidity at the time of the new timepoint
/// @param averageTick The average tick at the time of the new timepoint
/// @param volumePerLiquidity The gmean(volumes)/liquidity at the time of the new timepoint
/// @return Timepoint The newly populated timepoint
function createNewTimepoint(
Timepoint memory last,
uint32 blockTimestamp,
int24 tick,
int24 prevTick,
uint128 liquidity,
int24 averageTick,
uint128 volumePerLiquidity
) private pure returns (Timepoint memory) {
uint32 delta = blockTimestamp - last.blockTimestamp;
last.initialized = true;
last.blockTimestamp = blockTimestamp;
last.tickCumulative += int56(tick) * delta;
last.secondsPerLiquidityCumulative += ((uint160(delta) << 128) / (liquidity > 0 ? liquidity : 1)); // just timedelta if liquidity == 0
last.volatilityCumulative += uint88(_volatilityOnRange(delta, prevTick, tick, last.averageTick, averageTick)); // always fits 88 bits
last.averageTick = averageTick;
last.volumePerLiquidityCumulative += volumePerLiquidity;
return last;
}
/// @notice comparator for 32-bit timestamps
/// @dev safe for 0 or 1 overflows, a and b _must_ be chronologically before or equal to currentTime
/// @param a A comparison timestamp from which to determine the relative position of `currentTime`
/// @param b From which to determine the relative position of `currentTime`
/// @param currentTime A timestamp truncated to 32 bits
/// @return res Whether `a` is chronologically <= `b`
function lteConsideringOverflow(
uint32 a,
uint32 b,
uint32 currentTime
) private pure returns (bool res) {
res = a > currentTime;
if (res == b > currentTime) res = a <= b; // if both are on the same side
}
/// @dev guaranteed that the result is within the bounds of int24
/// returns int256 for fuzzy tests
function _getAverageTick(
Timepoint[UINT16_MODULO] storage self,
uint32 time,
int24 tick,
uint16 index,
uint16 oldestIndex,
uint32 lastTimestamp,
int56 lastTickCumulative
) internal view returns (int256 avgTick) {
uint32 oldestTimestamp = self[oldestIndex].blockTimestamp;
int56 oldestTickCumulative = self[oldestIndex].tickCumulative;
if (lteConsideringOverflow(oldestTimestamp, time - WINDOW, time)) {
if (lteConsideringOverflow(lastTimestamp, time - WINDOW, time)) {
index -= 1; // considering underflow
Timepoint storage startTimepoint = self[index];
avgTick = startTimepoint.initialized
? (lastTickCumulative - startTimepoint.tickCumulative) / (lastTimestamp - startTimepoint.blockTimestamp)
: tick;
} else {
Timepoint memory startOfWindow = getSingleTimepoint(self, time, WINDOW, tick, index, oldestIndex, 0);
// current-WINDOW last current
// _________*____________*_______*_
// ||||||||||||
avgTick = (lastTickCumulative - startOfWindow.tickCumulative) / (lastTimestamp - time + WINDOW);
}
} else {
avgTick = (lastTimestamp == oldestTimestamp) ? tick : (lastTickCumulative - oldestTickCumulative) / (lastTimestamp - oldestTimestamp);
}
}
/// @notice Fetches the timepoints beforeOrAt and atOrAfter a target, i.e. where [beforeOrAt, atOrAfter] is satisfied.
/// The result may be the same timepoint, or adjacent timepoints.
/// @dev The answer must be contained in the array, used when the target is located within the stored timepoint
/// boundaries: older than the most recent timepoint and younger, or the same age as, the oldest timepoint
/// @param self The stored dataStorage array
/// @param time The current block.timestamp
/// @param target The timestamp at which the reserved timepoint should be for
/// @param lastIndex The index of the timepoint that was most recently written to the timepoints array
/// @param oldestIndex The index of the oldest timepoint in the timepoints array
/// @return beforeOrAt The timepoint recorded before, or at, the target
/// @return atOrAfter The timepoint recorded at, or after, the target
function binarySearch(
Timepoint[UINT16_MODULO] storage self,
uint32 time,
uint32 target,
uint16 lastIndex,
uint16 oldestIndex
) private view returns (Timepoint storage beforeOrAt, Timepoint storage atOrAfter) {
uint256 left = oldestIndex; // oldest timepoint
uint256 right = lastIndex >= oldestIndex ? lastIndex : lastIndex + UINT16_MODULO; // newest timepoint considering one index overflow
uint256 current = (left + right) >> 1; // "middle" point between the boundaries
do {
beforeOrAt = self[uint16(current)]; // checking the "middle" point between the boundaries
(bool initializedBefore, uint32 timestampBefore) = (beforeOrAt.initialized, beforeOrAt.blockTimestamp);
if (initializedBefore) {
if (lteConsideringOverflow(timestampBefore, target, time)) {
// is current point before or at `target`?
atOrAfter = self[uint16(current + 1)]; // checking the next point after "middle"
(bool initializedAfter, uint32 timestampAfter) = (atOrAfter.initialized, atOrAfter.blockTimestamp);
if (initializedAfter) {
if (lteConsideringOverflow(target, timestampAfter, time)) {
// is the "next" point after or at `target`?
return (beforeOrAt, atOrAfter); // the only fully correct way to finish
}
left = current + 1; // "next" point is before the `target`, so looking in the right half
} else {
// beforeOrAt is initialized and <= target, and next timepoint is uninitialized
// should be impossible if initial boundaries and `target` are correct
return (beforeOrAt, beforeOrAt);
}
} else {
right = current - 1; // current point is after the `target`, so looking in the left half
}
} else {
// we've landed on an uninitialized timepoint, keep searching higher
// should be impossible if initial boundaries and `target` are correct
left = current + 1;
}
current = (left + right) >> 1; // calculating the new "middle" point index after updating the bounds
} while (true);
atOrAfter = beforeOrAt; // code is unreachable, to suppress compiler warning
assert(false);
}
/// @dev Reverts if an timepoint at or before the desired timepoint timestamp does not exist.
/// 0 may be passed as `secondsAgo' to return the current cumulative values.
/// If called with a timestamp falling between two timepoints, returns the counterfactual accumulator values
/// at exactly the timestamp between the two timepoints.
/// @param self The stored dataStorage array
/// @param time The current block timestamp
/// @param secondsAgo The amount of time to look back, in seconds, at which point to return an timepoint
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param oldestIndex The index of the oldest timepoint
/// @param liquidity The current in-range pool liquidity
/// @return targetTimepoint desired timepoint or it's approximation
function getSingleTimepoint(
Timepoint[UINT16_MODULO] storage self,
uint32 time,
uint32 secondsAgo,
int24 tick,
uint16 index,
uint16 oldestIndex,
uint128 liquidity
) internal view returns (Timepoint memory targetTimepoint) {
uint32 target = time - secondsAgo;
// if target is newer than last timepoint
if (secondsAgo == 0 || lteConsideringOverflow(self[index].blockTimestamp, target, time)) {
Timepoint memory last = self[index];
if (last.blockTimestamp == target) {
return last;
} else {
// otherwise, we need to add new timepoint
int24 avgTick = int24(_getAverageTick(self, time, tick, index, oldestIndex, last.blockTimestamp, last.tickCumulative));
int24 prevTick = tick;
{
if (index != oldestIndex) {
Timepoint memory prevLast;
Timepoint storage _prevLast = self[index - 1]; // considering index underflow
prevLast.blockTimestamp = _prevLast.blockTimestamp;
prevLast.tickCumulative = _prevLast.tickCumulative;
prevTick = int24((last.tickCumulative - prevLast.tickCumulative) / (last.blockTimestamp - prevLast.blockTimestamp));
}
}
return createNewTimepoint(last, target, tick, prevTick, liquidity, avgTick, 0);
}
}
require(lteConsideringOverflow(self[oldestIndex].blockTimestamp, target, time), 'OLD');
(Timepoint memory beforeOrAt, Timepoint memory atOrAfter) = binarySearch(self, time, target, index, oldestIndex);
if (target == atOrAfter.blockTimestamp) {
return atOrAfter; // we're at the right boundary
}
if (target != beforeOrAt.blockTimestamp) {
// we're in the middle
uint32 timepointTimeDelta = atOrAfter.blockTimestamp - beforeOrAt.blockTimestamp;
uint32 targetDelta = target - beforeOrAt.blockTimestamp;
// For gas savings the resulting point is written to beforeAt
beforeOrAt.tickCumulative += ((atOrAfter.tickCumulative - beforeOrAt.tickCumulative) / timepointTimeDelta) * targetDelta;
beforeOrAt.secondsPerLiquidityCumulative += uint160(
(uint256(atOrAfter.secondsPerLiquidityCumulative - beforeOrAt.secondsPerLiquidityCumulative) * targetDelta) / timepointTimeDelta
);
beforeOrAt.volatilityCumulative += ((atOrAfter.volatilityCumulative - beforeOrAt.volatilityCumulative) / timepointTimeDelta) * targetDelta;
beforeOrAt.volumePerLiquidityCumulative +=
((atOrAfter.volumePerLiquidityCumulative - beforeOrAt.volumePerLiquidityCumulative) / timepointTimeDelta) *
targetDelta;
}
// we're at the left boundary or at the middle
return beforeOrAt;
}
/// @notice Returns the accumulator values as of each time seconds ago from the given time in the array of `secondsAgos`
/// @dev Reverts if `secondsAgos` > oldest timepoint
/// @param self The stored dataStorage array
/// @param time The current block.timestamp
/// @param secondsAgos Each amount of time to look back, in seconds, at which point to return an timepoint
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param liquidity The current in-range pool liquidity
/// @return tickCumulatives The tick * time elapsed since the pool was first initialized, as of each `secondsAgo`
/// @return secondsPerLiquidityCumulatives The cumulative seconds / max(1, liquidity) since the pool was first initialized, as of each `secondsAgo`
/// @return volatilityCumulatives The cumulative volatility values since the pool was first initialized, as of each `secondsAgo`
/// @return volumePerAvgLiquiditys The cumulative volume per liquidity values since the pool was first initialized, as of each `secondsAgo`
function getTimepoints(
Timepoint[UINT16_MODULO] storage self,
uint32 time,
uint32[] memory secondsAgos,
int24 tick,
uint16 index,
uint128 liquidity
)
internal
view
returns (
int56[] memory tickCumulatives,
uint160[] memory secondsPerLiquidityCumulatives,
uint112[] memory volatilityCumulatives,
uint256[] memory volumePerAvgLiquiditys
)
{
tickCumulatives = new int56[](secondsAgos.length);
secondsPerLiquidityCumulatives = new uint160[](secondsAgos.length);
volatilityCumulatives = new uint112[](secondsAgos.length);
volumePerAvgLiquiditys = new uint256[](secondsAgos.length);
uint16 oldestIndex;
// check if we have overflow in the past
uint16 nextIndex = index + 1; // considering overflow
if (self[nextIndex].initialized) {
oldestIndex = nextIndex;
}
Timepoint memory current;
for (uint256 i = 0; i < secondsAgos.length; i++) {
current = getSingleTimepoint(self, time, secondsAgos[i], tick, index, oldestIndex, liquidity);
(tickCumulatives[i], secondsPerLiquidityCumulatives[i], volatilityCumulatives[i], volumePerAvgLiquiditys[i]) = (
current.tickCumulative,
current.secondsPerLiquidityCumulative,
current.volatilityCumulative,
current.volumePerLiquidityCumulative
);
}
}
/// @notice Returns average volatility in the range from time-WINDOW to time
/// @param self The stored dataStorage array
/// @param time The current block.timestamp
/// @param tick The current tick
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param liquidity The current in-range pool liquidity
/// @return volatilityAverage The average volatility in the recent range
/// @return volumePerLiqAverage The average volume per liquidity in the recent range
function getAverages(
Timepoint[UINT16_MODULO] storage self,
uint32 time,
int24 tick,
uint16 index,
uint128 liquidity
) internal view returns (uint88 volatilityAverage, uint256 volumePerLiqAverage) {
uint16 oldestIndex;
Timepoint storage oldest = self[0];
uint16 nextIndex = index + 1; // considering overflow
if (self[nextIndex].initialized) {
oldest = self[nextIndex];
oldestIndex = nextIndex;
}
Timepoint memory endOfWindow = getSingleTimepoint(self, time, 0, tick, index, oldestIndex, liquidity);
uint32 oldestTimestamp = oldest.blockTimestamp;
if (lteConsideringOverflow(oldestTimestamp, time - WINDOW, time)) {
Timepoint memory startOfWindow = getSingleTimepoint(self, time, WINDOW, tick, index, oldestIndex, liquidity);
return (
(endOfWindow.volatilityCumulative - startOfWindow.volatilityCumulative) / WINDOW,
uint256(endOfWindow.volumePerLiquidityCumulative - startOfWindow.volumePerLiquidityCumulative) >> 57
);
} else if (time != oldestTimestamp) {
uint88 _oldestVolatilityCumulative = oldest.volatilityCumulative;
uint144 _oldestVolumePerLiquidityCumulative = oldest.volumePerLiquidityCumulative;
return (
(endOfWindow.volatilityCumulative - _oldestVolatilityCumulative) / (time - oldestTimestamp),
uint256(endOfWindow.volumePerLiquidityCumulative - _oldestVolumePerLiquidityCumulative) >> 57
);
}
}
/// @notice Initialize the dataStorage array by writing the first slot. Called once for the lifecycle of the timepoints array
/// @param self The stored dataStorage array
/// @param time The time of the dataStorage initialization, via block.timestamp truncated to uint32
/// @param tick Initial tick
function initialize(
Timepoint[UINT16_MODULO] storage self,
uint32 time,
int24 tick
) internal {
require(!self[0].initialized);
self[0].initialized = true;
self[0].blockTimestamp = time;
self[0].averageTick = tick;
}
/// @notice Writes an dataStorage timepoint to the array
/// @dev Writable at most once per block. Index represents the most recently written element. index must be tracked externally.
/// @param self The stored dataStorage array
/// @param index The index of the timepoint that was most recently written to the timepoints array
/// @param blockTimestamp The timestamp of the new timepoint
/// @param tick The active tick at the time of the new timepoint
/// @param liquidity The total in-range liquidity at the time of the new timepoint
/// @param volumePerLiquidity The gmean(volumes)/liquidity at the time of the new timepoint
/// @return indexUpdated The new index of the most recently written element in the dataStorage array
function write(
Timepoint[UINT16_MODULO] storage self,
uint16 index,
uint32 blockTimestamp,
int24 tick,
uint128 liquidity,
uint128 volumePerLiquidity
) internal returns (uint16 indexUpdated) {
Timepoint storage _last = self[index];
// early return if we've already written an timepoint this block
if (_last.blockTimestamp == blockTimestamp) {
return index;
}
Timepoint memory last = _last;
// get next index considering overflow
indexUpdated = index + 1;
uint16 oldestIndex;
// check if we have overflow in the past
if (self[indexUpdated].initialized) {
oldestIndex = indexUpdated;
}
int24 avgTick = int24(_getAverageTick(self, blockTimestamp, tick, index, oldestIndex, last.blockTimestamp, last.tickCumulative));
int24 prevTick = tick;
if (index != oldestIndex) {
Timepoint storage _prevLast = self[index - 1]; // considering index underflow
uint32 _prevLastBlockTimestamp = _prevLast.blockTimestamp;
int56 _prevLastTickCumulative = _prevLast.tickCumulative;
prevTick = int24((last.tickCumulative - _prevLastTickCumulative) / (last.blockTimestamp - _prevLastBlockTimestamp));
}
self[indexUpdated] = createNewTimepoint(last, blockTimestamp, tick, prevTick, liquidity, avgTick, volumePerLiquidity);
}
}
// SPDX-License-Identifier: MIT
pragma solidity ^0.4.0 || ^0.5.0 || ^0.6.0 || ^0.7.0;
/// @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 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 = a * b; // Least significant 256 bits of the product
uint256 prod1; // Most significant 256 bits of the product
assembly {
let mm := mulmod(a, b, not(0))
prod1 := sub(sub(mm, prod0), lt(mm, prod0))
}
// Make sure the result is less than 2**256.
// Also prevents denominator == 0
require(denominator > prod1);
// Handle non-overflow cases, 256 by 256 division
if (prod1 == 0) {
assembly {
result := div(prod0, denominator)
}
return result;
}
///////////////////////////////////////////////
// 512 by 256 division.
///////////////////////////////////////////////
// Make division exact by subtracting the remainder from [prod1 prod0]
// Compute remainder using mulmod
// Subtract 256 bit remainder from 512 bit number
assembly {
let remainder := mulmod(a, b, denominator)
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;
// 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 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 * 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) {
if (a == 0 || ((result = a * b) / a == b)) {
require(denominator > 0);
assembly {
result := add(div(result, denominator), gt(mod(result, denominator), 0))
}
} else {
result = mulDiv(a, b, denominator);
if (mulmod(a, b, denominator) > 0) {
require(result < type(uint256).max);
result++;
}
}
}
/// @notice Returns ceil(x / y)
/// @dev division by 0 has unspecified behavior, and must be checked externally
/// @param x The dividend
/// @param y The divisor
/// @return z The quotient, ceil(x / y)
function divRoundingUp(uint256 x, uint256 y) internal pure returns (uint256 z) {
assembly {
z := add(div(x, y), gt(mod(x, y), 0))
}
}
}
// SPDX-License-Identifier: GPL-3.0-or-later
pragma solidity ^0.5.0 || ^0.6.0 || ^0.7.0 || ^0.8.0;
library Sqrt {
/// @notice Gets the square root of the absolute value of the parameter
function sqrtAbs(int256 _x) internal pure returns (uint256 result) {
// get abs value
int256 mask = _x >> (256 - 1);
uint256 x = uint256((_x ^ mask) - mask);
if (x == 0) result = 0;
else {
uint256 xx = x;
uint256 r = 1;
if (xx >= 0x100000000000000000000000000000000) {
xx >>= 128;
r <<= 64;
}
if (xx >= 0x10000000000000000) {
xx >>= 64;
r <<= 32;
}
if (xx >= 0x100000000) {
xx >>= 32;
r <<= 16;
}
if (xx >= 0x10000) {
xx >>= 16;
r <<= 8;
}
if (xx >= 0x100) {
xx >>= 8;
r <<= 4;
}
if (xx >= 0x10) {
xx >>= 4;
r <<= 2;
}
if (xx >= 0x8) {
r <<= 1;
}
r = (r + x / r) >> 1;
r = (r + x / r) >> 1;
r = (r + x / r) >> 1;
r = (r + x / r) >> 1;
r = (r + x / r) >> 1;
r = (r + x / r) >> 1;
r = (r + x / r) >> 1; // @dev Seven iterations should be enough.
uint256 r1 = x / r;
result = r < r1 ? r : r1;
}
}
}