Math
Author: Axicon Labs Limited
Contains general math helpers and functions
State Variables
MAX_UINT256
This is equivalent to type(uint256).max — used in assembly blocks as a replacement.
uint256 internal constant MAX_UINT256 = 2 ** 256 - 1;
MAX_UINT128
This is equivalent to type(uint128).max — used in assembly blocks as a replacement.
uint256 internal constant MAX_UINT128 = 2 ** 128 - 1;
Functions
min24
Compute the min of the incoming int24s a and b.
function min24(int24 a, int24 b) internal pure returns (int24);
Parameters
| Name | Type | Description |
|---|---|---|
a | int24 | The first number |
b | int24 | The second number |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int24 | The min of a and b: min(a, b), e.g.: min(4, 1) = 1 |
max24
Compute the max of the incoming int24s a and b.
function max24(int24 a, int24 b) internal pure returns (int24);
Parameters
| Name | Type | Description |
|---|---|---|
a | int24 | The first number |
b | int24 | The second number |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int24 | The max of a and b: max(a, b), e.g.: max(4, 1) = 4 |
min
Compute the min of the incoming a and b.
function min(uint256 a, uint256 b) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The first number |
b | uint256 | The second number |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The min of a and b: min(a, b), e.g.: min(4, 1) = 1 |
min
Compute the min of the incoming a and b.
function min(int256 a, int256 b) internal pure returns (int256);
Parameters
| Name | Type | Description |
|---|---|---|
a | int256 | The first number |
b | int256 | The second number |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int256 | The min of a and b: min(a, b), e.g.: min(4, 1) = 1 |
max
Compute the max of the incoming a and b.
function max(uint256 a, uint256 b) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The first number |
b | uint256 | The second number |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The max of a and b: max(a, b), e.g.: max(4, 1) = 4 |
max
Compute the max of the incoming a and b.
function max(int256 a, int256 b) internal pure returns (int256);
Parameters
| Name | Type | Description |
|---|---|---|
a | int256 | The first number |
b | int256 | The second number |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int256 | The max of a and b: max(a, b), e.g.: max(4, 1) = 4 |
abs
Compute the absolute value of an integer (int256).
Does not support type(int256).min and will revert (type(int256).max = abs(type(int256).min) - 1).
function abs(int256 x) internal pure returns (int256);
Parameters
| Name | Type | Description |
|---|---|---|
x | int256 | The incoming signed integer to take the absolute value of |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int256 | The absolute value of x, e.g. abs(-4) = 4 |
absUint
Compute the absolute value of an integer (int256).
Supports type(int256).min because the corresponding value can fit in a uint (unlike type(int256).max).
function absUint(int256 x) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
x | int256 | The incoming signed integer to take the absolute value of |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The absolute value of x, e.g. abs(-4) = 4 |
mostSignificantNibble
Returns the index of the most significant nibble of the 160-bit number, where the least significant nibble is at index 0 and the most significant nibble is at index 39.
function mostSignificantNibble(uint160 x) internal pure returns (uint256 r);
Parameters
| Name | Type | Description |
|---|---|---|
x | uint160 | The value for which to compute the most significant nibble |
Returns
| Name | Type | Description |
|---|---|---|
r | uint256 | The index of the most significant nibble (default: 0) |
getApproxTickWithMaxAmount
Computes a tick that will require approximately amount of currency0 to create a tickSpacing-wide position with maxLiquidityPerTick at tickLower = tick in Uniswap.
This function can have a maximum of two ticks of error from one of the ticks with amount(tickA) < amount < amount(tickA + 1 = tickB).
`tickSpacing is assumed to be within the range (0, 32768)
maxLiquidityPerTick for s=tickSpacing should be defined by (2^128 - 1) / ((887272/s) - (-887272/s) + 1)
function getApproxTickWithMaxAmount(uint256 amount, int24 tickSpacing, uint256 maxLiquidityPerTick)
internal
pure
returns (int24);
Parameters
| Name | Type | Description |
|---|---|---|
amount | uint256 | The desired amount of currency0 required to fill the returned tick |
tickSpacing | int24 | The spacing between initializable ticks in the Uniswap pool |
maxLiquidityPerTick | uint256 | The maximum liquidity that can reference any given tick in the Uniswap pool |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int24 | A tick that will require approximately amount of currency0 to create a tickSpacing-wide position with maxLiquidityPerTick at tickLower = tick |
getMaxLiquidityPerTick
Computes the maximum liquidity that is allowed to reference any given tick in a Uniswap V4 pool with tickSpacing.
function getMaxLiquidityPerTick(int24 tickSpacing) internal pure returns (uint128 maxLiquidityPerTick);
Parameters
| Name | Type | Description |
|---|---|---|
tickSpacing | int24 | The spacing between initializable ticks in the Uniswap V4 pool |
Returns
| Name | Type | Description |
|---|---|---|
maxLiquidityPerTick | uint128 | The maximum liquidity that can reference any given tick in the Uniswap V4 pool |
getSqrtRatioAtTick
Calculates 1.0001^(tick/2) as an X96 number.
Will revert if abs(tick) > 887272.
function getSqrtRatioAtTick(int24 tick) internal pure returns (uint160);
Parameters
| Name | Type | Description |
|---|---|---|
tick | int24 | Value of the tick for which sqrt(1.0001^tick) is calculated |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint160 | A Q64.96 number representing the sqrt price at the provided tick |
log_Sqrt1p0001
Approximates the log base sqrt(1.0001) of argX128/2^128 with precision bits of precision.
Validated for argX128 in the range [18447437466114719744, 6276865796315986613307619852238232712866172378830163935232).
Validated for precision in the range [0, 20].
function log_Sqrt1p0001(uint256 argX128, uint256 precision) internal pure returns (int256);
Parameters
| Name | Type | Description |
|---|---|---|
argX128 | uint256 | The Q128.128 fixed-point number to calculate the log of |
precision | uint256 | The bits of precision with which to compute the result, max 64 (err <≈ 2^-precision * log₂(√1.0001)⁻¹) |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int256 | The log with base sqrt(1.0001) of argX128/2^128 |
getAmount0ForLiquidity
Calculates the amount of currency0 received for a given LiquidityChunk.
function getAmount0ForLiquidity(LiquidityChunk liquidityChunk) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
liquidityChunk | LiquidityChunk | A specification for a liquidity chunk in Uniswap containing liquidity, tickLower, and tickUpper |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The amount of currency0 represented by liquidityChunk when currentTick < tickLower |
getAmount1ForLiquidity
Calculates the amount of currency1 received for a given LiquidityChunk.
function getAmount1ForLiquidity(LiquidityChunk liquidityChunk) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
liquidityChunk | LiquidityChunk | A specification for a liquidity chunk in Uniswap containing liquidity, tickLower, and tickUpper |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The amount of currency1 represented by liquidityChunk when currentTick > tickUpper |
getAmountsForLiquidity
Calculates the amount of currency0 and currency1 received for a given LiquidityChunk at the provided currentTick.
function getAmountsForLiquidity(int24 currentTick, LiquidityChunk liquidityChunk)
internal
pure
returns (uint256 amount0, uint256 amount1);
Parameters
| Name | Type | Description |
|---|---|---|
currentTick | int24 | The tick at which to evaluate liquidityChunk |
liquidityChunk | LiquidityChunk | A specification for a liquidity chunk in Uniswap containing liquidity, tickLower, and tickUpper |
Returns
| Name | Type | Description |
|---|---|---|
amount0 | uint256 | The amount of currency0 represented by liquidityChunk at currentTick |
amount1 | uint256 | The amount of currency1 represented by liquidityChunk at currentTick |
getLiquidityForAmount0
Returns a LiquidityChunk at the provided tick range with liquidity corresponding to amount0.
function getLiquidityForAmount0(int24 tickLower, int24 tickUpper, uint256 amount0)
internal
pure
returns (LiquidityChunk);
Parameters
| Name | Type | Description |
|---|---|---|
tickLower | int24 | The lower tick of the chunk |
tickUpper | int24 | The upper tick of the chunk |
amount0 | uint256 | The amount of currency0 |
Returns
| Name | Type | Description |
|---|---|---|
<none> | LiquidityChunk | A LiquidityChunk with tickLower, tickUpper, and the calculated amount of liquidity for amount0 |
getLiquidityForAmount1
Returns a LiquidityChunk at the provided tick range with liquidity corresponding to amount1.
function getLiquidityForAmount1(int24 tickLower, int24 tickUpper, uint256 amount1)
internal
pure
returns (LiquidityChunk);
Parameters
| Name | Type | Description |
|---|---|---|
tickLower | int24 | The lower tick of the chunk |
tickUpper | int24 | The upper tick of the chunk |
amount1 | uint256 | The amount of currency1 |
Returns
| Name | Type | Description |
|---|---|---|
<none> | LiquidityChunk | A LiquidityChunk with tickLower, tickUpper, and the calculated amount of liquidity for amount1 |
toUint128
Downcast uint256 to uint128. Revert on overflow or underflow.
function toUint128(uint256 toDowncast) internal pure returns (uint128 downcastedInt);
Parameters
| Name | Type | Description |
|---|---|---|
toDowncast | uint256 | The uint256 to be downcasted |
Returns
| Name | Type | Description |
|---|---|---|
downcastedInt | uint128 | toDowncast downcasted to uint128 |
toUint128Capped
Downcast uint256 to uint128, but cap at type(uint128).max on overflow.
function toUint128Capped(uint256 toDowncast) internal pure returns (uint128 downcastedInt);
Parameters
| Name | Type | Description |
|---|---|---|
toDowncast | uint256 | The uint256 to be downcasted |
Returns
| Name | Type | Description |
|---|---|---|
downcastedInt | uint128 | toDowncast downcasted to uint128 |
toInt128
Downcast uint128 to int128.
function toInt128(uint128 toCast) internal pure returns (int128 downcastedInt);
Parameters
| Name | Type | Description |
|---|---|---|
toCast | uint128 | The uint256 to be downcasted |
Returns
| Name | Type | Description |
|---|---|---|
downcastedInt | int128 | toDowncast downcasted to int128 |
toInt128
Cast an int256 to an int128, revert on overflow or underflow.
function toInt128(int256 toCast) internal pure returns (int128 downcastedInt);
Parameters
| Name | Type | Description |
|---|---|---|
toCast | int256 | the int256 to be downcasted |
Returns
| Name | Type | Description |
|---|---|---|
downcastedInt | int128 | toCast downcasted to int128 |
toInt256
Cast a uint256 to an int256, revert on overflow.
function toInt256(uint256 toCast) internal pure returns (int256);
Parameters
| Name | Type | Description |
|---|---|---|
toCast | uint256 | The uint256 to be downcasted |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int256 | toCast downcasted to int256 |
mulDiv
Calculates floor(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0.
Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv for this and all following mulDiv functions
function mulDiv(uint256 a, uint256 b, uint256 denominator) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
denominator | uint256 | The divisor |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
mulDivCapped
Calculates min(floor(a×b÷denominator), 2^256-1) with full precision.
function mulDivCapped(uint256 a, uint256 b, uint256 denominator) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
denominator | uint256 | The divisor |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
mulDivRoundingUp
Calculates ceil(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0.
function mulDivRoundingUp(uint256 a, uint256 b, uint256 denominator) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
denominator | uint256 | The divisor |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
mulDiv64
Calculates floor(a×b÷2^64) with full precision. Throws if result overflows a uint256.
function mulDiv64(uint256 a, uint256 b) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The 256-bit result |
mulDiv96
Calculates floor(a×b÷2^96) with full precision. Throws if result overflows a uint256.
function mulDiv96(uint256 a, uint256 b) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The 256-bit result |
mulDiv96RoundingUp
Calculates ceil(a×b÷2^96) with full precision. Throws if result overflows a uint256.
function mulDiv96RoundingUp(uint256 a, uint256 b) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
mulDiv128
Calculates floor(a×b÷2^128) with full precision. Throws if result overflows a uint256.
function mulDiv128(uint256 a, uint256 b) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The 256-bit result |
mulDiv128RoundingUp
Calculates ceil(a×b÷2^128) with full precision. Throws if result overflows a uint256.
function mulDiv128RoundingUp(uint256 a, uint256 b) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
mulDiv192
Calculates floor(a×b÷2^192) with full precision. Throws if result overflows a uint256.
function mulDiv192(uint256 a, uint256 b) internal pure returns (uint256);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
<none> | uint256 | The 256-bit result |
mulDiv192RoundingUp
Calculates ceil(a×b÷2^192) with full precision.
function mulDiv192RoundingUp(uint256 a, uint256 b) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The multiplicand |
b | uint256 | The multiplier |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
unsafeDivRoundingUp
Calculates ceil(a÷b), returning 0 if b == 0.
function unsafeDivRoundingUp(uint256 a, uint256 b) internal pure returns (uint256 result);
Parameters
| Name | Type | Description |
|---|---|---|
a | uint256 | The numerator |
b | uint256 | The denominator |
Returns
| Name | Type | Description |
|---|---|---|
result | uint256 | The 256-bit result |
quickSort
QuickSort is a sorting algorithm that employs the Divide and Conquer strategy. It selects a pivot element and arranges the given array around this pivot by correctly positioning it within the sorted array.
function quickSort(int256[] memory arr, int256 left, int256 right) internal pure;
Parameters
| Name | Type | Description |
|---|---|---|
arr | int256[] | The elements that must be sorted |
left | int256 | The starting index |
right | int256 | The ending index |
sort
Calls quickSort with default starting index of 0 and ending index of the last element in the array.
function sort(int256[] memory data) internal pure returns (int256[] memory);
Parameters
| Name | Type | Description |
|---|---|---|
data | int256[] | The elements that must be sorted |
Returns
| Name | Type | Description |
|---|---|---|
<none> | int256[] | The sorted array |