Math

Title: Core math library.

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

WAD

uint256 constant WAD = 1e18

WAD_INT

int256 constant WAD_INT = int256(1e18)

LN_2_INT

ln(2).

int256 internal constant LN_2_INT = 0.693147180559945309 ether

LN_WEI_INT

ln(1e-18).

int256 internal constant LN_WEI_INT = -41.446531673892822312 ether

WEXP_UPPER_BOUND

Above this bound, wExp is clipped to avoid overflowing when multiplied with 1 ether.

This upper bound corresponds to: ln(type(int256).max / 1e36) (scaled by WAD, floored).

int256 internal constant WEXP_UPPER_BOUND = 93.859467695000404319 ether

WEXP_UPPER_VALUE

The value of wExp(WEXP_UPPER_BOUND).

int256 internal constant WEXP_UPPER_VALUE = 57716089161558943949701069502944508345128.422502756744429568 ether

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.

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.

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 token0 to create a tickSpacing-wide position with maxLiquidityPerTick at tickUpper = tick in Uniswap.

This function can have a maximum of two ticks of error from one of the ticks with amount(tickRes + 2) < amount < amount(tickRes - 2).

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 token0 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 token0 to create a `tickSpacing`-wide position with `maxLiquidityPerTick` at `tickUpper = tick`

getMaxLiquidityPerTick

Computes the maximum liquidity that is allowed to reference any given tick in a Uniswap V3 pool with tickSpacing.

function getMaxLiquidityPerTick(int24 tickSpacing) internal pure returns (uint128);

Parameters

Name	Type	Description
`tickSpacing`	`int24`	The spacing between initializable ticks in the Uniswap V3 pool

Returns

Name	Type	Description
`<none>`	`uint128`	The maximum liquidity that can reference any given tick in the Uniswap V3 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_Sqrt1p0001MantissaRect

Approximates the absolute value of log base sqrt(1.0001) for a number in (0, 1) (argX128/2^128) with precision bits of precision.

function log_Sqrt1p0001MantissaRect(uint256 argX128, uint256 precision) internal pure returns (uint256);

Parameters

Name	Type	Description
`argX128`	`uint256`	The Q128.128 fixed-point number in the range (0, 1) to calculate the log of
`precision`	`uint256`	The bits of precision with which to compute the result, max 63 (`err <≈ 2^-precision * log₂(√1.0001)⁻¹`)

Returns

Name	Type	Description
`<none>`	`uint256`	The absolute value of log with base `sqrt(1.0001)` for `argX128/2^128`

getAmount0ForLiquidityUp

Calculates the amount of token0 received for a given LiquidityChunk.

function getAmount0ForLiquidityUp(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 token0 represented by `liquidityChunk` when `currentTick < tickLower`

getAmount1ForLiquidityUp

Calculates the amount of token1 received for a given LiquidityChunk.

function getAmount1ForLiquidityUp(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 token1 represented by `liquidityChunk` when `currentTick > tickUpper`

getAmount0ForLiquidity

Calculates the amount of token0 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 token0 represented by `liquidityChunk` when `currentTick < tickLower`

getAmount1ForLiquidity

Calculates the amount of token1 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 token1 represented by `liquidityChunk` when `currentTick > tickUpper`

getAmountsForLiquidity

Calculates the amount of token0 and token1 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 token0 represented by `liquidityChunk` at `currentTick`
`amount1`	`uint256`	The amount of token1 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 token0

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 token1

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

mulDivCapped

Calculates min(floor(a×b÷denominator), 2^power-1) with full precision.

function mulDivCapped(uint256 a, uint256 b, uint256 denominator, uint256 power)
    internal
    pure
    returns (uint256 result);

Parameters

Name	Type	Description
`a`	`uint256`	The multiplicand
`b`	`uint256`	The multiplier
`denominator`	`uint256`	The divisor
`power`	`uint256`	The upper bound of the open interval representing the range of this function, given by `2^power`

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

mulDivWad

Calculates floor(a×b÷10^18) with full precision. Throws if result overflows a uint256.

Optimized version of mulDiv for WAD (10^18) denominator

function mulDivWad(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

mulDivWadRoundingUp

Calculates ceil(a×b÷10^18) with full precision.

function mulDivWadRoundingUp(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

wTaylorCompounded

Returns the sum of the first three non-zero terms of a Taylor expansion of e^(nx) - 1, to approximate a continuous compound interest rate. Source: https://github.com/morpho-org/morpho-blue/blob/main/src/libraries/MathLib.sol

function wTaylorCompounded(uint256 x, uint256 n) internal pure returns (uint256);

sTaylorCompounded

Returns the sum of the first three non-zero terms of a Taylor expansion of e^(nx), to approximate a continuous compound interest rate for a custom scale s. Source: https://github.com/morpho-org/morpho-blue/blob/main/src/libraries/MathLib.sol

function sTaylorCompounded(uint256 x, uint256 s) internal pure returns (uint256);

wMulToZero

Returns the multiplication of x by y (in WAD) rounded towards 0.

function wMulToZero(int256 x, int256 y) internal pure returns (int256);

wDivToZero

Returns the division of x by y (in WAD) rounded towards 0.

function wDivToZero(int256 x, int256 y) internal pure returns (int256);

bound

Bounds x between low and high.

Assumes that low <= high. If it is not the case it returns low.

function bound(int256 x, int256 low, int256 high) internal pure returns (int256 z);

wExp

Returns an approximation of exp.

function wExp(int256 x) internal pure returns (int256);

Math

State Variables​

MAX_UINT256​

WAD​

WAD_INT​

LN_2_INT​

LN_WEI_INT​

WEXP_UPPER_BOUND​

WEXP_UPPER_VALUE​

Functions​

min24​

max24​

min​

min​

max​

max​

abs​

absUint​

mostSignificantNibble​

getApproxTickWithMaxAmount​

getMaxLiquidityPerTick​

getSqrtRatioAtTick​

log_Sqrt1p0001MantissaRect​

getAmount0ForLiquidityUp​

getAmount1ForLiquidityUp​

getAmount0ForLiquidity​

getAmount1ForLiquidity​

getAmountsForLiquidity​

getLiquidityForAmount0​

getLiquidityForAmount1​

toUint128​

toUint128Capped​

toInt128​

toInt128​

toInt256​

mulDiv​

mulDivCapped​

mulDivCapped​

mulDivRoundingUp​

mulDiv64​

mulDiv96​

mulDiv96RoundingUp​

mulDiv128​

mulDiv128RoundingUp​

mulDiv192​

mulDiv192RoundingUp​

mulDivWad​

mulDivWadRoundingUp​

unsafeDivRoundingUp​

quickSort​

sort​

wTaylorCompounded​

sTaylorCompounded​

wMulToZero​

wDivToZero​

bound​

wExp​

State Variables

MAX_UINT256

WAD

WAD_INT

LN_2_INT

LN_WEI_INT

WEXP_UPPER_BOUND

WEXP_UPPER_VALUE

Functions

min24

max24

min

min

max

max

abs

absUint

mostSignificantNibble

getApproxTickWithMaxAmount

getMaxLiquidityPerTick

getSqrtRatioAtTick

log_Sqrt1p0001MantissaRect

getAmount0ForLiquidityUp

getAmount1ForLiquidityUp

getAmount0ForLiquidity

getAmount1ForLiquidity

getAmountsForLiquidity

getLiquidityForAmount0

getLiquidityForAmount1

toUint128

toUint128Capped

toInt128

toInt128

toInt256

mulDiv

mulDivCapped

mulDivCapped

mulDivRoundingUp

mulDiv64

mulDiv96

mulDiv96RoundingUp

mulDiv128

mulDiv128RoundingUp

mulDiv192

mulDiv192RoundingUp

mulDivWad

mulDivWadRoundingUp

unsafeDivRoundingUp

quickSort

sort

wTaylorCompounded

sTaylorCompounded

wMulToZero

wDivToZero

bound

wExp