MathLibs

Git Source Author: @oscarsernarosero @mpetersoCode55 @cirsteve current implementation is using the FixedPointMathLib library from Solady and the Solidity_Uint512 library Wrapper functions to act as an abstraction layer between Equations and the Math library we’re using.

State Variables

WAD

uint256 constant WAD = FixedPointMathLib.WAD;

Functions

divideToWAD

expects the parameters to not have been multiplied by WAD, and the result will be a WAD number Provides an abstraction layer for division between liquidity-altbc and the underlying math library

function divideToWAD(uint256 numerator, uint256 denominator) internal pure returns (uint256 result);

Parameters

Name	Type	Description
`numerator`	`uint256`	the numerator for the division
`denominator`	`uint256`	the denominator for the division

Returns

Name	Type	Description
`result`	`uint256`	of the division operation

uncheckedMultiply

expects the parameters to not have been multiplied by WAD, and the result will be a WAD number Provides an abstraction layer for multiplication between liquidity-altbc and the underlying math library

function uncheckedMultiply(uint256 x, uint256 y) internal pure returns (uint256 result);

Parameters

Name	Type	Description
`x`	`uint256`	one of the factors for the multiplication
`y`	`uint256`	the other factors for the multiplication

Returns

Name	Type	Description
`result`	`uint256`	of the multiplication operation

powWad

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

convertToWAD

Converts a raw number to a WAD number

function convertToWAD(uint256 value) internal pure returns (uint256 result);

Parameters

Name	Type	Description
`value`	`uint256`	The number to be converted

Returns

Name	Type	Description
`result`	`uint256`	resulting WAD number

convertToRaw

Converts a WAD number to a raw number

function convertToRaw(uint256 value) internal pure returns (uint256 result);

Parameters

Name	Type	Description
`value`	`uint256`	The number to be converted

Returns

Name	Type	Description
`result`	`uint256`	resulting raw number

mul256x256

Calculates the product of two uint256 Used the chinese remainder theoreme

function mul256x256(uint256 a, uint256 b) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a`	`uint256`	A uint256 representing the first factor
`b`	`uint256`	A uint256 representing the second factor

Returns

Name	Type	Description
`r0`	`uint256`	The result as an uint512. r0 contains the lower bits
`r1`	`uint256`	The higher bits of the result

mul512x256

Calculates the product of two uint512 and uint256 Used the chinese remainder theoreme

function mul512x256(uint256 a0, uint256 a1, uint256 b) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing lower bits of the first factor
`a1`	`uint256`	A uint256 representing higher bits of the first factor
`b`	`uint256`	A uint256 representing the second factor

Returns

Name	Type	Description
`r0`	`uint256`	The result as an uint512. r0 contains the lower bits
`r1`	`uint256`	The higher bits of the result

mul512x256In768

Calculates the product of a uint512 and uint256. The result is a uint768. Used the chinese remainder theoreme

function mul512x256In768(uint256 a0, uint256 a1, uint256 b)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the first factor
`a1`	`uint256`	A uint256 representing the higher bits of the first factor
`b`	`uint256`	A uint256 representing the second factor

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The higher bits of the result
`r2`	`uint256`	The highest bits of the result

mul512x512In1024

Calculates the product of two uint512. The result is a uint1024. Used the chinese remainder theoreme

function mul512x512In1024(uint256 a0, uint256 a1, uint256 b0, uint256 b1)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2, uint256 r3);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the first factor
`a1`	`uint256`	A uint256 representing the higher bits of the first factor
`b0`	`uint256`	A uint256 representing the lower bits of the second factor
`b1`	`uint256`	A uint256 representing the higher bits of the second factor

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The high bits of the result
`r2`	`uint256`	The higher bits of the result
`r3`	`uint256`	The highest bits of the result

mulMod256x256

Calculates the product and remainder of two uint256 Used the chinese remainder theoreme

function mulMod256x256(uint256 a, uint256 b, uint256 c) internal pure returns (uint256 r0, uint256 r1, uint256 r2);

Parameters

Name	Type	Description
`a`	`uint256`	A uint256 representing the first factor
`b`	`uint256`	A uint256 representing the second factor
`c`	`uint256`

Returns

Name	Type	Description
`r0`	`uint256`	The result as an uint512. r0 contains the lower bits
`r1`	`uint256`	The higher bits of the result
`r2`	`uint256`	The remainder

add512x512

Calculates the difference of two uint512

function add512x512(uint256 a0, uint256 a1, uint256 b0, uint256 b1) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the first addend
`a1`	`uint256`	A uint256 representing the higher bits of the first addend
`b0`	`uint256`	A uint256 representing the lower bits of the seccond addend
`b1`	`uint256`	A uint256 representing the higher bits of the seccond addend

Returns

Name	Type	Description
`r0`	`uint256`	The result as an uint512. r0 contains the lower bits
`r1`	`uint256`	The higher bits of the result

add768x768

Calculates the sum of two uint768. The result is a uint768.

function add768x768(uint256 a0, uint256 a1, uint256 a2, uint256 b0, uint256 b1, uint256 b2)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the first addend
`a1`	`uint256`	A uint256 representing the higher bits of the first addend
`a2`	`uint256`	A uint256 representing the highest bits of the first addend
`b0`	`uint256`	A uint256 representing the lower bits of the second addend
`b1`	`uint256`	A uint256 representing the higher bits of the second addend
`b2`	`uint256`	A uint256 representing the highest bits of the second addend

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The higher bits of the result
`r2`	`uint256`	The highest bits of the result

add1024x1024

Calculates the sum of two Uint1024. The result is a Uint1024.

function add1024x1024(uint256 a0, uint256 a1, uint256 a2, uint256 a3, uint256 b0, uint256 b1, uint256 b2, uint256 b3)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2, uint256 r3);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the first addend
`a1`	`uint256`	A uint256 representing the high bits of the first addend
`a2`	`uint256`	A uint256 representing the higher bits of the first addend
`a3`	`uint256`	A uint256 representing the highest bits of the first addend
`b0`	`uint256`	A uint256 representing the lower bits of the second addend
`b1`	`uint256`	A uint256 representing the high bits of the second addend
`b2`	`uint256`	A uint256 representing the higher bits of the second addend
`b3`	`uint256`	A uint256 representing the highest bits of the second addend

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The high bits of the result
`r2`	`uint256`	The higher bits of the result
`r3`	`uint256`	The highest bits of the result

sub512x512

Calculates the difference of two uint512

function sub512x512(uint256 a0, uint256 a1, uint256 b0, uint256 b1) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the minuend
`a1`	`uint256`	A uint256 representing the higher bits of the minuend
`b0`	`uint256`	A uint256 representing the lower bits of the subtrahend
`b1`	`uint256`	A uint256 representing the higher bits of the subtrahend

Returns

Name	Type	Description
`r0`	`uint256`	The result as an uint512. r0 contains the lower bits
`r1`	`uint256`	The higher bits of the result

sub768x768

Calculates the difference of two uint768. The result is a uint768.

function sub768x768(uint256 a0, uint256 a1, uint256 a2, uint256 b0, uint256 b1, uint256 b2)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the minuend
`a1`	`uint256`	A uint256 representing the higher bits of the minuend
`a2`	`uint256`	A uint256 representing the highest bits of the minuend
`b0`	`uint256`	A uint256 representing the lower bits of the subtrahend
`b1`	`uint256`	A uint256 representing the higher bits of the subtrahend
`b2`	`uint256`	A uint256 representing the highest bits of the subtrahend

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The higher bits of the result
`r2`	`uint256`	The highest bits of the result

sub1024x1024

Calculates the difference of two Uint1024. The result is a Uint1024.

function sub1024x1024(uint256 a0, uint256 a1, uint256 a2, uint256 a3, uint256 b0, uint256 b1, uint256 b2, uint256 b3)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2, uint256 r3);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the minuend
`a1`	`uint256`	A uint256 representing the high bits of the minuend
`a2`	`uint256`	A uint256 representing the higher bits of the minuend
`a3`	`uint256`	A uint256 representing the highest bits of the minuend
`b0`	`uint256`	A uint256 representing the lower bits of the subtrahend
`b1`	`uint256`	A uint256 representing the high bits of the subtrahend
`b2`	`uint256`	A uint256 representing the higher bits of the subtrahend
`b3`	`uint256`	A uint256 representing the highest bits of the subtrahend

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The high bits of the result
`r2`	`uint256`	The higher bits of the result
`r3`	`uint256`	The highest bits of the result

divRem512x256

Calculates the division of a 512 bit unsigned integer by a 256 bit integer. It requires the remainder to be known and the result must fit in a 256 bit integer For a detailed explaination see: https://www.researchgate.net/internalation/235765881_Efficient_long_division_via_Montgomery_multiply

function divRem512x256(uint256 a0, uint256 a1, uint256 b, uint256 rem) internal pure returns (uint256 r);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the nominator
`a1`	`uint256`	A uint256 representing the high bits of the nominator
`b`	`uint256`	A uint256 representing the denominator
`rem`	`uint256`	A uint256 representing the remainder of the devision. The algorithm is cheaper to compute if the remainder is known. The remainder often be retreived cheaply using the mulmod and addmod operations

Returns

Name	Type	Description
`r`	`uint256`	The result as an uint256. Result must have at most 256 bit

div512x256

Calculates the division of a 512 bit unsigned integer by a 256 bit integer. It requires the result to fit in a 256 bit integer For a detailed explaination see: https://www.researchgate.net/internalation/235765881_Efficient_long_division_via_Montgomery_multiply

function div512x256(uint256 a0, uint256 a1, uint256 b) internal pure returns (uint256 r);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the nominator
`a1`	`uint256`	A uint256 representing the high bits of the nominator
`b`	`uint256`	A uint256 representing the denominator

Returns

Name	Type	Description
`r`	`uint256`	The result as an uint256. Result must have at most 256 bit

sqrt256

Calculates the square root of x, rounding down Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method

function sqrt256(uint256 x) internal pure returns (uint256 s);

Parameters

Name	Type	Description
`x`	`uint256`	The uint256 number for which to calculate the square root

Returns

Name	Type	Description
`s`	`uint256`	The square root as an uint256

sqrt512

Calculates the square root of a 512 bit unsigned integer, rounding down Uses the Karatsuba Square Root method. See https://hal.inria.fr/inria-00072854/document for details

function sqrt512(uint256 a0, uint256 a1) internal pure returns (uint256 s);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the input
`a1`	`uint256`	A uint256 representing the high bits of the input

Returns

Name	Type	Description
`s`	`uint256`	The square root as an uint256. Result has at most 256 bit

gt512

x > y Tells if x is greater than y where x and y are 512 bit numbers

function gt512(uint256 x0, uint256 x1, uint256 y0, uint256 y1) internal pure returns (bool gt);

Parameters

Name	Type	Description
`x0`	`uint256`	lower bits of x
`x1`	`uint256`	higher bits of y
`y0`	`uint256`	lower bits of y
`y1`	`uint256`	higher bits of y

Returns

Name	Type	Description
`gt`	`bool`	boolean. True if x > y

eq512

x == y Tells if x is equal to y where x and y are 512 bit numbers

function eq512(uint256 x0, uint256 x1, uint256 y0, uint256 y1) internal pure returns (bool eq);

Parameters

Name	Type	Description
`x0`	`uint256`	lower bits of x
`x1`	`uint256`	higher bits of y
`y0`	`uint256`	lower bits of y
`y1`	`uint256`	higher bits of y

Returns

Name	Type	Description
`eq`	`bool`	boolean. True if x = y

ge512

x >= y Tells if x is greater or equal than y where x and y are 512 bit numbers

function ge512(uint256 x0, uint256 x1, uint256 y0, uint256 y1) internal pure returns (bool ge);

Parameters

Name	Type	Description
`x0`	`uint256`	lower bits of x
`x1`	`uint256`	higher bits of y
`y0`	`uint256`	lower bits of y
`y1`	`uint256`	higher bits of y

Returns

Name	Type	Description
`ge`	`bool`	boolean. True if x >= y

lt512

x < y Tells if x is less than y where x and y are 512 bit numbers

function lt512(uint256 x0, uint256 x1, uint256 y0, uint256 y1) internal pure returns (bool lt);

Parameters

Name	Type	Description
`x0`	`uint256`	lower bits of x
`x1`	`uint256`	higher bits of y
`y0`	`uint256`	lower bits of y
`y1`	`uint256`	higher bits of y

Returns

Name	Type	Description
`lt`	`bool`	boolean. True if x < y

lt768

Checks the minuend(a0-a2) is greater than the right operand(b0-b2) Used as an underflow/negative result indicator for subtraction methods

function lt768(uint256 a0, uint256 a1, uint256 a2, uint256 b0, uint256 b1, uint256 b2) internal pure returns (bool);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the minuend
`a1`	`uint256`	A uint256 representing the high bits of the minuend
`a2`	`uint256`	A uint256 representing the higher bits of the minuend
`b0`	`uint256`	A uint256 representing the lower bits of the subtrahend
`b1`	`uint256`	A uint256 representing the high bits of the subtrahend
`b2`	`uint256`	A uint256 representing the higher bits of the subtrahend

Returns

Name	Type	Description
`<none>`	`bool`	Returns true if there would be an underflow/negative result

lt1024

Checks the minuend(a0-a3) is greater than the right operand(b0-b3) Used as an underflow/negative result indicator for subtraction methods

function lt1024(uint256 a0, uint256 a1, uint256 a2, uint256 a3, uint256 b0, uint256 b1, uint256 b2, uint256 b3)
    internal
    pure
    returns (bool);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the minuend
`a1`	`uint256`	A uint256 representing the high bits of the minuend
`a2`	`uint256`	A uint256 representing the higher bits of the minuend
`a3`	`uint256`	A uint256 representing the highest bits of the minuend
`b0`	`uint256`	A uint256 representing the lower bits of the subtrahend
`b1`	`uint256`	A uint256 representing the high bits of the subtrahend
`b2`	`uint256`	A uint256 representing the higher bits of the subtrahend
`b3`	`uint256`	A uint256 representing the highest bits of the subtrahend

Returns

Name	Type	Description
`<none>`	`bool`	Returns true if there would be an underflow/negative result

div512ByPowerOf2

Calculates the division of a 512 bit unsigned integer by a denominator which is a power of 2. It doesn’t require the result to be a uint256. very useful if a division of a 512 is expected to be also a 512.

function div512ByPowerOf2(uint256 a0, uint256 a1, uint8 n)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 remainder);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the numerator
`a1`	`uint256`	A uint256 representing the high bits of the numerator
`n`	`uint8`	the power of 2 that the division will be carried out by (demominator = 2**n).

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The higher bits of the result
`remainder`	`uint256`	of the division

div512x256ResultIn512

this function requires a0 to be greater than 0 and less or equal than 2**254 Calculates the division of a 512-bit unsigned integer by a 256-bit uint where the result can be a 512 bit unsigned integer.

function div512x256ResultIn512(uint256 a0, uint256 a1, uint256 b) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the numerator
`a1`	`uint256`	A uint256 representing the high bits of the numerator
`b`	`uint256`	A uint256 representing the denominator

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The highre bits of the result

div512x512

Calculates the division of a 512-bit unsigned integer by a 256-bit uint where the result can be a 512 bit unsigned integer.

function div512x512(uint256 a0, uint256 a1, uint256 b0, uint256 b1) internal pure returns (uint256 result);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the numerator
`a1`	`uint256`	A uint256 representing the high bits of the numerator
`b0`	`uint256`	A uint256 representing thehigh bits of the denominator
`b1`	`uint256`	A uint256 representing the low bits of the denominator

Returns

Name	Type	Description
`result`	`uint256`	result

div768x256

Used long division Calculates the division of a uint768 by a uint256. The result is a uint768.

function div768x256(uint256 a0, uint256 a1, uint256 a2, uint256 b)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lower bits of the first factor
`a1`	`uint256`	A uint256 representing the middle bits of the first factor
`a2`	`uint256`	A uint256 representing the higher bits of the first factor
`b`	`uint256`	A uint256 representing the divisor

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The middle bits of the result
`r2`	`uint256`	The higher bits of the result

div768ByPowerOf2

Calculates the division of a 768-bit unsigned integer by a denominator which is a power of 2 less than 256.

function div768ByPowerOf2(uint256 a0, uint256 a1, uint256 a2, uint8 n)
    internal
    pure
    returns (uint256 r0, uint256 r1, uint256 r2, uint256 remainder);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the numerator
`a1`	`uint256`	A uint256 representing the middle bits of the numerator
`a2`	`uint256`	A uint256 representing the high bits of the numerator
`n`	`uint8`	the power of 2 that the division will be carried out by (demominator = 2**n).

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The middle bits of the result
`r2`	`uint256`	The higher bits of the result
`remainder`	`uint256`	of the division

divRem1024x512In512

it requires to previously know the remainder of the division Calculates the division a / b where a is a 1024-bit unsigned integer and b is a uint512.

function divRem1024x512In512(
    uint256 a0,
    uint256 a1,
    uint256 a2,
    uint256 a3,
    uint256 b0,
    uint256 b1,
    uint256 rem0,
    uint256 rem1
) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the lowest bits of a
`a1`	`uint256`	A uint256 representing the mid-lower bits of a
`a2`	`uint256`	A uint256 representing the mid-higher bits of a
`a3`	`uint256`	A uint256 representing the highest bits of a
`b0`	`uint256`	A uint256 representing the lower bits of b
`b1`	`uint256`	A uint256 representing the higher bits of b
`rem0`	`uint256`	A uint256 representing the lower bits of the remainder of the division
`rem1`	`uint256`	A uint256 representing the higher bits of the remainder of the division

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The high bits of the result

convertToRaw512

this function requires a0 to be greater than 0 and less or equal than 2**254 Divides a 512 bit unsigned integer by WAD where the result can be a 512 bit unsigned integer.

function convertToRaw512(uint256 a0, uint256 a1) internal pure returns (uint256 r0, uint256 r1);

Parameters

Name	Type	Description
`a0`	`uint256`	A uint256 representing the low bits of the numerator
`a1`	`uint256`	A uint256 representing the high bits of the numerator

Returns

Name	Type	Description
`r0`	`uint256`	The lower bits of the result
`r1`	`uint256`	The highre bits of the result

lnWAD2Negative

function lnWAD2Negative(uint256 x) internal pure returns (uint256 result);

Parameters

Name	Type	Description
`x`	`uint256`	the number to take the natural log of. Expressed as a WAD ** 2

Returns

Name	Type	Description
`result`	`uint256`	the ln of x multiplied by -1. Expressed as a WAD ** 2

findWADsToSlashTo0

this function tells how many WADs a number needs to be divided by to get to 0

function findWADsToSlashTo0(uint256 x) internal pure returns (uint256 precisionSlashingFactor);

Parameters

Name	Type	Description
`x`	`uint256`	the number to be divided

Returns

Name	Type	Description
`precisionSlashingFactor`	`uint256`	the number of WADs needed to be divided to get to 0

Reference

​MathLibs

​State Variables

​WAD

​Functions

​divideToWAD

​uncheckedMultiply

​powWad

​convertToWAD

​convertToRaw

​mul256x256

​mul512x256

​mul512x256In768

​mul512x512In1024

​mulMod256x256

​add512x512

​add768x768

​add1024x1024

​sub512x512

​sub768x768

​sub1024x1024

​divRem512x256

​div512x256

​sqrt256

​sqrt512

​gt512

​eq512

​ge512

​lt512

​lt768

​lt1024

​div512ByPowerOf2

​div512x256ResultIn512

​div512x512

​div768x256

​div768ByPowerOf2

​divRem1024x512In512

​convertToRaw512

​lnWAD2Negative

​findWADsToSlashTo0

MathLibs

State Variables

WAD

Functions

divideToWAD

uncheckedMultiply

powWad

convertToWAD

convertToRaw

mul256x256

mul512x256

mul512x256In768

mul512x512In1024

mulMod256x256

add512x512

add768x768

add1024x1024

sub512x512

sub768x768

sub1024x1024

divRem512x256

div512x256

sqrt256

sqrt512

gt512

eq512

ge512

lt512

lt768

lt1024

div512ByPowerOf2

div512x256ResultIn512

div512x512

div768x256

div768ByPowerOf2

divRem1024x512In512

convertToRaw512

lnWAD2Negative

findWADsToSlashTo0