ABDKMath64x64

• Smart contract library of mathematical functions operating with signed 64.64-bit fixed point numbers. Signed 64.64-bit fixed point number is basically a simple fraction whose numerator is signed 128-bit integer and denominator is 2^64. As long as denominator is always the same, there is no need to store it, thus in Solidity signed 64.64-bit fixed point numbers are represented by int128 type holding only the numerator. /

Functions

uint256toInt128

function uint256toInt128(
    uint256 input
) internal returns (int128)

Parameters

Name	Type	Description
input	uint256

int128toUint256

function int128toUint256(
    int128 input
) internal returns (uint256)

Parameters

Name	Type	Description
input	int128

int128toUint64

function int128toUint64(
    int128 input
) internal returns (uint64)

Parameters

Name	Type	Description
input	int128

fromInt

function fromInt(
    int256 x
) internal returns (int128)

• Convert signed 256-bit integer number into signed 64.64-bit fixed point number. Revert on overflow.

Parameters

Name	Type	Description
x	int256	signed 256-bit integer number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

toInt

function toInt(
    int128 x
) internal returns (int64)

• Convert signed 64.64 fixed point number into signed 64-bit integer number rounding down.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int64	signed 64-bit integer number
/

fromUInt

function fromUInt(
    uint256 x
) internal returns (int128)

• Convert unsigned 256-bit integer number into signed 64.64-bit fixed point number. Revert on overflow.

Parameters

Name	Type	Description
x	uint256	unsigned 256-bit integer number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

fromScaled

function fromScaled(
    uint256 x,
    uint256 decimal
) internal returns (int128)

• Convert unsigned 256-bit integer number scaled with 10^decimals into signed 64.64-bit fixed point number. Revert on overflow.

Parameters

Name	Type	Description
x	uint256	unsigned 256-bit integer number
decimal	uint256	scale of the number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

toUInt

function toUInt(
    int128 x
) internal returns (uint64)

• Convert signed 64.64 fixed point number into unsigned 64-bit integer number rounding down. Revert on underflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	uint64	unsigned 64-bit integer number
/

from128x128

function from128x128(
    int256 x
) internal returns (int128)

• Convert signed 128.128 fixed point number into signed 64.64-bit fixed point number rounding down. Revert on overflow.

Parameters

Name	Type	Description
x	int256	signed 128.128-bin fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

to128x128

function to128x128(
    int128 x
) internal returns (int256)

• Convert signed 64.64 fixed point number into signed 128.128 fixed point number.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int256	signed 128.128 fixed point number
/

add

function add(
    int128 x,
    int128 y
) internal returns (int128)

• Calculate x + y. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

sub

function sub(
    int128 x,
    int128 y
) internal returns (int128)

• Calculate x - y. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

mul

function mul(
    int128 x,
    int128 y
) internal returns (int128)

• Calculate x * y rounding down. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

muli

function muli(
    int128 x,
    int256 y
) internal returns (int256)

• Calculate x * y rounding towards zero, where x is signed 64.64 fixed point number and y is signed 256-bit integer number. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64 fixed point number
y	int256	signed 256-bit integer number

Return Values

Name	Type	Description
[0]	int256	signed 256-bit integer number
/

mulu

function mulu(
    int128 x,
    uint256 y
) internal returns (uint256)

• Calculate x * y rounding down, where x is signed 64.64 fixed point number and y is unsigned 256-bit integer number. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64 fixed point number
y	uint256	unsigned 256-bit integer number

Return Values

Name	Type	Description
[0]	uint256	unsigned 256-bit integer number
/

div

function div(
    int128 x,
    int128 y
) internal returns (int128)

• Calculate x / y rounding towards zero. Revert on overflow or when y is zero.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

divi

function divi(
    int256 x,
    int256 y
) internal returns (int128)

• Calculate x / y rounding towards zero, where x and y are signed 256-bit integer numbers. Revert on overflow or when y is zero.

Parameters

Name	Type	Description
x	int256	signed 256-bit integer number
y	int256	signed 256-bit integer number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

divu

function divu(
    uint256 x,
    uint256 y
) internal returns (int128)

• Calculate x / y rounding towards zero, where x and y are unsigned 256-bit integer numbers. Revert on overflow or when y is zero.

Parameters

Name	Type	Description
x	uint256	unsigned 256-bit integer number
y	uint256	unsigned 256-bit integer number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

neg

function neg(
    int128 x
) internal returns (int128)

• Calculate -x. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

abs

function abs(
    int128 x
) internal returns (int128)

• Calculate |x|. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

inv

function inv(
    int128 x
) internal returns (int128)

• Calculate 1 / x rounding towards zero. Revert on overflow or when x is zero.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

avg

function avg(
    int128 x,
    int128 y
) internal returns (int128)

• Calculate arithmetics average of x and y, i.e. (x + y) / 2 rounding down.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

gavg

function gavg(
    int128 x,
    int128 y
) internal returns (int128)

• Calculate geometric average of x and y, i.e. sqrt (x y) rounding down. Revert on overflow or in case x y is negative.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

pow

function pow(
    int128 x,
    uint256 y
) internal returns (int128)

• Calculate x^y assuming 0^0 is 1, where x is signed 64.64 fixed point number and y is unsigned 256-bit integer number. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number
y	uint256	uint256 value

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

sqrt

function sqrt(
    int128 x
) internal returns (int128)

• Calculate sqrt (x) rounding down. Revert if x < 0.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

log_2

function log_2(
    int128 x
) internal returns (int128)

• Calculate binary logarithm of x. Revert if x <= 0.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

ln

function ln(
    int128 x
) internal returns (int128)

• Calculate natural logarithm of x. Revert if x <= 0.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

exp_2

function exp_2(
    int128 x
) internal returns (int128)

• Calculate binary exponent of x. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

exp

function exp(
    int128 x
) internal returns (int128)

• Calculate natural exponent of x. Revert on overflow.

Parameters

Name	Type	Description
x	int128	signed 64.64-bit fixed point number

Return Values

Name	Type	Description
[0]	int128	signed 64.64-bit fixed point number
/

divuu

function divuu(
    uint256 x,
    uint256 y
) private returns (uint128)

• Calculate x / y rounding towards zero, where x and y are unsigned 256-bit integer numbers. Revert on overflow or when y is zero.

Parameters

Name	Type	Description
x	uint256	unsigned 256-bit integer number
y	uint256	unsigned 256-bit integer number

Return Values

Name	Type	Description
[0]	uint128	unsigned 64.64-bit fixed point number
/

powu

function powu(
    uint256 x,
    uint256 y
) private returns (uint256)

• Calculate x^y assuming 0^0 is 1, where x is unsigned 129.127 fixed point number and y is unsigned 256-bit integer number. Revert on overflow.

Parameters

Name	Type	Description
x	uint256	unsigned 129.127-bit fixed point number
y	uint256	uint256 value

Return Values

Name	Type	Description
[0]	uint256	unsigned 129.127-bit fixed point number
/

sqrtu

function sqrtu(
    uint256 x
) private returns (uint128)

• Calculate sqrt (x) rounding down, where x is unsigned 256-bit integer number.

Parameters

Name	Type	Description
x	uint256	unsigned 256-bit integer number

Return Values

Name	Type	Description
[0]	uint128	unsigned 128-bit integer number
/