FixidityLib

An overflow-protected fixed-point arithmetic library for Solidity.

Testing

yarn In a separate terminal: yarn start:ganache:local npx truffle compile npx truffle migrate npx truffle test

Usage

Fixidity implements fixed point arithmetic by keeping a "virtual comma" in an int256.

With the current implementation that uses 24 digits, 1 in Fixidity is 1 * 1024. In a similar manner, 0.5 would be 5 * 1023.

Fixidity stores fixed point numbers in an int256, and it is up to the developer to remember whether a specific int256 is a Fixidity fixed point number or not.

To create Fixidity fixed point numbers use toFixed().

using FixidityLib for uint256;
uint256 x = 1;
int256 xFixed;
xFixed = x.toFixed();
...

To convert Fixidity fixed point number back to uint use toUint().

...
using FixidityLib for uint256;
using FixidityLib for int256;
uint256 x = 1;
uint256 y = 1;
int256 xFixed = x.toFixed();
int256 yFixed = y.toFixed();
int256 zFixed = x.add(y);
uint256 z = zFixed.toUint();

FixidityLib can keep 24 digits, when using toUint() only the integer part will be returned.

Fixed Point Math with Tokens

Ether and the ERC20 implementation use an implicit fixed point representation, although no arithmetic comes bundled up with them.

The decimals() field from ERC20Detailed states how many digits are in the fractional part of a token. For Ether the fractional part of a token takes 18 digits.

When using Fixidity to operate on wei or ERC20Detailed tokens, it is recommended to remember that these already have an implicit fractional part. When converting an uint to a fixed point number Fixidity will just add 24 zeros at the end. If the uint contains a number representing a large number of tokens an overflow will be more likely that it needs to be.

When converting an ether wei amount to Fixidity fixed point use toFixed(18). When converting from Fixidity fixed point to we use toUint(18). When converting an ERC20 wei amount to Fixidity fixed point use toFixed(token.decimals()). When converting from Fixidity fixed point to we use toUint(token.decimals()).

FixidityLib.sol

This library implements addition, subtraction, multiplication and division, along with the related constants and limits.

function digits() public pure returns(uint8) Number of positions that the comma is shifted to the right. Default: 24

function fixed1() public pure returns(int256) This is 1 in the fixed point units used in this library. Calculated as fixed1() = equals 10^digits() Default: 1000000000000000000000000000000000000

function mulPrecision() public pure returns(int256) The amount of decimals lost on each multiplication operand. Calculated as mulPrecision() = sqrt(fixed1) Default: 1000000000000000000

function maxInt256() public pure returns(int256) Maximum value that can be represented in an int256 Calculated as maxInt256() = 2^255 -1 Default: 57896044618658097711785492504343953926634992332820282019728792003956564819967

function minInt256() public pure returns(int256) Minimum value that can be represented in an int256 Calculated as minInt256() = (2^255) * (-1) Default: -57896044618658097711785492504343953926634992332820282019728792003956564819968

function maxNewFixed() public pure returns(int256) Maximum value that can be converted to fixed point. Default: 57896044618658097711785492504343953926634

function minNewFixed() public pure returns(int256) Maximum value that can be converted to fixed point. Calculated as minNewFixed() = -(maxInt256()) / fixed1() Default: -57896044618658097711785492504343953926634

function maxFixedAdd() public pure returns(int256) Maximum value that can be safely used as an addition operand. Additions with one operand over this value might overflow, but not necessarily so. Calculated as maxFixedAdd() equals maxInt256()-1 / 2 Default: 28948022309329048855892746252171976963317496166410141009864396001978282409983

function maxFixedSub() public pure returns(int256) Maximum negative value that can be safely subtracted. Operations where values larger than maxFixedSub() are subtracted might overflow, but not necessarily so. Calculated as maxFixedSub() = minInt256() / 2 Default: -28948022309329048855892746252171976963317496166410141009864396001978282409984

function maxFixedMul() public pure returns(int256) Maximum value that can be safely used as a multiplication operator. Divisions where a value is divided by another over this value might overflow, but not necessarily so. Calculated as maxFixedMul() = sqrt(maxNewFixed())*fixed1(). Default: 240615969168004511545033772477625056927114980741063

function maxFixedDiv() public pure returns(int256) Maximum value that can be safely used as a dividend. Operations where values larger than maxFixedDiv() are divided might overflow, but not necessarily so. Calculated as divide(maxFixedDiv,newFixedFraction(1,fixed1())) = maxInt256(). Default: 57896044618658097711785492504343953926634 }

function maxFixedDivisor() public pure returns(int256) Maximum value that can be safely used as a divisor. The divide(x, y) function uses reciprocal(y), and numbers above maxFixedDivisor() will cause a division by zero. Calculated as maxFixedDivisor() = fixed1()*fixed1() Default: 1000000000000000000000000000000000000000000000000000000000000000000000000

function convertFixed(int256 x, uint8 _originDigits, uint8 _destinationDigits) Converts an int256 which is already in some fixed point representation to a different fixed precision representation. Both the origin and destination precisions must be 38 or less digits. Origin values with a precision higher than the destination precision will be truncated accordingly.

function toFixed(int256 x) Converts an int256 to fixed point units, equivalent to multiplying by 10^digits().

function toFixed(uint256 x) Converts an uint256 to fixed point units, equivalent to multiplying by 10^digits().

function toFixed(int256 x, uint8 _originDigits) Converts an int256 which is already in some fixed point representation to that of this library. The _originDigits parameter is the precision of x. Values with a precision higher than FixidityLib.digits() will be truncated accordingly.

function toFixed(uint256 x, uint8 _originDigits) Converts an uint256 which is already in some fixed point representation to that of this library. The _originDigits parameter is the precision of x. Values with a precision higher than FixidityLib.digits() will be truncated accordingly.

function toUint(int256 x) Converts an int256 in the fixed point representation of this library to an uint. All decimal digits will be truncated.

function toInt(int256 x) Converts an int256 in the fixed point representation of this library to an uint. All decimal digits will be truncated.

function toUint(int256 x, uint8 _destinationDigits) Converts an int256 in the fixed point representation of this library to a different fixed point representation stored in an uint256. The _destinationDigits parameter is the precision of the output x. Values with a precision below than FixidityLib.digits() will be truncated accordingly.

function toInt(int256 x, uint8 _destinationDigits) Converts an int256 in the fixed point representation of this library to a different fixed point representation stored in uint256. The _destinationDigits parameter is the precision of the output x. Values with a precision below than FixidityLib.digits() will be truncated accordingly.

function integer(int256 x) public pure returns (int256) Returns the integer part of a fixed point number, still in fixed point format.

function fractional(int256 x) public pure returns (int256) Returns the fractional part of a fixed point number. In the case of a negative number the fractional is also negative.

function abs(int256 x) public pure returns (int256) Converts to positive if negative. Due to int256 having one more negative number than positive numbers abs(minInt256) reverts.

function add(int256 x, int256 y) public pure returns (int256) x+y. If any operator is higher than maxFixedAdd() it might overflow.

function subtract(int256 x, int256 y) public pure returns (int256) x-y. You can use add(x,-y) instead.

function multiply(int256 x, int256 y) public pure returns (int256) x*y. If any of the operators is higher than maxFixedMul() it might overflow.

function reciprocal(int256 x) public pure returns (int256) 1/x.

function divide(int256 x, int256 y) public pure returns (int256) x/y. If the dividend is higher than maxFixedDiv() it might overflow. You can use multiply(x,reciprocal(y)) instead. There is a loss of precision on division for the lower mulPrecision() decimals.

** Obsolete functions kept for backwards compatibility**

function newFixed(int256 x) Converts an int256 to fixed point units, equivalent to multiplying by 10^digits().

function fromFixed(int256 x) Converts an int256 in the fixed point representation of this library to a non decimal. All decimal digits will be truncated.

function newFixed(int256 x, uint8 _originDigits) Converts an int256 which is already in some fixed point representation to that of this library. The _originDigits parameter is the precision of x. Values with a precision higher than FixidityLib.digits() will be truncated accordingly.

function fromFixed(int256 x, uint8 _destinationDigits) Converts an int256 in the fixed point representation of this library to a different representation. The _destinationDigits parameter is the precision of the output x. Values with a precision below than FixidityLib.digits() will be truncated accordingly.

function newFixedFraction(int256 numerator, int256 denominator) Converts two int256 representing a fraction to fixed point units, equivalent to multiplying dividend and divisor by 10^digits() and then dividing them.

LogarithmLib.sol

This library extends FixidityLib by implementing logarithms, along with the related constants and limits.

function fixedE() public pure returns(int256) This is e in the fixed point units used in this library. Default: 27182818284590452353602874713526624977572470936999595749669676277240766303535/fixed1()

function fixedLn1_5() public pure returns(int256) ln(1.5) Default: 405465108108164381978013115464349137;

function fixedLn10() public pure returns (int256) ln(10) Default: 2302585092994045684017991454684364208;

function ln(int256 value) public pure returns (int256) ln(x). This function has a 1/50 deviation close to ln(-1), 1/maxFixedMul() deviation at fixedE()**2, but diverges to 10x deviation at maxNewFixed().

function log_b(int256 b, int256 x) public pure returns (int256) log_b(x)

• int256 b Base in fixed point representation.
• int256 x Value to calculate the logarithm for in fixed point representation.