@prb/math

PRBMath

Solidity library for advanced fixed-point math that operates with signed 59.18-decimal fixed-point and unsigned 60.18-decimal fixed-point numbers. The name of the number format is due to the integer part having up to 59/60 decimals and the fractional part having up to 18 decimals. The numbers are bound by the minimum and the maximum values permitted by the Solidity types int256 and uint256.

• Operates with signed and unsigned denary fixed-point numbers, with 18 trailing decimals
• Offers advanced math functions like logarithms, exponentials, powers and square roots
• Provides type safety via user-defined value types
• Gas efficient, but still user-friendly
• Ergonomic developer experience thanks to using free functions instead of libraries
• Bakes in overflow-safe multiplication and division via mulDiv
• Reverts with custom errors instead of reason strings
• Well-documented with NatSpec comments
• Built and tested with Foundry

I created this because I wanted a fixed-point math library that is at the same time intuitive, efficient and safe. I looked at ABDKMath64x64, which is fast, but it uses binary numbers which are counter-intuitive and non-familiar to humans. Then, I looked at Fixidity, which operates with denary numbers and has wide precision, but is slow and susceptible to phantom overflow. Finally, I looked at Solmate, which checks all the boxes mentioned thus far, but it doesn't offer type safety.

Install

All users are recommended to install PRBMath as a Node.js package:

pnpm add @prb/math

This example uses Pnpm, but using Yarn or Npm is also possible.

Foundry

If you're using Foundry, you have to add this to your remappings.txt file:

@prb/math/=lib/prb-math/

Usage

There are two user-defined value types:

1. SD59x18 (signed)
2. UD60x18 (unsigned)

If you don't know what a user-defined value type is, check out this blog post.

If you don't need negative numbers, there's no point in using the signed flavor SD59x18. The unsigned flavor UD60x18 is more gas efficient.

Note that PRBMath is not a library in the Solidity sense. It's just a collection of free functions.

Importing

It is recommended that you import PRBMath using specific symbols. Importing full files can result in Solidity complaining about duplicate definitions and static analyzers like Slither erroring, especially as repos grow and have more dependencies with overlapping names.

pragma solidity >=0.8.19;

import { SD59x18 } from "@prb/math/src/SD59x18.sol";
import { UD60x18 } from "@prb/math/src/UD60x18.sol";

Any function that is not available in the types directly has to be imported explicitly. Here's an example for the sd and the ud functions:

pragma solidity >=0.8.19;

import { SD59x18, sd } from "@prb/math/src/SD59x18.sol";
import { UD60x18, ud } from "@prb/math/src/UD60x18.sol";

Note that PRBMath can only be used in Solidity v0.8.19 and above.

SD59x18

// SPDX-License-Identifier: UNLICENSED
pragma solidity >=0.8.19;

import { SD59x18, sd } from "@prb/math/src/SD59x18.sol";

contract SignedConsumer {
  /// @notice Calculates 5% of the given signed number.
  /// @dev Try this with x = 400e18.
  function signedPercentage(SD59x18 x) external pure returns (SD59x18 result) {
    SD59x18 fivePercent = sd(0.05e18);
    result = x.mul(fivePercent);
  }

  /// @notice Calculates the binary logarithm of the given signed number.
  /// @dev Try this with x = 128e18.
  function signedLog2(SD59x18 x) external pure returns (SD59x18 result) {
    result = x.log2();
  }
}

UD60x18

// SPDX-License-Identifier: UNLICENSED
pragma solidity >=0.8.19;

import { UD60x18, ud } from "@prb/math/src/UD60x18.sol";

contract UnsignedConsumer {
  /// @notice Calculates 5% of the given signed number.
  /// @dev Try this with x = 400e18.
  function unsignedPercentage(UD60x18 x) external pure returns (UD60x18 result) {
    UD60x18 fivePercent = ud(0.05e18);
    result = x.mul(fivePercent);
  }

  /// @notice Calculates the binary logarithm of the given signed number.
  /// @dev Try this with x = 128e18.
  function unsignedLog2(UD60x18 x) external pure returns (UD60x18 result) {
    result = x.log2();
  }
}

Features

Because there's significant overlap between the features available in SD59x18 and UD60x18, there is only one table per section. If in doubt, refer to the source code, which is well-documented with NatSpec comments.

Mathematical Functions

Name	Operator	Description
abs	N/A	Absolute value
avg	N/A	Arithmetic average
ceil	N/A	Smallest whole number greater than or equal to x
div	/	Fixed-point division
exp	N/A	Natural exponential e^x
exp2	N/A	Binary exponential 2^x
floor	N/A	Greatest whole number less than or equal to x
frac	N/A	Fractional part
gm	N/A	Geometric mean
inv	N/A	Inverse 1÷x
ln	N/A	Natural logarithm ln(x)
log10	N/A	Common logarithm log10(x)
log2	N/A	Binary logarithm log2(x)
mul	*	Fixed-point multiplication
pow	N/A	Power function x^y
powu	N/A	Power function x^y with y simple integer
sqrt	N/A	Square root

Adjacent Value Types

PRBMath provides adjacent value types that serve as abstractions over other vanilla types such as int64. The types currently available are:

Value Type	Underlying Type
SD1x18	int64
UD2x18	uint64

These are useful if you want to save gas by using a lower bit width integer, e.g. in a struct.

Note that these types don't have any mathematical functionality. To do math with them, you will have to unwrap them into a simple integer and then to the core types SD59x18 and UD60x18.

Casting Functions

All PRBMath types have casting functions to and from all other types, including a few basic types like uint128 and uint40.

Name	Description
intoSD1x18	Casts a number to SD1x18
intoSD59x18	Casts a number to SD59x18
intoUD2x18	Casts a number to UD2x18
intoUD60x18	Casts a number to UD60x18
intoUint256	Casts a number to uint256
intoUint128	Casts a number to uint128
intoUint40	Casts a number to uint40
sd1x18	Alias for SD1x18.wrap
sd59x18	Alias for SD59x18.wrap
ud2x18	Alias for UD2x18.wrap
ud60x18	Alias for UD60x18.wrap

Conversion Functions

The difference between "conversion" and "casting" is that conversion functions multiply or divide the inputs, whereas casting functions simply cast them.

Name	Description
convert(SD59x18)	Converts an SD59x18 number to a simple integer by dividing it by 1e18
convert(UD60x18)	Converts a UD60x18 number to a simple integer by dividing it by 1e18
convert(int256)	Converts a simple integer to SD59x18 by multiplying it by 1e18
convert(uint256)	Converts a simple integer to UD60x18 type by multiplying it by 1e18

Helper Functions

In addition to offering mathematical, casting, and conversion functions, PRBMath provides numerous helper functions for user-defined value types:

Name	Operator	Description
add	+	Checked addition
and	&	Logical AND
eq	==	Equality
gt	>	Greater than operator
gte	>=	Greater than or equal to
isZero	N/A	Check if a number is zero
lshift	N/A	Bitwise left shift
lt	<	Less than
lte	<=	Less than or equal to
mod	%	Modulo
neq	!=	Not equal operator
not	~	Negation operator
or	|	Logical OR
rshift	N/A	Bitwise right shift
sub	-	Checked subtraction
unary	-	Checked unary
uncheckedAdd	N/A	Unchecked addition
uncheckedSub	N/A	Unchecked subtraction
xor	^	Exclusive or (XOR)

These helpers are designed to streamline basic operations such as addition and equality checks, eliminating the need to constantly unwrap and re-wrap variables. However, it is important to be aware that utilizing these functions may result in increased gas costs compared to unwrapping and directly using the vanilla types.

// SPDX-License-Identifier: UNLICENSED
pragma solidity >=0.8.19;

import { UD60x18, ud } from "@prb/math/src/UD60x18.sol";

function addRshiftEq() pure returns (bool result) {
  UD60x18 x = ud(1e18);
  UD60x18 y = ud(3e18);
  y = y.add(x); // also: y = y + x
  y = y.rshift(2);
  result = x.eq(y); // also: y == x
}

Assertions

PRBMath comes with typed assertions that you can use for writing tests with PRBTest, which is based on Foundry. This is useful if, for example, you would like to assert that two UD60x18 numbers are equal.

pragma solidity >=0.8.19;

import { UD60x18, ud } from "@prb/math/src/UD60x18.sol";
import { Assertions as PRBMathAssertions } from "@prb/math/test/Assertions.sol";
import { PRBTest } from "@prb/math/src/test/PRBTest.sol";

contract MyTest is PRBTest, PRBMathAssertions {
  function testAdd() external {
    UD60x18 x = ud(1e18);
    UD60x18 y = ud(2e18);
    UD60x18 z = ud(3e18);
    assertEq(x.add(y), z);
  }
}

Gas Efficiency

PRBMath is faster than ABDKMath for abs, exp, exp2, gm, inv, ln, log2, but it is slower than ABDKMath for avg, div, mul, powu and sqrt.

The main reason why PRBMath lags behind ABDKMath's mul and div functions is that it operates with 256-bit word sizes, and so it has to account for possible intermediary overflow. ABDKMath, on the other hand, operates with 128-bit word sizes.

Note: I did not find a good way to automatically generate gas reports for PRBMath. See the #134 discussion for more details about this issue.

PRBMath

Gas estimations based on the v2.0.1 and the v3.0.0 releases.

SD59x18	Min	Max	Avg		UD60x18	Min	Max	Avg
abs	68	72	70		n/a	n/a	n/a	n/a
avg	95	105	100		avg	57	57	57
ceil	82	117	101		ceil	78	78	78
div	431	483	451		div	205	205	205
exp	38	2797	2263		exp	1874	2742	2244
exp2	63	2678	2104		exp2	1784	2652	2156
floor	82	117	101		floor	43	43	43
frac	23	23	23		frac	23	23	23
gm	26	892	690		gm	26	893	691
inv	40	40	40		inv	40	40	40
ln	463	7306	4724		ln	419	6902	3814
log10	104	9074	4337		log10	503	8695	4571
log2	377	7241	4243		log2	330	6825	3426
mul	455	463	459		mul	219	275	247
pow	64	11338	8518		pow	64	10637	6635
powu	293	24745	5681		powu	83	24535	5471
sqrt	140	839	716		sqrt	114	846	710

ABDKMath64x64

Gas estimations based on the v3.0 release of ABDKMath. See my abdk-gas-estimations repo.

Method	Min	Max	Avg
abs	88	92	90
avg	41	41	41
div	168	168	168
exp	77	3780	2687
exp2	77	3600	2746
gavg	166	875	719
inv	157	157	157
ln	7074	7164	7126
log2	6972	7062	7024
mul	111	111	111
pow	303	4740	1792
sqrt	129	809	699

Contributing

Feel free to dive in! Open an issue, start a discussion or submit a PR.

Pre Requisites

You will need the following software on your machine:

• Git
• Foundry
• Node.Js
• Pnpm

In addition, familiarity with Solidity is requisite.

Set Up

Clone this repository including submodules:

$ git clone --recurse-submodules -j8 git@github.com:PaulRBerg/prb-math.git

Then, inside the project's directory, run this to install the Node.js dependencies:

$ pnpm install

Now you can start making changes.

Syntax Highlighting

You will need the following VSCode extensions:

• hardhat-solidity
• vscode-tree-language

Security

While I set a high bar for code quality and test coverage, you should not assume that this project is completely safe to use. PRBMath has not yet been audited by a third-party security researcher.

Caveat Emptor

This is experimental software and is provided on an "as is" and "as available" basis. I do not give any warranties and will not be liable for any loss, direct or indirect through continued use of this codebase.

Contact

If you discover any bugs or security issues, please report them via Telegram.

Acknowledgments

• Mikhail Vladimirov for the insights he shared in the Math in Solidity article series.
• Remco Bloemen for his work on overflow-safe multiplication and division, and for responding to the questions I asked him while developing the library.
• Everyone who has contributed a PR to this repository.

License

This project is licensed under MIT.