PRBMath $GitHub Actions$

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
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

Foundry

First, run the install step:

forge install --no-commit paulrberg/prb-math@v3

Then, add this to your remappings.txt file:

prb-math/=lib/prb-math/src/

Node.js

yarn add @prb/math
# or
npm install @prb/math

Usage

PRBMath comes in two flavors:

SD59x18 (signed)
UD60x18 (unsigned)

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

Importing

It is recommended that you import PRBMath in the global scope, because you will often need multiple symbols:

pragma solidity >=0.8.13;

import "@prb/math/SD59x18.sol";
import "@prb/math/UD60x18.sol";

Note that PRBMath can only be used in Solidity v0.8.13 or above.

SD59x18

// SPDX-License-Identifier: UNLICENSED
pragma solidity >=0.8.13;

import "@prb/math/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 = log2(x);
  }
}

UD60x18

// SPDX-License-Identifier: UNLICENSED
pragma solidity >=0.8.13;

import "@prb/math/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).div(UNIT); // UNIT = 1e18
  }

  /// @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 = log2(x);
  }
}

Features

There's significant overlap between the functions available in SD59x18 and UD60x18, so I did not duplicate the functions tables below. If in doubt, refer to the source code, which is well-documented with NatSpec comments.

Mathematical Functions

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

Conversion Functions

Name	Description
`fromSD59x18`	Converts an SD59x18 number to a simple integer by dividing it by 1e18
`fromUD60x18`	Converts an UD60x18 number to a simple integer by dividing it by 1e18
`sd`	Wraps a simple integer into SD59x18
`sd59x18`	Wraps a simple integer into SD59x18
`toSD59x18`	Converts a simple integer to SD59x18 by multiplying it by 1e18
`toSD59x18`	Converts a simple integer to UD60x18 by multiplying it by 1e18
`ud`	Wraps a simple integer into UD60x18
`ud60x18`	Wraps a simple integer into UD60x18
`unwrap`	Unwrap an SD59x18 or UD60x18 number into a simple integer
`wrap`	Wraps a simple integer into either SD59x18 or UD60x18

Helper Functions

In addition to the mathematical and the conversion functions, PRBMath provides many other helpers for the user-defined value types, such as add,eq, and rshift. These functions are not part of the core API and are frequently updated, so I invite you to take a look at the source code to see the full list.

The goal with these helpers is not have to always unwrap and re-wrap variables to perform such basic operations as addition and equality checks. However, you should note that using these functions instead of the vanilla operators (e.g. +, ==, and >>) will result in a higher gas cost.

// SPDX-License-Identifier: UNLICENSED
pragma solidity >=0.8.13;

import "@prb/math/UD60x18.sol";

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

Assertions

PRBMath is shipped 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 SD59x18 or UD60x18 numbers are equal.

pragma solidity >=0.8.13;

import "@prb/math/UD60x18.sol";
import { Assertions as PRBMathAssertions } from "prb-math/test/Assertions.sol";
import { PRBTest } from "@prb/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:

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:

$ yarn install

Now you can start making changes.

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 been audited by a 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.

Acknowledgements

Mikhail Vladimirov for the insights he shared in his Math in Solidity 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 contributed a PR to this repository.

License

MIT

joeblau/prb-math

PRBMath

Install

Foundry

Node.js

Usage

Importing

SD59x18

UD60x18

Features

Mathematical Functions

Conversion Functions

Helper Functions

Assertions

Gas Efficiency

PRBMath

ABDKMath64x64

Contributing

Pre Requisites

Set Up

Security

Caveat Emptor

Contact

Acknowledgements

License

PRBMath $GitHub Actions$