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

Node.js

This is the recommended approach.

Install PRBMath using your favorite package manager, e.g., with Bun:

bun add @prb/math

Then, if you are using Foundry, you need to add this to your remappings.txt file:

@prb/math/=node_modules/@prb/math/

Git Submodules

This installation method is not recommended, but it is available for those who prefer it.

First, install the submodule using Forge:

forge install --no-commit PaulRBerg/prb-math@release-v4

Your .gitmodules file should now contain the following entry:

[submodule "lib/prb-math"]
  branch = "release-v4"
  path = "lib/prb-math"
  url = "https://github.com/PaulRBerg/prb-math"

Finally, 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:

| Value Type | Underlying Type | | ---------- | --------------- | | SD1x18 | int64 | | SD21x18 | int128 | | UD2x18 | uint64 | | UD21x18 | uint128 |

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
• Bun

In addition, familiarity with Solidity is requisite.

Set Up

Clone this repository including submodules:

$ git clone --recurse-submodules -j8 [email protected]:PaulRBerg/prb-math.git

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

$ bun install

Now you can start making changes.

Syntax Highlighting

You will need the following VSCode extensions:

• hardhat-solidity
• vscode-tree-language

Security

The codebase has undergone audits by leading security experts from Cantina and Certora. For a comprehensive list of all audits conducted, see the SECURITY file.

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.