# @prb/math

4.0.2 • Public • Published

# 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
• 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

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 {
UD60x18 x = ud(1e18);
UD60x18 y = ud(2e18);
UD60x18 z = ud(3e18);
}
}```

## 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:

`\$ pnpm install`

Now you can start making changes.

### Syntax Highlighting

You will need the following VSCode extensions:

## 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.

## Package Sidebar

### Install

`npm i @prb/math`

### Repository

github.com/PaulRBerg/prb-math

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4.0.2

MIT

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