# Uint128x128Math

[Git Source](https://github.com/traderjoe-xyz/joe-v2/blob/16f011d25e6bf6d0a0c479974345b623d491104f/src/libraries/math/Uint128x128Math.sol)

**Author:**\
Trader Joe

Helper contract used for power and log calculations

## State Variables

### LOG\_SCALE\_OFFSET

```solidity
uint256 constant LOG_SCALE_OFFSET = 127;
```

### LOG\_SCALE

```solidity
uint256 constant LOG_SCALE = 1 << LOG_SCALE_OFFSET;
```

### LOG\_SCALE\_SQUARED

```solidity
uint256 constant LOG_SCALE_SQUARED = LOG_SCALE * LOG_SCALE;
```

## Functions

### log2

Calculates the binary logarithm of x.

\*Based on the iterative approximation algorithm.\
<https://en.wikipedia.org/wiki/Binary\\_logarithm#Iterative\\_approximation\\>
Requirements:

* x must be greater than zero.\
  Caveats:
* The results are not perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation\
  Also because x is converted to an unsigned 129.127-binary fixed-point number during the operation to optimize the multiplication\*

```solidity
function log2(uint256 x) internal pure returns (int256 result);
```

**Parameters**

| Name | Type      | Description                                                                                 |
| ---- | --------- | ------------------------------------------------------------------------------------------- |
| `x`  | `uint256` | The unsigned 128.128-binary fixed-point number for which to calculate the binary logarithm. |

**Returns**

| Name     | Type     | Description                                                         |
| -------- | -------- | ------------------------------------------------------------------- |
| `result` | `int256` | The binary logarithm as a signed 128.128-binary fixed-point number. |

### pow

Returns the value of x^y. It calculates `1 / x^abs(y)` if x is bigger than 2^128.\
At the end of the operations, we invert the result if needed.

```solidity
function pow(uint256 x, int256 y) internal pure returns (uint256 result);
```

**Parameters**

| Name | Type      | Description                                                                     |
| ---- | --------- | ------------------------------------------------------------------------------- |
| `x`  | `uint256` | The unsigned 128.128-binary fixed-point number for which to calculate the power |
| `y`  | `int256`  | A relative number without any decimals, needs to be between ]-2^21; 2^21\[      |

## Errors

### Uint128x128Math\_\_LogUnderflow

```solidity
error Uint128x128Math__LogUnderflow();
```

### Uint128x128Math\_\_PowUnderflow

```solidity
error Uint128x128Math__PowUnderflow(uint256 x, int256 y);
```


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