# FullMath

Facilitates multiplication and division that can have overflow of an intermediate value without any loss of precision

Handles "phantom overflow" i.e., allows multiplication and division where an intermediate value overflows 256 bits

### Functions[​](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#functions) <a href="#functions" id="functions"></a>

#### mulDiv[​](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#muldiv) <a href="#muldiv" id="muldiv"></a>

```solidity
  function mulDiv(
    uint256 a,
    uint256 b,
    uint256 denominator
  ) internal pure returns (uint256 result)
```

Calculates floor(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0

Credit to Remco Bloemen under MIT license <https://xn--2-umb.com/21/muldiv>

**Parameters:**[**​**](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#parameters)

| Name          | Type    | Description      |
| ------------- | ------- | ---------------- |
| `a`           | uint256 | The multiplicand |
| `b`           | uint256 | The multiplier   |
| `denominator` | uint256 | The divisor      |

**Return Values:**[**​**](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#return-values)

| Name     | Type    | Description        |
| -------- | ------- | ------------------ |
| `result` | uint256 | The 256-bit result |

#### mulDivRoundingUp[​](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#muldivroundingup) <a href="#muldivroundingup" id="muldivroundingup"></a>

```solidity
  function mulDivRoundingUp(
    uint256 a,
    uint256 b,
    uint256 denominator
  ) internal pure returns (uint256 result)
```

Calculates ceil(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0

**Parameters:**[**​**](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#parameters-1)

| Name          | Type    | Description      |
| ------------- | ------- | ---------------- |
| `a`           | uint256 | The multiplicand |
| `b`           | uint256 | The multiplier   |
| `denominator` | uint256 | The divisor      |

**Return Values:**[**​**](https://docs.uniswap.org/contracts/v3/reference/core/libraries/FullMath#return-values-1)

| Name     | Type    | Description        |
| -------- | ------- | ------------------ |
| `result` | uint256 | The 256-bit result |


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