# MathLib Library Explanation

This **Solidity library** provides implementations of:

1. \( \log(1 + x) \) for small \( x \) (in 1e18 fixed-point).
2. \( \exp(x / 1e18) \) for small \( x \) (in 1e18 fixed-point).

---

- **Fixed-Point Math**: All calculations assume **1e18** scale, meaning 1 represents 1.0 in typical decimal notation.
  - `log1p(x)` is calculated using the first two terms of its Taylor series expansion:  
    \[
      \log(1 + x) \approx x - \frac{x^2}{2}.
    \]
  - `expWad(x)` is calculated by the first three terms of the Taylor series for \(e^x\) or \(e^{-x}\):
    \[
      e^x \approx 1 + x + \frac{x^2}{2}, \quad |x| < 1e16.
    \]

---


### Logic

1. **Require**: `x < 1e14`  
   - If `x >= 1e14`, it reverts:  
     ```solidity
     require(x < 1e14, "x too large for log");
     ```


2. **Taylor Expansion**:  
   \[
     \log(1 + x) \approx x - \frac{x^2}{2}
   \]
   - In 1e18 format, `x^2` is in 1e36, so we divide by 1e18 to get back to 1e18 scale.
   - The final result is in 1e18 scale.

#### Example

If `x = 1e13 (0.00001)`, then:
- \(x^2 / 1e18 = 1e26 / 1e18 = 1e8\)  
- \(\log1p(x) \approx x - x^2/2\)

---

## 3. `expWad(int256 x)`

### Signature

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

Logic

1. Require: x > -1e16 && x < 1e16

   • If x is outside ((-1e16, +1e16)), it reverts:

     require(x > -1e16 && x < 1e16, "x out of range");

2. Taylor Expansion:

   • For x >= 0:
     [ e^x \approx 1 + x + \frac{x^2}{2} ]
   • For x < 0:
     [ e^{-x} \approx 1 - x + \frac{x^2}{2} ]
   • Again, in 1e18 precision:
     • Compute x^2 in 1e36.
     • Divide by 1e18 to return to 1e18 scale.
     • Add or subtract from 1e18 as needed.

Example

If x = 5e15 (which is 0.005 in decimal):

• (x^2 / 1e18 = (5e15)^2 / 1e18 = 25e30 / 1e18 = 25e12 = 2.5e13)
• (e^x \approx 1e18 + 5e15 + (2.5e13 / 2))

---

5. Summary

MathLib provides:

• log1p(x): Calculates (\log(1 + x)) for x < 1e14, ensuring the series expansion is valid and reverts otherwise.
• expWad(x): Calculates (\exp(x)) (or (\exp(-x))) for |x| < 1e16, returning results in 1e18 fixed-point.