# EXPTRANS: Exponential Function

> *The exponential function is the only function that is its own derivative—a mathematical curiosity that makes it indispensable for modeling growth, decay, and everything compounding.*

| Property         | Value                            |
| ---------------- | -------------------------------- |
| **Category**     | Numeric                        |
| **Inputs**       | Source (close)                          |
| **Parameters**   | None                      |
| **Outputs**      | Single series (EXPTRANS)                       |
| **Output range** | Varies (see docs)                     |
| **Warmup**       | `0` bars                          |
| **PineScript**   | [exptrans.pine](exptrans.pine)                       |

- The Exponential (EXP) transformer applies the natural exponential function $e^x$ to each value in a time series.
- No configurable parameters; computation is stateless per bar.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Exponential (EXP) transformer applies the natural exponential function $e^x$ to each value in a time series. As the inverse of the natural logarithm, it converts additive relationships back to multiplicative ones, making it essential for reconstructing price levels from log-returns and implementing models that assume log-normal distributions.

## Mathematical Foundation

### Core Formula

$$
\text{EXP}_t = e^{x_t}
$$

where:
- $x_t$ is the input value at time $t$
- $e \approx 2.71828...$ is Euler's number

### Key Properties

| Property | Formula | Description |
|:---------|:--------|:------------|
| **Inverse of Log** | $e^{\ln(x)} = x$ | Undoes natural logarithm |
| **Product Rule** | $e^{a+b} = e^a \cdot e^b$ | Additive inputs → multiplicative outputs |
| **Quotient Rule** | $e^{a-b} = e^a / e^b$ | Differences → ratios |
| **Power Rule** | $e^{n \cdot x} = (e^x)^n$ | Scaling in exponent → power |
| **Identity** | $e^0 = 1$ | Zero maps to unity |
| **Base Value** | $e^1 = e \approx 2.71828$ | Unit exponent gives $e$ |

### Domain and Range

| | Value |
|:--|:--|
| **Domain** | $(-\infty, +\infty)$ |
| **Range** | $(0, +\infty)$ |

The exponential function accepts any real number but always produces strictly positive outputs.

## Financial Applications

### Log-Return to Price Reconstruction

Given cumulative log-returns, reconstruct price levels:

$$
P_t = P_0 \cdot e^{\sum_{i=1}^{t} r_i}
$$

where $r_i$ are log-returns.

### Volatility Scaling

Convert log-volatility to multiplicative factors:

$$
\text{VolFactor} = e^{\sigma \sqrt{T}}
$$

### Compound Growth

Model continuous compounding:

$$
A = P \cdot e^{rt}
$$

where $r$ is the continuous rate and $t$ is time.

### Option Pricing

The exponential appears throughout Black-Scholes:

$$
C = S \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)
$$

## Implementation Details

### Overflow Handling

For large positive inputs, $e^x$ can overflow to infinity:
- $e^{709}$ ≈ $8.2 \times 10^{307}$ (near double max)
- $e^{710}$ → overflow

The implementation substitutes the last valid value when overflow occurs.

### Precision Considerations

| Input Range | Relative Precision |
|:------------|:-------------------|
| $|x| < 1$ | Full 15-16 digits |
| $|x| < 20$ | Full precision |
| $|x| > 700$ | Overflow risk |

### Streaming Characteristics

| Metric | Value |
|:-------|:------|
| **Warmup Period** | 0 |
| **Memory** | O(1) |
| **Complexity** | O(1) per update |

## Performance Profile

### Operation Count (Scalar)

| Operation | Count | Notes |
|:----------|:-----:|:------|
| EXP | 1 | Hardware instruction |
| **Total** | ~20 cycles | Platform dependent |

### Quality Metrics

| Metric | Score | Notes |
|:-------|:-----:|:------|
| **Accuracy** | 10/10 | IEEE 754 compliant |
| **Timeliness** | 10/10 | Zero lag |
| **Smoothness** | N/A | Transform preserves input characteristics |

## Usage Examples

### Basic Usage

```csharp
// Create EXP transformer
var exp = new Exptrans();

// Transform log-returns back to growth factors
var logReturn = new TValue(DateTime.UtcNow, 0.05);
var growthFactor = exp.Update(logReturn);  // ≈ 1.0513
```

### Reconstructing Prices from Log-Returns

```csharp
var logReturns = new TSeries();
// ... populate with cumulative log-returns

var cumulativeExp = new Exptrans();
var priceRatios = cumulativeExp.Update(logReturns);

// Multiply by initial price to get price levels
var initialPrice = 100.0;
var prices = priceRatios.Select(v => v * initialPrice);
```

### Undoing Log Transform

```csharp
var log = new Logtrans();
var exp = new Exptrans();

// Round-trip: price → log → exp → price
var price = new TValue(DateTime.UtcNow, 150.0);
var logPrice = log.Update(price);     // ≈ 5.0106
var recovered = exp.Update(logPrice); // ≈ 150.0
```

## Common Pitfalls

1. **Overflow Risk**: Input values above ~709 cause overflow. Monitor input ranges when working with cumulative sums.

2. **Magnitude Explosion**: Small additive changes in the exponent create large multiplicative changes in output. A change of 1.0 in the exponent multiplies the output by $e$ ≈ 2.72.

3. **Inverse Relationship**: EXP undoes LOG, but only if the original values were positive. Negative prices cannot be recovered through log-exp round-trip.

4. **Scale Sensitivity**: Unlike LOG which compresses ranges, EXP expands them dramatically. Ensure downstream consumers can handle the output magnitudes.

## Validation

| Test | Status |
|:-----|:------:|
| **Math.Exp Parity** | ✅ |
| **Known Values (e⁰=1, e¹=e)** | ✅ |
| **Inverse of Log** | ✅ |
| **Product Rule** | ✅ |
| **Quotient Rule** | ✅ |
| **Power Rule** | ✅ |

## References

- Euler, L. (1748). *Introductio in analysin infinitorum*.
- Maor, E. (1994). *e: The Story of a Number*. Princeton University Press.
- Hull, J. (2018). *Options, Futures, and Other Derivatives*. Pearson. (Black-Scholes applications)