> ## Documentation Index
> Fetch the complete documentation index at: https://doc.blueapi.ir/llms.txt
> Use this file to discover all available pages before exploring further.

# SLERP Math: Formula, Boundaries, and Fallback

> SLERP mathematical specification for EMEP. Defines the spherical linear interpolation formula, shape expectations, boundary cases near theta = 0 and theta = pi, and fallback to lerp. Reference: Shoemake 1985.

SLERP (Spherical Linear Interpolation) interpolates between two parameter vectors along the shortest arc on a hypersphere. This page defines the formula, boundary cases, and fallback rules used by the TensorEngine.

## Formula

Given two parameter vectors A and B in R^d, and interpolation parameter t in \[0, 1]:

```text theme={null}
theta = arccos( (A dot B) / (||A||_2 * ||B||_2) )

SLERP(A, B, t) = (sin((1 - t) * theta) / sin(theta)) * A
               + (sin(t * theta) / sin(theta)) * B
```

Normalization is applied per parameter tensor, not globally across the model.

## Symbol Table

| Symbol | Meaning                               |   |   |     |         |
| ------ | ------------------------------------- | - | - | --- | ------- |
| A, B   | Parameter vectors (flattened tensors) |   |   |     |         |
| t      | Interpolation parameter in \[0, 1]    |   |   |     |         |
| theta  | Angle between A and B in radians      |   |   |     |         |
|        |                                       | . |   | \_2 | L2 norm |
| dot    | Dot product                           |   |   |     |         |
| sin    | Sine function                         |   |   |     |         |
| arccos | Inverse cosine                        |   |   |     |         |

## Shape Expectations

SLERP operates on flattened 1-D vectors. For a weight tensor of shape (H, I), both A and B are reshaped to length H \* I before interpolation. The result is reshaped back to (H, I).

| Input Shape | Flattened Length | Output Shape |
| ----------- | ---------------- | ------------ |
| (H, I)      | H \* I           | (H, I)       |
| (V, H)      | V \* H           | (V, H)       |
| (H,)        | H                | (H,)         |

## Boundary Cases

### Theta Near Zero

When A and B are nearly parallel, theta approaches 0 and sin(theta) approaches 0. Division by sin(theta) becomes numerically unstable.

| Condition     | Action                                  |
| ------------- | --------------------------------------- |
| theta \< 1e-6 | Fallback to linear interpolation (lerp) |

Fallback formula:

```text theme={null}
SLERP(A, B, t) = (1 - t) * A + t * B
```

### Theta Near Pi

When A and B are nearly opposite, theta approaches pi. The shortest path is ambiguous: both arcs have equal length.

| Condition | Action     |         |                  |
| --------- | ---------- | ------- | ---------------- |
|           | theta - pi | \< 1e-6 | Fallback to lerp |

Engineering assumption: in practice, parameter vectors from fine-tuned variants of the same base model rarely diverge to theta near pi. This case is logged but not specially handled.

## Numerical Stability

All dot products and norms are computed in fp64. The result is cast to fp32 before the final weighted sum. This prevents loss of precision when A and B are nearly orthogonal.

## Example

For two 3-D vectors:

```text theme={null}
A = [1, 0, 0]
B = [0, 1, 0]
t = 0.5

theta = arccos(0) = pi / 2

SLERP(A, B, 0.5) = (sin(pi/4) / sin(pi/2)) * A + (sin(pi/4) / sin(pi/2)) * B
                 = (sqrt(2)/2) * A + (sqrt(2)/2) * B
                 = [0.707, 0.707, 0]
```

## Cross-Links

* [Merge Math](/math/merge-math) for shared notation and weight shapes.
* [Merge Strategies](/merge/merge-strategies) for when SLERP is selected.
* [Tensor Operations](/merge/tensor-operations) for the TensorEngine implementation.