THE SEARCH QUERY POSITION

Difficulty: Medium Tags: Transformers, Positional Encoding, Sequence Models

Story

A document transformer must add sinusoidal positional encodings to token embeddings. A refactor mistakenly uses the feature index where the pair index belongs, shifting every frequency and degrading long-document retrieval.

The Math

For position pos and embedding dimension index i (0 <= i < d), with k = i // 2:

PE(pos, 2k)   = sin(pos / 10000^(2k/d))
PE(pos, 2k+1) = cos(pos / 10000^(2k/d))

For an odd d, the final even-indexed component is still defined by the sine formula.

Input Format

pos (int), d (int).

Output Format

List of d positional-encoding components in index order (6 decimal places).

Constraints

• 0 <= pos <= 10^9, 1 <= d <= 512.
• Time limit: 1.0 second.

Example

Input

pos = 1, d = 4

Output

[0.841471, 0.540302, 0.010000, 0.999950]

Explanation: For d=4, the first pair uses angle 1; the second pair uses angle 1/100, so its sine and cosine are approximately 0.010000 and 0.999950.