THE RETRIEVAL COSINE GATE

Difficulty: Medium Tags: Embeddings, Retrieval, Similarity Search

Story

A retrieval service ranks document embeddings against a query. One code path uses raw dot product while another uses cosine similarity, causing inconsistent ranking when vector norms differ. Compute cosine similarity for each document vector.

The Math

For query q and document vector d_i:

s_i = (q . d_i) / (||q||_2 * ||d_i||_2)

If either vector has zero norm, define its similarity as 0.

Input Format

query (length d), documents (n x d).

Output Format

n cosine similarities, 6 decimal places.

Constraints

• 1 <= n <= 10^4, 1 <= d <= 1000, component magnitudes at most 10^6.
• Time limit: 2.0 seconds.

Example

Input

query = [1, 0], documents = [[3, 0], [1, 1]]

Output

1.000000 0.707107

Explanation: The first document points in the same direction as the query (similarity 1). The second forms a 45-degree angle with it.