The TF-IDF version of the document scoring formula computes the cosine similarity between the TF-IDF vector of a query $q$ and the TF-IDF vector of a document $d$:

$score(q,d) = \sum_{t \in q} \frac{tf-idf(t,q)}{\sqrt{\sum_{q_i \in q} tf-idf^2(q_i,q)}} \cdot \frac{tf-idf(t,d)}{\sqrt{\sum_{d_i \in d} tf-idf^2(d_i,d)}}$

Where:

• $t$ is a term present in the query $q$ and the document $d$.
• $tf-idf(t,q)$ is the TF-IDF weight of term $t$ in query $q$.
• $tf-idf(t,d)$ is the TF-IDF weight of term $t$ in document $d$.
• The denominators represent the Euclidean ($L_2$) norms of the query and document vectors, respectively, which normalize the score to be independent of document and query lengths.