--- title: "Reciprocal rank fusion formula" canonical: "https://xerj.org/science/reciprocal-rank-fusion" updated: "2026-11-14" source: "content/science/reciprocal-rank-fusion.md" --- # Reciprocal rank fusion formula **Short answer** (updated 2026-11-14): What is the reciprocal rank fusion formula? For each list an item appears in, add 1 / (k + rank), then add those values. The lists do not need a shared score scale. Cormack, Clarke and Buettcher found k = 60 near-optimal in their 2009 pilots. XERJ uses k = 60 by default and accepts another value. **TL;DR** Reciprocal rank fusion merges two or more ranked lists by adding 1 / (k + rank) for each list an item appears in. The lists do not need a shared score scale. ## Agent prompt ```text Act as a coding agent. Read https://xerj.org/llms.txt, install XERJ, start a local node, index one folder I point you to, and run one search so I can see a cited passage. ``` *Figure: The formula and the choice k = 60 come from Cormack, Clarke and Buettcher, Reciprocal Rank Fusion outperforms Condorcet and individual Rank Learning Methods, SIGIR 2009. The three documents are a teaching example, not a XERJ run.* ## How it works You start with two or more ranked lists, for example a BM25 list and a vector list. For each document, look up its rank in each list. Add 1 / (k + rank) for every list it appears in. Missing from a list means that list adds nothing. Sort by the total. The table above is a labeled example with three documents and k = 60. It is not a measurement on a XERJ corpus. In that example, a document that ranks first in one list and second in the other ties with a document that does the reverse. A document that appears in only one list ranks lower. k is a constant that softens the gap between early and later ranks. The 2009 paper reports k = 60 as near-optimal in the authors' pilots. ## The formula The score for a document is the sum, over every ranked list it appears in, of: `score(d) = 1 / (k + rank_i(d))` `rank_i(d)` is the 1-based rank of `d` in list `i`. If the document is missing from a list, that list adds nothing. `k` is a constant. A larger `k` flattens the gap between rank 1 and later ranks. In their 2009 pilots, Cormack, Clarke and Buettcher found `k = 60` near-optimal. That value is also the published XERJ default. The table above is a labeled example, not a measurement on a XERJ corpus. Here is the same example in Python: ```python # Example only, not a XERJ measurement. # k = 60 comes from Cormack, Clarke and Buettcher, SIGIR 2009. def rrf_score(ranks, k=60): return sum(1.0 / (k + rank) for rank in ranks) # Document X ranks 1 in list A and 2 in list B: print(round(rrf_score([1, 2]), 4)) # 0.0325 ``` ## How XERJ uses it XERJ uses reciprocal rank fusion to merge a lexical arm with a vector arm in [hybrid search](/science/hybrid-search). The product default is k = 60, and the request can set another value. A weighted linear sum is also available when you want to keep raw scores, after normalization. The practical guide is [when to use reciprocal rank fusion](/answers/reciprocal-rank-fusion-when-to-use). The request shape is in [how to run hybrid search in XERJ](/answers/what-is-hybrid-search). ## History and research The method was introduced by Cormack, Clarke and Buettcher, [Reciprocal Rank Fusion outperforms Condorcet and individual Rank Learning Methods](https://cormack.uwaterloo.ca/cormacksigir09-rrf.pdf), SIGIR 2009, pages 758 to 759. An ACM page also lists the paper at [doi 10.1145/1571941.1572114](https://dl.acm.org/doi/10.1145/1571941.1572114). The k = 60 figure on this page is theirs, from those pilots. It is also the published XERJ default. This page does not add a new fusion experiment. ## Guides See [when to use reciprocal rank fusion](/answers/reciprocal-rank-fusion-when-to-use), [how XERJ combines search](/answers/how-xerj-combines-search) and the [hybrid search recipe](/docs/recipes/hybrid-search). ## Related terms [Hybrid search](/science/hybrid-search), [BM25](/science/bm25), and [semantic search](/science/semantic-search). ## FAQ ### What is the reciprocal rank fusion formula? For each list an item appears in, add 1 / (k + rank). Then add those values. A common choice of k is 60, from the 2009 paper. ### Why use ranks instead of raw scores? BM25 scores and vector similarities sit on different scales. Ranks do not, so you can fuse the lists without a shared unit. ### What does the constant k do? A larger k flattens the gap between rank 1 and later ranks. In their 2009 pilots, Cormack, Clarke and Buettcher found k = 60 near-optimal. XERJ uses k = 60 by default and accepts another value. ### How does XERJ use RRF? As the usual fusion method in hybrid search. The how-to, and when to pick a weighted sum instead, is the hybrid search answer and the when-to-use RRF answer. ## Related - [What is hybrid search?](/science/hybrid-search) - [What is Reciprocal Rank Fusion?](/answers/reciprocal-rank-fusion-when-to-use) - [How to run hybrid search in XERJ](/answers/what-is-hybrid-search) - [How does XERJ combine BM25 and kNN?](/answers/how-xerj-combines-search) - [What is agentic memory?](/science/agentic-memory) (same intent: Science) - [BM25 vs vector search](/science/bm25) (same intent: Science) - [What is context engineering?](/science/context-engineering) (same intent: Science)