Update k-NN Cosine formula

_search-plugins/knn/approximate-knn.md

@@ -402,7 +402,7 @@ Not every method supports each of these spaces. Be sure to check out [the method
 | `l1`  | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n \lvert x_i - y_i \rvert $$ | $$ score = {1 \over {1 + d} } $$ |
 | `l2`  | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n (x_i - y_i)^2 $$ | $$ score = {1 \over 1 + d } $$ |
 | `linf` | $$ d(\mathbf{x}, \mathbf{y}) = max(\lvert x_i - y_i \rvert) $$ | $$ score = {1 \over 1 + d } $$ |
-| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$, <br> where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | **NMSLIB** and **Faiss**:<br>$$ score = {1 \over 1 + d } $$  <br><br>**Lucene**:<br>$$ score = {2 - d \over 2}$$ |
+| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$, <br> where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | $$ score = {2 - d \over 2} $$ |
 | `innerproduct` (supported for Lucene in OpenSearch version 2.13 and later) | **NMSLIB** and **Faiss**:<br> $$ d(\mathbf{x}, \mathbf{y}) = - {\mathbf{x} \cdot \mathbf{y}} = - \sum_{i=1}^n x_i y_i $$  <br><br>**Lucene**:<br> $$ d(\mathbf{x}, \mathbf{y}) = {\mathbf{x} \cdot \mathbf{y}} = \sum_{i=1}^n x_i y_i $$ | **NMSLIB** and **Faiss**:<br> $$ \text{If} d \ge 0,  score = {1 \over 1 + d }$$ <br> $$\text{If} d < 0, score = −d + 1$$  <br><br>**Lucene:**<br> $$ \text{If} d > 0, score = d + 1 $$ <br> $$\text{If} d \le 0, score = {1 \over 1 + (-1 \cdot d) }$$ |
 | `hamming` (supported for binary vectors in OpenSearch version 2.16 and later) | $$ d(\mathbf{x}, \mathbf{y}) = \text{countSetBits}(\mathbf{x} \oplus \mathbf{y})$$ | $$ score = {1 \over 1 + d } $$ |
 

_search-plugins/knn/knn-score-script.md

@@ -293,7 +293,7 @@ A _space_ corresponds to the function used to measure the distance between two p
 | `l1`  | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n \lvert x_i - y_i \rvert $$ | $$ score = {1 \over {1 + d} } $$ |
 | `l2`  | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n (x_i - y_i)^2 $$ | $$ score = {1 \over 1 + d } $$ |
 | `linf` | $$ d(\mathbf{x}, \mathbf{y}) = max(\lvert x_i - y_i \rvert) $$ | $$ score = {1 \over 1 + d } $$ |
-| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$, <br> where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | $$ score = 2 - d $$ |
+| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$, <br> where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | $$ score = {2 - d \over 2 } $$ |
 | `innerproduct` (supported for Lucene in OpenSearch version 2.13 and later) | $$ d(\mathbf{x}, \mathbf{y}) = - {\mathbf{x} \cdot \mathbf{y}} = - \sum_{i=1}^n x_i y_i $$ | $$ \text{If} d \ge 0,  score = {1 \over 1 + d }$$ <br> $$\text{If} d < 0, score = −d + 1$$ |
 | `hammingbit` (supported for binary and long vectors) <br><br>`hamming` (supported for binary vectors in OpenSearch version 2.16 and later) | $$ d(\mathbf{x}, \mathbf{y}) = \text{countSetBits}(\mathbf{x} \oplus \mathbf{y})$$ | $$ score = {1 \over 1 + d } $$ |