Update vector similarity examples to avoid negative scores. (#40493)

Negative scores are no longer allowed, but the cosine similarity between two
vectors lies in the range [-1, 1], and dot products can also be negative. This commit
updates the documentation with an example of how to avoid negative scores.

docs/reference/query-dsl/script-score-query.asciidoc

@@ -36,6 +36,10 @@ GET /_search
 // CONSOLE
 // TEST[setup:twitter]
 
+NOTE: The values returned from `script_score` cannot be negative. In general,
+Lucene requires the scores produced by queries to be non-negative in order to
+support certain search optimizations.
+
 ==== Accessing the score of a document within a script
 
 Within a script, you can
@@ -92,9 +96,9 @@ cosine similarity between a given query vector and document vectors.
         "match_all": {}
       },
       "script": {
-        "source": "cosineSimilarity(params.queryVector, doc['my_dense_vector'])",
+        "source": "cosineSimilarity(params.query_vector, doc['my_dense_vector']) + 1.0" <1>,
         "params": {
-          "queryVector": [4, 3.4, -0.2]  <1>
+          "query_vector": [4, 3.4, -0.2]  <2>
         }
       }
     }
@@ -102,7 +106,8 @@ cosine similarity between a given query vector and document vectors.
 }
 --------------------------------------------------
 // NOTCONSOLE
-<1> To take advantage of the script optimizations, provide a query vector as a script parameter.
+<1> The script adds 1.0 to the cosine similarity to prevent the score from being negative.
+<2> To take advantage of the script optimizations, provide a query vector as a script parameter.
 
 Similarly, for sparse_vector fields, `cosineSimilaritySparse` calculates cosine similarity
 between a given query vector and document vectors.
@@ -116,9 +121,9 @@ between a given query vector and document vectors.
         "match_all": {}
       },
       "script": {
-        "source": "cosineSimilaritySparse(params.queryVector, doc['my_sparse_vector'])",
+        "source": "cosineSimilaritySparse(params.query_vector, doc['my_sparse_vector']) + 1.0",
         "params": {
-          "queryVector": {"2": 0.5, "10" : 111.3, "50": -1.3, "113": 14.8, "4545": 156.0}
+          "query_vector": {"2": 0.5, "10" : 111.3, "50": -1.3, "113": 14.8, "4545": 156.0}
         }
       }
     }
@@ -139,9 +144,12 @@ dot product between a given query vector and document vectors.
         "match_all": {}
       },
       "script": {
-        "source": "dotProduct(params.queryVector, doc['my_dense_vector'])",
+        "source": """
+          double value = dotProduct(params.query_vector, doc['my_vector']);
+          return sigmoid(1, Math.E, -value); <1>
+        """,
         "params": {
-          "queryVector": [4, 3.4, -0.2]
+          "query_vector": [4, 3.4, -0.2]
         }
       }
     }
@@ -150,6 +158,8 @@ dot product between a given query vector and document vectors.
 --------------------------------------------------
 // NOTCONSOLE
 
+<1> Using the standard sigmoid function prevents scores from being negative.
+
 Similarly, for sparse_vector fields, `dotProductSparse` calculates dot product
 between a given query vector and document vectors.
 
@@ -162,9 +172,12 @@ between a given query vector and document vectors.
         "match_all": {}
       },
       "script": {
-        "source": "dotProductSparse(params.queryVector, doc['my_sparse_vector'])",
-        "params": {
-          "queryVector": {"2": 0.5, "10" : 111.3, "50": -1.3, "113": 14.8, "4545": 156.0}
+        "source": """
+          double value = dotProductSparse(params.query_vector, doc['my_sparse_vector']);
+          return sigmoid(1, Math.E, -value);
+        """,
+         "params": {
+          "query_vector": {"2": 0.5, "10" : 111.3, "50": -1.3, "113": 14.8, "4545": 156.0}
         }
       }
     }