Elasticsearch semi-equivalent of Solr "bf=product(query('cat'),query('dog'))"

gistfile1.txt

GET /my_index/my_type/_search
{
  "query": {
    "function_score": {
        "query": { <ORIGINAL QUERY> },
        "functions": [
            {
                "filter" : {
                    "term" : { "_all" : "cat"}
                }, 
                "script_score": {
                  "params": {
                        "field": "_all",
                        "term": "cat"
                  },
                  "script": "tf = _index[field][term].tf(); idf = (1 + log ( _index.numDocs() / (_index[field][term].df() + 1))); return sqrt(tf) * pow(idf,2)"
                }
            },
            {
                "filter" : {
                    "term" : { "_all" : "dog"}
                }, 
                "script_score": {
                  "params": {
                        "field": "_all",
                        "term": "dog"
                  },
                  "script": "tf = _index[field][term].tf(); idf = (1 + log ( _index.numDocs() / (_index[field][term].df() + 1))); return sqrt(tf) * pow(idf,2)"
                }
            }
        ],
        "score_mode": "multiply",
        "boost_mode": "sum"
    }
  }
}

"score_mode": "multiply" will take the product of all the functions defined in the "functions": [...] block and then "boost_mode": "sum" will add that to the score of the <ORIGINAL QUERY>

bf=... in Solr also adds the function result to the original query score.

Solr's query() will also perform field-length normalization and query normalization which this will not.
(see Lucene's Practical Scoring Function)
(Elasticsearch doesn't seem to provide scripting access to the field-length encoding provided in norms).

The scoring behavior here should be similar to the OP's Solr query if omitNorms=true were specified on the fields searched.