Function update_cache 

pub fn update_cache(
    cache: Option<&dyn BaseCache>,
    existing_prompts: &mut HashMap<usize, Vec<Generation>>,
    llm_string: &str,
    missing_prompt_idxs: &[usize],
    new_results: &LLMResult,
    prompts: &[String],
) -> Option<HashMap<String, Value>>

Helper function to update cache with new results.