Emergency Fund vs Investing Explained

You want your money to grow but you also know an empty rainy-day fund is asking for trouble, and the advice online contradicts itself. Give it your savings, income and expenses and it explains the right order of operations for you: how big the emergency fund should really be, and when it becomes rational to start investing rather than saving more.

I want to start investing but I also know I should have an emergency fund. I currently have {CURRENT_SAVINGS} saved and earn {MONTHLY_INCOME} with {MONTHLY_EXPENSES} in expenses.

Explain the right order of operations for someone in my situation. How much emergency fund is actually the right amount and why? At what point is it rational to start investing rather than building the fund further? What is the cost in expected returns of waiting to invest while building the fund? Give me a concrete answer for my numbers, not a general principle.