It is really not hard. Heat pump coefficients of efficiency floor at 1, but typically range between ~2.5 and 7. That is, for every joule of energy they consume, they pump 2.5 to 7 joules of heat into the conditioned space.

So you have to just look at efficiencies involved.

• The electric heat pump (theoretically ~250-700% efficient at warming a space)
• The efficiency of energy transmission on the electrical grid (maybe around 90% ish? Depends on a lot of factors)
• The fossil fuel furnace (theoretically something like 90-100% efficient)
• The efficiency of the residential fossil fuel delivery system (last I read, something around 5% of natural gas ends up just leaking)
• The efficiency at conversion of chemical to electrical energy by fossil production plants (and this one is rough -- I see numbers like 40-60% efficient)

Still, we're just summing stuff. And while I won't pretend any napkin math here is the same as a formal research project, we can plainly see that the HUGE energy efficiency of heat pumps can easily eclipse the inefficiency of fossil electrical production, all else being equal. Of course, whether it actually WILL be better than a fossil furnace will depend on local factors, but these places are increasingly becoming edge cases. And then, on top of that, you unlock future potential to seamlessly switch fuel sources from fossils to renewables, which becomes very important in lifecycle cost analysis.