So run that system backwards for heating, and you would at only use twice the energy for heating. That said, you can typically cut your heating bill in half using ASHP, with it losing efficiency as the outside temperature decreases. It rarely goes below a COP of 1, so it makes sense to use in place of resistance heating.
However that assumes that you must use electricity for heating. Using wood for heating, you are using a renewable resource that is cheaper for every unit of heat you get.
Even then you have to be willing to keep the thermostat set at 55F/13C or the resistive heaters are going to kick in.
That is not to say gas isn't cheaper.
Be careful, you're angering the thermo gods! If that was true, I would simply use an air-conditioner to heat my house. Cool, the outside and move the heat inside. In reality, you have the efficiencies backwards. Cooling is more energy intensive than heating.
It's true, and people do. Install a heat pump, and save your traditional heat source for when it's sufficiently cold outside.
So if you are cooling a house to 70 degrees in 90 degree weather, you are fighting a 20 degree difference.
If you are heating a house to 70 degrees in 30 degree weather, you are fighting a 40 degree difference.
In a severe heatwave, you might spend weeks with much of the day at 100 degrees; during a cold snap last winter, there was a span where I live where the temperature was usually around 0, and never higher than 10 degrees.
Meanwhile, a furnace works by directly converting energy to heat; for simplicity, let's assume a heater generates 1 unit of heat for 1 unit of energy (it's 100% efficient).
An air-conditioner, however, moves heat, moving heat from inside to outside. An air conditioner doesn't use 1 unit of energy to move 1 unit of heat; an air conditioner might move 3 units of heat using 1 unit of energy.
So putting the two together, it takes roughly 6 times as much energy to heat my house on the coldest days as it does to cool my house on the hottest days.
The full answer is a lot more complicated than that, because other things in houses are generating waste heat and you need to look at the deviations above and below your target temperature range over the full year and across many different places. Someone else has already posted a link to US energy usage showing that total energy spent on air conditioning is a fraction of that spent on heating, but from the above you should be able to get an intuitive feel for why that's so.