Which is more sensible: heat pump or insulation?

Question why?- If I heat 10 liters of water from 20° to 40°, the heat pump needs, for example, X watts // I can measure all 3 values - temperature - kW electricity - time
Whether I then use the water as shower water or pump it into the heating system - I thought - would be the same?? 40° is 40°
Maybe I don’t see something quite right. I’m always on the fast trial side --- (the heat pump hardly cost me anything)
And as a small note on skepticism: A long time ago, I was supposed to use a new control system recommended by Viessmann, which would save energy!

For efficiency, the temperature difference is decisive. Heating from 10 to 20 degrees requires more energy than twice from 10 to 15 degrees.
It is indeed irrelevant whether you operate the heating or the shower with 40° warm water, but it makes a difference if you operate the heating at 35° and the shower at 50°.

The maximum achievable efficiency is given by COP = T_warm / (T_warm - T_cold). In other words, your efficiency decreases with increasing temperature lift. In addition, the devices do not operate at the theoretical limit and become additionally less efficient at larger lifts.

Since the outside temperature is already in the lower range, and therefore the electric heating rods in the heat pumps also switch on

No, if planned properly, the heating rod (in normal areas, maybe not in the Black Forest at 1,000m altitude) does not switch on at all. Only during the defrosting process. has already written something about the efficiency of the WWWP. Simplified, it can also be compared to a car. Driving steadily on the highway at 90-110 km/h is significantly more efficient than constantly accelerating from 80 to 200 and then braking again.

:)Thanks, understood. I have packed up my stuff for now, as the weather has slowed down my activity.

From a purely financial perspective, it is worth waiting for the upcoming funding programs. The coalition agreement contains quite a bit about the renovation of the building stock.

Is there an update on this? I am planning to insulate the facade. Better to wait then?