But this is the NOW generation, and waiting will waste a lot of water. So, in the light of more perfect instant gratification, and perhaps conserving some water (though maybe not any power) let us design a different system.

Rather than routing the water up next to the tub, the underbasin water heater routes the water through a winding pattern of pipes beneath the tub. Around these twisting pipes (and carefully sealed off from the water in them!) is a series of heating elements.

As the heating elements start working, both the tub, and the water being pumped into it will get heated, thus making the shower warm faster, without having to run the water.

As an added bonus, the same elements can provide at least some heat to the bathtub while you take a bath, so you won't need to refill with warm water so often.

Some prospective customers will be wary of electricity in their bath room of course. The Japanese have been electrically heating bath water for years without many safety problems. Of course, everything great is made in japan, so if this isn't already made there, we'll just put our factories there if we can get the patents.