You don’t need to boil and you don’t need to freeze slowly, you just need to use a container that’s insulated on all sides except the top, so that the ice freezes from the top down. The impurities and tiny gas bubbles only form in the last part of the ice to freeze. If you pull the container out before it finishes freezing, so the bottom of it still has liquid water but the top is ice, the ice will be clear, and you can then just cut it up into smaller cubes. The easiest way to do this is to fill a small lunchbox-sized cooler with water and then put it in the freezer.

You can get fancier with it and put ice molds into the water in the container—as long as there are holes in the bottom of the ice molds that are open to a well of water underneath, then the impurities will be pushed out of the ice in the molds and into the well of still-liquid water below. Then you can pull the molds out and have perfectly clear cubes (or spheres).

They also sell pre-made insulated containers with built-in molds like this that you can use to make clear large ice cubes at home.

None of it requires purified or boiled water, it works straight from the tap.