Alchemiracles are just like ordinary alchemy except from you can get a chance of getting a 5 star equipment. To be lucky and get 5 star equipment, you need 3 agate of evolutions, 3 coloured orbs (depending on the item you want) and a other piece of equiptment that are around 4 star. You have to save before you do the alchemiracle so you can't trick the game. You normally get around a 15% chance of an alchemiracle happening. This normally takes ages to get. Such as for example the Hypernova sword which is 5 star.
Hypernova Sword = 1 Nebula Sword, 3 Agate of Evolution, 3 Silver Orbs = 20% of success = 5 star http://www.ualberta.ca/~fqureshi/Hypernova.png
Supernova Sword = 1 Nebula Sword, 3 Agate of Evolution, 3 Silver Orbs = 80% of success = 4 star