Chang 'e is the descendant of Chang 'e Xi, also called Heng E, and the wife of Hou Yi. Hou Yi is a sharpshooter. Legend has it that 10 suns appeared in the ancient sky at the same time, and the earth was scorched. Hou Yi killed nine suns for the people. The heavenly queen gave him the elixir of life. His wife, Chang 'e, took advantage of Houyi's carelessness to steal these medicines, and then she became a fairy and flew to the Heavenly Palace. The heavenly queen punished her. Let her think in the Guanghan Palace. This is the story of Chang 'e.
Wu Gang, on the other hand, is a character in another fairy tale. In the Tang Dynasty, he wrote Youyang Miscellanies, saying only that WU GANG was a fairy and was punished for cutting down osmanthus trees on the moon. As for WU GANG's "learning immortals", how can there be "passers-by"? What do you mean "end"? The language is unclear. After sorting out the ancient books, later generations have a process of understanding the myth of WU GANG.
The story goes like this: WU GANG went to study immortals and was away from home for three years. A man named Buling, the grandson of Emperor Yan and the sun god, took the opportunity to have sex with WU GANG's wife A Nv and gave birth to three children. When WU GANG came back, he was very angry and killed Bering. Moreover, he killed Sun Boling and died miserably (probably related to clan vendetta). Emperor Yan, the sun god, was furious, but because his grandson was at fault first and couldn't kill WU GANG, he sent WU GANG to cut down osmanthus trees on the moon, so that the trees could grow together, and WU GANG's ineffective labor had to go on forever.
WU GANG killed the "adulterer" but didn't punish his wife A Nv, probably because A Nv's wife is a woman after all and can't resist violence. A Nv's wife also felt guilty, so she sent the children to the moon, turned them into toads and rabbits, and accompanied their nominal father. This is the origin of the toad and the rabbit on the moon.
As for the later myth, why did the rabbit go to Chang 'e? This should be the result of the late integration and development of fairy tales. Maybe WU GANG is busy logging every day and has no time to look after the children, and Chang 'e stays on the moon all day, so let Chang 'e take care of it. It can also be seen that the ancients in China were very human, and two irrelevant mythical figures were linked by toad and sons of liberty. This is a kind of mutual care and help.