burst forth vs leap out

burst forth and leap out both are verbs.

Word	Noun	Adjective	Verb	Adverb
burst forth	No	No	Yes	No
leap out	No	No	Yes	No

As verbs, burst forth and leap out are synonyms defined as:

burst forth and leap out: jump out from a hiding place and surprise (someone)

Other synonyms of burst forth include rush out, sally out.

burst forth (verb)	leap out (verb)
be unleashed; emerge with violence or noise	jump out from a hiding place and surprise (someone)
come into or as if into flower	be highly noticeable
jump out from a hiding place and surprise (someone)