1. in an extremely enthusiastic or passionate way
2. in a frenzied or uncontrolled manner
3. in an insane or mentally unstable way

Examples

• She was madly in love with her college sweetheart.
• The fans cheered madly when their team scored.
• He searched madly through his papers for the missing document.
• The dog ran madly around the yard chasing its tail.
• She was typing madly to finish her report before the deadline.
• The character in the story laughed madly at his own jokes.