If you choose not to cast a card with madness when the madness triggered ability resolves, it's put into your graveyard. Madness doesn't give you another chance to cast it later.
Call to the Netherworld
If you discard a card with madness while a spell or ability is resolving, it moves immediately to exile. Continue resolving that spell or ability, noting that the card you discarded is not in your graveyard at this time. Its madness triggered ability will be placed onto the stack once that spell or ability has completely resolved.
A card with madness that's discarded counts as having been discarded even though it's put into exile rather than a graveyard. If it was discarded to pay a cost, that cost is still paid. Abilities that trigger when a card is discarded will still trigger.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Vance Kovacs
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
