Can you return a card to your hand with a madness ability if you discard it as part of a cost?

Madness (CR 702.34) triggers when you discard a card that has madness. Instead of going to your graveyard, the card is exiled. You then have the opportunity to cast it for its madness cost. If you choose not to cast it, it moves from exile to your graveyard. At no point does the rules text allow you to return the card to your hand.

The key mechanic is a replacement effect combined with a triggered ability. CR 702.34a states that if you discard a card with madness, it is exiled rather than put into your graveyard, and the madness ability triggers, giving you the option to cast it. Returning it to your hand is simply not one of the offered choices.

Discarding a card as a cost is still discarding for the purposes of madness (CR 702.34b). So madness does trigger when a card is discarded as a cost — you just can't choose to put it back in your hand as a result of that trigger.

Concrete example: You activate Seasoned Pyromancer's ability, discarding Fiery Temper (which has madness). Fiery Temper is exiled and its madness ability triggers. You may cast it for its madness cost of {R}, or let it go to the graveyard. You cannot choose to put Fiery Temper back into your hand.

If you want to return a discarded card to your hand, you would need a separate card or effect that specifically does so, such as Elixir of Immortality or a card with a triggered ability that retrieves cards from the graveyard — not madness itself.