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 spell cast for its madness cost is put onto the stack like any other spell. It can be countered, copied, and so on. As it resolves, it's put onto the battlefield if it's a permanent card or into its owner's graveyard if it's an instant or sorcery card.

Casting a spell with madness ignores the timing rules based on the card's card type. For example, you can cast a sorcery with madness if you discard it during an opponent's turn.

To determine the total cost of a spell, start with the mana cost or alternative cost (such as a madness cost) you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.

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.

If you discard a card with madness to pay the cost of a spell or activated ability, that card's madness triggered ability (and the spell that card becomes, if you choose to cast it) will resolve before the spell or ability the discard paid for.

Cards are discarded in a Magic game only from a player's hand. Effects that put cards into a player's graveyard from anywhere else do not cause those cards to be discarded.

Madness works independently of why you're discarding the card. You could discard it to pay a cost, because a spell or ability tells you to, or because you have too many cards in your hand during your cleanup step. You can't discard a card with madness just because you want to, though.

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.

The source's controller must either sacrifice permanents or lose life. They can't choose to sacrifice some permanents and lose some life.

If the player doesn't control enough permanents that they can sacrifice, they must lose life.

The player may always choose not to sacrifice permanents, even if they will end up losing more life than they have or if an effect (such as that of Platinum Emperion) says that they can't lose life.

The player who loses life or permanents is the player who controls the source as Archfiend of Spite's triggered ability resolves. In some rare cases, this may not be the same player who controlled the source as it dealt damage. If the source is no longer in the zone it was in as it dealt damage, the source's last known information is used to determine who controlled it.

Archfiend of Spite's triggered ability will trigger even if it is dealt lethal damage. For example, if Archfiend of Spite blocks a 7/7 creature, its ability will trigger when combat damage is dealt and the attacking player will sacrifice seven permanents or lose 7 life.