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I'm not so sure it's easy to represent the exact behavior of a complex system.

What would your estimate be to: "rewrite facebook in language L, with the same behavior, bugs and all".

Doing an incremental approximation is very much possible. And it's up to the customer to say when to stop. And also to judge the sufficiency of the speed of convergence.

An incremental approximation is unacceptable for payroll. It legally has to be 100% accurate.

No, you misunderstood. It means you have two systems, Old and New. You use Old and start development of New. And you do the payroll on both, and if there's a difference you debug that and modify New. After a while it's good enough. And you stop using Old.

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