There are a variety of different ways of regarding computations as somehow the same. For Turing machines the most basic form is to regard them as the same if they recognize the same languages (that is halt on the same inputs, accept the same inputs and reject the same inputs). This is probably too broad a definition for your purposes and doesn’t fit most of our intuitions. Thus, for example, two algorithms for determining if an integer is prime, one that uses brute-force, and one that does something quick and clever (e.g. Agrawal’s polynomial time algorithm) should probably not be equivalent for most purposes.
I’m not aware of a better definition that captures precisely what you want.
There are a variety of different ways of regarding computations as somehow the same. For Turing machines the most basic form is to regard them as the same if they recognize the same languages (that is halt on the same inputs, accept the same inputs and reject the same inputs). This is probably too broad a definition for your purposes and doesn’t fit most of our intuitions. Thus, for example, two algorithms for determining if an integer is prime, one that uses brute-force, and one that does something quick and clever (e.g. Agrawal’s polynomial time algorithm) should probably not be equivalent for most purposes.
I’m not aware of a better definition that captures precisely what you want.