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For example, the PyFloat_Type has many operations in tp_as_number. When this type object gets initialized, all these operations will be written into tp_dict of it with slots.

On the other hand, when I write a custom class with __add__ in it, it's tp_dict will have __add__. This __add__ function will be written into tp_as_number of the type object with slots when the type object gets initialized.

I think tp_dict has recorded all the information we need. Why do we need other members such as tp_as_number? Is this just a historical issue?

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Because when a C level function is using your type, it isn't performing lookup by name in tp_dict (slow), it's directly pulling the pointer from tp_as_number (fast).

A couple pointer dereferences is a trivial cost, requires no reference counting boilerplate, etc.; the cost is measured in single digit cycles most of the time, where a dict lookup is a couple orders of magnitude more expensive or so.

The common case is actually tp_as_number, since a + b will actually use tp_as_number (or tp_as_sequence) knowing that it is synchronized with the equivalent entries in tp_dict; if simple addition required a dict lookup, CPython would be much slower than it is.

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Thank you very much. So, tp_as_number is used because of lookup speed, and tp_dict is used because sometimes we want do things such as super(MyFloat, self).__add__(other). Is that right?
@delphifirst: Yup. Or reference the __add__ for some other purpose, e.g. to generate values that are some other value + 1 efficiently, map((1).__add__, mynums).

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