How do we use sin,cos,tan generically (including user-defined types) in Python?

Question

Edit: Let me try to reword and improve my question. The old version is attached at the bottom.

What I am looking for is a way to express and use free functions in a type-generic way. Examples:

abs(x)  # maps to x.__abs__()
next(x) # maps to x.__next__() at least in Python 3
-x      # maps to x.__neg__()

In these cases the functions have been designed in a way that allows users with user-defined types to customize their behaviour by delegating the work to a non-static method call. This is nice. It allows us to write functions that don't really care about the exact parameter types as long as they "feel" like objects that model a certain concept.

Counter examples: Functions that can't be easily used generically:

math.exp  # only for reals
cmath.exp # takes complex numbers

Suppose, I want to write a generic function that applies exp on a list of number-like objects. What exp function should I use? How do I select the correct one?

def listexp(lst):
    return [math.exp(x) for x in lst]

Obviously, this won't work for lists of complex numbers even though there is an exp for complex numbers (in cmath). And it also won't work for any user-defined number-like type which might offer its own special exp function.

So, what I'm looking for is a way to deal with this on both sides -- ideally without special casing a lot of things. As a writer of some generic function that does not care about the exact types of parameters I want to use the correct mathematical functions that is specific to the types involved without having to deal with this explicitly. As a writer of a user-defined type, I would like to expose special mathematical functions that have been augmented to deal with additional data stored in those objects (similar to the imaginary part of complex numbers).

What is the preferred pattern/protocol/idiom for doing that? I did not yet test numpy. But I downloaded its source code. As far as I know, it offers a sin function for arrays. Unfortunately, I haven't found its implementation yet in the source code. But it would be interesting to see how they managed to pick the right sin function for the right type of numbers the array currently stores.

In C++ I would have relied on function overloading and ADL (argument-dependent lookup). With C++ being statically typed, it should come as no surprise that this (name lookup, overload resolution) is handled completely at compile-time. I suppose, I could emulate this at runtime with Python and the reflective tools Python has to offer. But I also know that trying to import a coding style into another language might be a bad idea and not very idiomatic in the new language. So, if you have a different idea for an approach, I'm all ears.

I guess, somewhere at some point I need to manually do some type-dependent dispatching in an extensible way. Maybe write a module "tgmath" (type generic math) that comes with support for real and complex support as well as allows others to register their types and special case functions... Opinions? What do the Python masters say about this?

TIA

Edit: Apparently, I'm not the only one who is interested in generic functions and type-dependent overloading. There is PEP 3124 but it is in draft state since 4 years ago.

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Old version of the question:

I have a strong background in Java and C++ and just recently started learning Python. What I'm wondering about is: How do we extend mathematical functions (at least their names) so they work on other user-defined types? Do these kinds of functions offer any kind of extension point/hook I can leverage (similar to the iterator protocol where next(obj) actually delegates to obj.__next__, etc) ?

In C++ I would have simply overloaded the function with the new parameter type and have the compiler figure out which of the functions was meant using the argument expressions' static types. But since Python is a very dynamic language there is no such thing as overloading. What is the preferred Python way of doing this?

Also, when I write custom functions, I would like to avoid long chains of

if isinstance(arg,someClass):
    suchandsuch
elif ...

What are the patterns I could use to make the code look prettier and more Pythonish?

I guess, I'm basically trying to deal with the lack of function overloading in Python. At least in C++ overloading and argument-dependent lookup is an important part of good C++ style.

Is it possible to make

x = udt(something)  # object of user-defined type that represents a number
y = sin(x)          # how do I make this invoke custom type-specific code for sin?
t = abs(x)          # works because abs delegates to __abs__() which I defined.

work? I know I could make sin a non-static method of the class. But then I lose genericity because for every other kind of number-like object it's sin(x) and not x.sin().

Adding a __float__ method is not acceptable since I keep additional information in the object such as derivatives for "automatic differentiation".

TIA

Edit: If you're curious about what the code looks like, check this out. In an ideal world I would be able to use sin/cos/sqrt in a type-generic way. I consider these functions part of the objects interface even if they are "free functions". In __somefunction I did not qualify the functions with math. nor __main__.. It just works because I manually fall back on math.sin (etc) in my custom functions via the decorator. But I consider this to be an ugly hack.

Answer 1

you can do this, but it works backwards. you implement __float__() in your new type and then sin() will work with your class.

in other words, you don't adapt sine to work on other types; you adapt those types so that they work with sine.

this is better because it forces consistency. if there is no obvious mapping from your object to a float then there probably isn't a reasonable interpretation of sin() for that type.

[sorry if i missed the "__float__ won't work" part earlier; perhaps you added that in response to this? anyway, for convincing proof that what you want isn't possible, python has the cmath library to add sin() etc for complex numbers...]

Answer 2

If you want the return type of math.sin() to be your user-defined type, you appear to be out of luck. Python's math library is basically a thin wrapper around a fast native IEEE 754 floating point math library. If you want to be internally consistent and duck-typed, you can at least put the extensibility shim that python is missing into your own code.

def sin(x):
    try:
        return x.__sin__()
    except AttributeError:
        return math.sin(x)

Now you can import this sin function and use it indiscriminately wherever you used math.sin previously. It's not quite as pretty as having math.sin pick up your duck-typing automatically but at least it can be consistent within your codebase.