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authorSebastian Berg <sebastian@sipsolutions.net>2020-10-13 18:36:47 -0500
committerGitHub <noreply@github.com>2020-10-13 18:36:47 -0500
commitdbc4f5d3fff330a842576427674ee4fc5de0164d (patch)
tree78bce212d7e47f29bd4d4e00a891196d7498f5cb
parent9af9a963b9b154dbc958f05480dfdee6a2965292 (diff)
downloadnumpy-dbc4f5d3fff330a842576427674ee4fc5de0164d.tar.gz
NEP: Initial draft for NEP 43 for extensible ufuncs (gh-16723)
* NEP: Initial draft for NEP 43 * NEP: Small updates to NEP 43 * Partial iteration on some content, mostly updates the calling convention Its not quite settled, I don't quite like the `npy_intp *` as a default "user-data" yet, but I think I may warm up to it, and I think some kind of default like that is really very convenient. Split out `user_data *`, since I think Marten has a point that separating "constant" and volatile data is conceptionally easier. * Add a bit more overview-intro at the start and refactor "bound" method section * Small fixups based on Marten and also Hameers comments * Revise the "promotion and dispatching" section a bit. * Fix back-compat section and add a tiny bit of implementation * WIP: Further updates nep 43 * Revise NEP 43 and expand on arraymethod (previously the dispatching/promotion was more fleshed out) * Update image and make sure it is included at the right place. * small fixups * Update doc/neps/nep-0043-extensible-ufuncs.rst Co-authored-by: Dave Hirschfeld <dave.hirschfeld@gmail.com> * Fix comment which was missing the final part of sentence * Pass through, up to "steps involved in a ufunc call" * Review up to ArrayMethod * Review up to ArrayMethod Specs * Review up to loop reuse * Review most of the remaining document * Fix title typo * Wording updates and minor fixups. * Apply suggestions from code review Co-authored-by: Ross Barnowski <rossbar@berkeley.edu> * Address Ross's additional comments Co-authored-by: Dave Hirschfeld <dave.hirschfeld@gmail.com> Co-authored-by: Ross Barnowski <rossbar@berkeley.edu>
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diff --git a/doc/neps/nep-0043-extensible-ufuncs.rst b/doc/neps/nep-0043-extensible-ufuncs.rst
new file mode 100644
index 000000000..008cfad5f
--- /dev/null
+++ b/doc/neps/nep-0043-extensible-ufuncs.rst
@@ -0,0 +1,1301 @@
+.. _NEP43:
+
+==============================================================================
+NEP 43 — Enhancing the Extensibility of UFuncs
+==============================================================================
+
+:title: Enhancing the Extensibility of UFuncs
+:Author: Sebastian Berg
+:Status: Draft
+:Type: Standard
+:Created: 2020-06-20
+
+
+.. note::
+
+ This NEP is fourth in a series:
+
+ - :ref:`NEP 40 <NEP40>` explains the shortcomings of NumPy's dtype implementation.
+
+ - :ref:`NEP 41 <NEP41>` gives an overview of our proposed replacement.
+
+ - :ref:`NEP 42 <NEP42>` describes the new design's datatype-related APIs.
+
+ - NEP 43 (this document) describes the new design's API for universal functions.
+
+
+******************************************************************************
+Abstract
+******************************************************************************
+
+The previous NEP 42 proposes the creation of new DTypes which can
+be defined by users outside of NumPy itself.
+The implementation of NEP 42 will enable users to create arrays with a custom dtype
+and stored values.
+This NEP outlines how NumPy will operate on arrays with custom dtypes in the future.
+The most important functions operating on NumPy arrays are the so called
+"universal functions" (ufunc) which include all math functions, such as
+``np.add``, ``np.multiply``, and even ``np.matmul``.
+These ufuncs must operate efficiently on multiple arrays with
+different datatypes.
+
+This NEP proposes to expand the design of ufuncs.
+It makes a new distinction between the ufunc which can operate
+on many different dtypes such as floats or integers,
+and a new ``ArrayMethod`` which defines the efficient operation for
+specific dtypes.
+
+.. note::
+
+ Details of the private and external APIs may change to reflect user
+ comments and implementation constraints. The underlying principles and
+ choices should not change significantly.
+
+
+******************************************************************************
+Motivation and scope
+******************************************************************************
+
+The goal of this NEP is to extend universal
+functions support the new DType system detailed in NEPs 41 and 42.
+While the main motivation is enabling new user-defined DTypes, this will
+also significantly simplify defining universal functions for NumPy strings or
+structured DTypes.
+Until now, these DTypes are not supported by any of NumPy's functions
+(such as ``np.add`` or ``np.equal``), due to difficulties arising from
+their parametric nature (compare NEP 41 and 42), e.g. the string length.
+
+Functions on arrays must handle a number of distinct steps which are
+described in more detail in section "`Steps involved in a UFunc call`_".
+The most important ones are:
+
+- Organizing all functionality required to define a ufunc call for specific
+ DTypes. This is often called the "inner-loop".
+- Deal with input for which no exact matching implementation is found.
+ For example when ``int32`` and ``float64`` are added, the ``int32``
+ is cast to ``float64``. This requires a distinct "promotion" step.
+
+After organizing and defining these, we need to:
+
+- Define the user API for customizing both of the above points.
+- Allow convenient reuse of existing functionality.
+ For example a DType representing physical units, such as meters,
+ should be able to fall back to NumPy's existing math implementations.
+
+This NEP details how these requirements will be achieved in NumPy:
+
+- All DTyper-specific functionality currently part of the ufunc
+ definition will be defined as part of a new `ArrayMethod`_ object.
+ This ``ArrayMethod`` object will be the new, preferred, way to describe any
+ function operating on arrays.
+
+- Ufuncs will dispatch to the ``ArrayMethod`` and potentially use promotion
+ to find the correct ``ArrayMethod`` to use.
+ This will be described in the `Promotion and dispatching`_ section.
+
+A new C-API will be outlined in each section. A future Python API is
+expected to be very similar and the C-API is presented in terms of Python
+code for readability.
+
+The NEP proposes a large, but necessary, refactor of the NumPy ufunc internals.
+This modernization will not affect end users directly and is not only a necessary
+step for new DTypes, but in itself a maintenance effort which is expected to
+help with future improvements to the ufunc machinery.
+
+While the most important restructure proposed is the new ``ArrayMethod``
+object, the largest long-term consideration is the API choice for
+promotion and dispatching.
+
+
+***********************
+Backwards Compatibility
+***********************
+
+The general backwards compatibility issues have also been listed
+previously in NEP 41.
+
+The vast majority of users should not see any changes beyond those typical
+for NumPy releases.
+There are three main users or use-cases impacted by the proposed changes:
+
+1. The Numba package uses direct access to the NumPy C-loops and modifies
+ the NumPy ufunc struct directly for its own purposes.
+2. Astropy uses its own "type resolver", meaning that a default switch over
+ from the existing type resolution to a new default Promoter requires care.
+3. It is currently possible to register loops for dtype *instances*.
+ This is theoretically useful for structured dtypes and is a resolution
+ step happening *after* the DType resolution step proposed here.
+
+This NEP will try hard to maintain backward compatibility as much as
+possible. However, both of these projects have signaled willingness to adapt
+to breaking changes.
+
+The main reason why NumPy will be able to provide backward compatibility
+is that:
+
+* Existing inner-loops can be wrapped, adding an indirection to the call but
+ maintaining full backwards compatibility.
+ The ``get_loop`` function can, in this case, search the existing
+ inner-loop functions (which are stored on the ufunc directly) in order
+ to maintain full compatibility even with potential direct structure access.
+* Legacy type resolvers can be called as a fallback (potentially caching
+ the result). The resolver may need to be called twice (once for the DType
+ resolution and once for the ``resolve_descriptor`` implementation).
+* The fallback to the legacy type resolver should in most cases handle loops
+ defined for such structured dtype instances. This is because if there is no
+ other ``np.Void`` implementation, the legacy fallback will retain the old
+ behaviour at least initially.
+
+The masked type resolvers specifically will *not* remain supported, but
+has no known users (including NumPy itself, which only uses the default
+version).
+
+While the above changes potentially break some workflows,
+we believe that the long-term improvements vastly outweigh this.
+Further, packages such as astropy and Numba are capable of adapting so that
+end-users may need to update their libraries but not their code.
+
+
+******************************************************************************
+Usage and impact
+******************************************************************************
+
+This NEP restructures how operations on NumPy arrays are defined both
+within NumPy and for external implementers.
+The NEP mainly concerns those who either extend ufuncs for custom DTypes
+or create custom ufuncs. It does not aim to finalize all
+potential use-cases, but rather restructure NumPy to be extensible and allow
+addressing new issues or feature requests as they arise.
+
+
+Overview and end user API
+=========================
+
+To give an overview of how this NEP proposes to structure ufuncs,
+the following describes the potential exposure of the proposed restructure
+to the end user.
+
+Universal functions are much like a Python method defined on the DType of
+the array when considering a ufunc with only a single input::
+
+ res = np.positive(arr)
+
+could be implemented (conceptually) as::
+
+ positive_impl = arr.dtype.positive
+ res = positive_impl(arr)
+
+However, unlike methods, ``positive_impl`` is not stored on the dtype itself.
+It is rather the implementation of ``np.positive`` for a specific DType.
+Current NumPy partially exposes this "choice of implementation" using
+the ``dtype`` (or more exact ``signature``) attribute in universal functions,
+although these are rarely used::
+
+ np.positive(arr, dtype=np.float64)
+
+forces NumPy to use the ``positive_impl`` written specifically for the Float64
+DType.
+
+This NEP makes the distinction more explicit, by creating a new object to
+represent ``positive_impl``::
+
+ positive_impl = np.positive.resolve_impl((type(arr.dtype), None))
+ # The `None` represents the output DType which is automatically chosen.
+
+While the creation of a ``positive_impl`` object and the ``resolve_impl``
+method is part of this NEP, the following code::
+
+ res = positive_impl(arr)
+
+may not be implemented initially and is not central to the redesign.
+
+In general NumPy universal functions can take many inputs.
+This requires looking up the implementation by considering all of them
+and makes ufuncs "multi-methods" with respect to the input DTypes::
+
+ add_impl = np.add.resolve_impl((type(arr1.dtype), type(arr2.dtype), None))
+
+This NEP defines how ``positive_impl`` and ``add_impl`` will be represented
+as a new ``ArrayMethod`` which can be implemented outside of NumPy.
+Further, it defines how ``resolve_impl`` will implement and solve dispatching
+and promotion.
+
+The reasons for this split may be more clear after reviewing the
+`Steps involved in a UFunc call`_ section.
+
+
+Defining a new ufunc implementation
+===================================
+
+The following is a mock-up of how a new implementation, in this case
+to define string equality, will be added to a ufunc.
+
+.. code-block:: python
+
+ class StringEquality(BoundArrayMethod):
+ nin = 1
+ nout = 1
+ DTypes = (String, String, Bool)
+
+ def resolve_descriptors(context, given_descrs):
+ """The strided loop supports all input string dtype instances
+ and always returns a boolean. (String is always native byte order.)
+
+ Defining this function is not necessary, since NumPy can provide
+ it by default.
+ """
+ assert isinstance(given_descrs[0], context.DTypes[0])
+ assert isinstance(given_descrs[1], context.DTypes[1])
+
+ # The operation is always "safe" casting (most ufuncs are)
+ return (given_descrs[0], given_descrs[1], context.DTypes[2]()), "safe"
+
+ def strided_loop(context, n, data, strides):
+ """The 1-D strided loop, similar to those used in current ufuncs"""
+ # n: Number of elements in the one dimensional loop
+ # data: Pointers to the array data.
+ # strides: strides to iterate all elements
+ num_chars1 = context.descriptors[0].itemsize
+ num_chars2 = context.descriptors[0].itemsize
+
+ # C code using the above information to compare the strings in
+ # both arrays. In particular, this loop requires the `num_chars1`
+ # and `num_chars2`. Information which is currently not easily
+ # available.
+
+ np.equal.register_impl(StringEquality)
+ del StringEquality # may be deleted.
+
+
+This definition will be sufficient to create a new loop, and the
+structure allows for expansion in the future; something that is already
+required to implement casting within NumPy itself.
+We use ``BoundArrayMethod`` and a ``context`` structure here. These
+are described and motivated in details later. Briefly:
+
+* ``context`` is a generalization of the ``self`` that Python passes to its
+ methods.
+* ``BoundArrayMethod`` is equivalent to the Python distinction that
+ ``class.method`` is a method, while ``class().method`` returns a "bound" method.
+
+
+Customizing Dispatching and Promotion
+=====================================
+
+Finding the correct implementation when ``np.positive.resolve_impl()`` is
+called is largely an implementation detail.
+But, in some cases it may be necessary to influence this process when no
+implementation matches the requested DTypes exactly:
+
+.. code-block:: python
+
+ np.multiple.resolve_impl((Timedelta64, Int8, None))
+
+will not have an exact match, because NumPy only has an implementation for
+multiplying ``Timedelta64`` with ``Int64``.
+In simple cases, NumPy will use a default promotion step to attempt to find
+the correct implementation, but to implement the above step, we will allow
+the following:
+
+.. code-block:: python
+
+ def promote_timedelta_integer(ufunc, dtypes):
+ new_dtypes = (Timdelta64, Int64, dtypes[-1])
+ # Resolve again, using Int64:
+ return ufunc.resolve_impl(new_dtypes)
+
+ np.multiple.register_promoter(
+ (Timedelta64, Integer, None), promote_timedelta_integer)
+
+Where ``Integer`` is an abstract DType (compare NEP 42).
+
+
+.. _steps_of_a_ufunc_call:
+
+****************************************************************************
+Steps involved in a UFunc call
+****************************************************************************
+
+Before going into more detailed API choices, it is helpful to review the
+steps involved in a call to a universal function in NumPy.
+
+A UFunc call is split into the following steps:
+
+1. *Handle ``__array_ufunc__`` protocol:*
+
+ * For array-likes such as a Dask arrays, NumPy can defer the operation.
+ This step is performed first, and unaffected by this NEP (compare :ref:`NEP18`).
+
+2. *Promotion and dispatching*
+
+ * Given the DTypes of all inputs, find the correct implementation.
+ E.g. an implementation for ``float64``, ``int64`` or a user-defined DType.
+
+ * When no exact implementation exists, *promotion* has to be performed.
+ For example, adding a ``float32`` and a ``float64`` is implemented by
+ first casting the ``float32`` to ``float64``.
+
+3. *Parametric ``dtype`` resolution:*
+
+ * In general, whenever an output DType is parametric the parameters have
+ to be found (resolved).
+ * For example, if a loop adds two strings, it is necessary to define the
+ correct output (and possibly input) dtypes. ``S5 + S4 -> S9``, while
+ an ``upper`` function has the signature ``S5 -> S5``.
+ * When they are not parametric, a default implementation is provided
+ which fills in the default dtype instances (ensuring for example native
+ byte order).
+
+4. *Preparing the iteration:*
+
+ * This step is largely handled by ``NpyIter`` internally (the iterator).
+ * Allocate all outputs and temporary buffers necessary to perform casts.
+ * Find the best iteration order, which includes information to efficiently
+ implement broadcasting. For example, adding a single value to an array
+ repeats the same value.
+
+5. *Setup and fetch the C-level function:*
+
+ * If necessary, allocate temporary working space.
+ * Find the C-implemented, light weight, inner-loop function.
+ Finding the inner-loop function can allow specialized implementations
+ in the future.
+ For example casting currently optimizes contiguous casts and
+ reductions have optimizations that are currently handled
+ inside the inner-loop function itself.
+ * Signal whether the inner-loop requires the Python API or whether
+ the GIL may be released (to allow threading).
+ * Clear floating point exception flags.
+
+6. *Perform the actual calculation:*
+
+ * Run the DType specific inner-loop function.
+ * The inner-loop may require access to additional data, such as dtypes or
+ additional data set in the previous step.
+ * The inner-loop function may be called an undefined number of times.
+
+7. *Finalize:*
+
+ * Free any temporary working space allocated in 5.
+ * Check for floating point exception flags.
+ * Return the result.
+
+The ``ArrayMethod`` provides a concept to group steps 3 to 6 and partially 7.
+However, implementers of a new ufunc or ``ArrayMethod`` do not need to
+customize the behaviour in steps 4 or 6, aside from the inner-loop function.
+For the ``ArrayMethod`` implementer, the central steps to have control over
+are step 3 and step 5 to provide the custom inner-loop function.
+Further customization is a potential future extension.
+
+Step 2. is promotion and dispatching which will also be restructured
+with new API which allows influencing the process where necessary.
+
+Step 1 is listed for completeness and is unaffected by this NEP.
+
+The following sketch provides an overview of step 2 to 6 with an emphasize
+of how dtypes are handled:
+
+.. figure:: _static/nep43-sketch.svg
+ :figclass: align-center
+
+
+*****************************************************************************
+ArrayMethod
+*****************************************************************************
+
+A central proposal of this NEP is the creation of the ``ArrayMethod`` as an object
+describing each implementation specific to a given set of DTypes.
+We use the ``class`` syntax to describe the information required to create
+a new ``ArrayMethod`` object:
+
+.. code-block:: python
+ :dedent: 0
+
+ class ArrayMethod:
+ name: str # Name, mainly useful for debugging
+
+ # Casting safety information (almost always "safe", necessary to
+ # unify casting and universal functions)
+ casting: Casting = "safe"
+
+ # More general flags:
+ flags: int
+
+ @staticmethod
+ def resolve_descriptors(
+ Context: context, Tuple[DType]: given_descrs)-> Casting, Tuple[DType]:
+ """Returns the safety of the operation (casting safety) and the
+ """
+ # A default implementation can be provided for non-parametric
+ # output dtypes.
+ raise NotImplementedError
+
+ @staticmethod
+ def get_loop(Context : context, strides, ...) -> strided_loop_function, flags:
+ """Returns the low-level C (strided inner-loop) function which
+ performs the actual operation.
+
+ This method may initially private, users will be able to provide
+ a set of optimized inner-loop functions instead:
+
+ * `strided_inner_loop`
+ * `contiguous_inner_loop`
+ * `unaligned_strided_loop`
+ * ...
+ """
+ raise NotImplementedError
+
+ @staticmethod
+ def strided_inner_loop(Context : context, data, strides,...):
+ """The inner-loop (equivalent to the current ufunc loop)
+ which is returned by the default `get_loop()` implementation."""
+ raise NotImplementedError
+
+With ``Context`` providing mostly static information about the function call:
+
+.. code-block:: python
+ :dedent: 0
+
+ class Context:
+ # The ArrayMethod object itself:
+ ArrayMethod : method
+
+ # Information about the caller, e.g. the ufunc, such as `np.add`:
+ callable : caller = None
+ # The number of input arguments:
+ int : nin = 1
+ # The number of output arguments:
+ int : nout = 1
+ # The DTypes this Method operates on/is defined for:
+ Tuple[DTypeMeta] : dtypes
+ # The actual dtypes instances the inner-loop operates on:
+ Tuple[DType] : descriptors
+
+ # Any additional information required. In the future, this will
+ # generalize or duplicate things currently stored on the ufunc:
+ # - The ufunc signature of generalized ufuncs
+ # - The identity used for reductions
+
+And ``flags`` stored properties, for whether:
+
+* the ``ArrayMethod`` supports unaligned input and output arrays
+* the inner-loop function requires the Python API (GIL)
+* NumPy has to check the floating point error CPU flags.
+
+*Note: More information is expected to be added as necessary.*
+
+
+The call ``Context``
+====================
+
+The "context" object is analogous to Python's ``self`` that is
+passed to all methods.
+To understand why the "context" object is necessary and its
+internal structure, it is helpful to remember
+that a Python method can be written in the following way
+(see also the `documentation of __get__
+<https://docs.python.org/3.8/reference/datamodel.html#object.__get__>`_):
+
+.. code-block:: python
+
+ class BoundMethod:
+ def __init__(self, instance, method):
+ self.instance = instance
+ self.method = method
+
+ def __call__(self, *args, **kwargs):
+ return self.method.function(self.instance, *args, **kwargs)
+
+
+ class Method:
+ def __init__(self, function):
+ self.function = function
+
+ def __get__(self, instance, owner=None):
+ assert instance is not None # unsupported here
+ return BoundMethod(instance, self)
+
+
+With which the following ``method1`` and ``method2`` below, behave identically:
+
+.. code-block:: python
+
+ def function(self):
+ print(self)
+
+ class MyClass:
+ def method1(self):
+ print(self)
+
+ method2 = Method(function)
+
+And both will print the same result:
+
+.. code-block:: python
+
+ >>> myinstance = MyClass()
+ >>> myinstance.method1()
+ <__main__.MyClass object at 0x7eff65436d00>
+ >>> myinstance.method2()
+ <__main__.MyClass object at 0x7eff65436d00>
+
+Here ``self.instance`` would be all information passed on by ``Context``.
+The ``Context`` is a generalization and has to pass additional information:
+
+* Unlike a method which operates on a single class instance, the ``ArrayMethod``
+ operates on many input arrays and thus multiple dtypes.
+* The ``__call__`` of the ``BoundMethod`` above contains only a single call
+ to a function. But an ``ArrayMethod`` has to call ``resolve_descriptors``
+ and later pass on that information to the inner-loop function.
+* A Python function has no state except that defined by its outer scope.
+ Within C, ``Context`` is able to provide additional state if necessary.
+
+Just as Python requires the distinction of a method and a bound method,
+NumPy will have a ``BoundArrayMethod``.
+This stores all of the constant information that is part of the ``Context``,
+such as:
+
+* the ``DTypes``
+* the number of input and ouput arguments
+* the ufunc signature (specific to generalized ufuncs, compare :ref:`NEP20`).
+
+Fortunately, most users and even ufunc implementers will not have to worry
+about these internal details; just like few Python users need to know
+about the ``__get__`` dunder method.
+The ``Context`` object or C-structure provides all necessary data to the
+fast C-functions and NumPy API creates the new ``ArrayMethod`` or
+``BoundArrayMethod`` as required.
+
+
+.. _ArrayMethod_specs:
+
+ArrayMethod Specifications
+==========================
+
+These specifications provide a minimal initial C-API, which shall be expanded
+in the future, for example to allow specialized inner-loops.
+
+Briefly, NumPy currently relies on strided inner-loops and this
+will be the only allowed method of defining a ufunc initially.
+We expect the addition of a ``setup`` function or exposure of ``get_loop``
+in the future.
+
+UFuncs require the same information as casting, giving the following
+definitions (see also :ref:`NEP 42 <NEP42>` ``CastingImpl``):
+
+* A new structure to be passed to the resolve function and inner-loop::
+
+ typedef struct {
+ PyObject *caller; /* The ufunc object */
+ PyArrayMethodObject *method;
+
+ int nin, nout;
+
+ PyArray_DTypeMeta **dtypes;
+ /* Operand descriptors, filled in by resolve_desciptors */
+ PyArray_Descr **descriptors;
+
+ void *reserved; // For Potential in threading (Interpreter state)
+ } PyArrayMethod_Context
+
+ This structure may be appended to include additional information in future
+ versions of NumPy and includes all constant loop metadata.
+
+ We could version this structure, although it may be simpler to version
+ the ``ArrayMethod`` itself.
+
+* Similar to casting, ufuncs may need to find the correct loop dtype
+ or indicate that a loop is only capable of handling certain instances of
+ the involved DTypes (e.g. only native byteorder). This is handled by
+ a ``resolve_descriptors`` function (identical to the ``resolve_descriptors``
+ of ``CastingImpl``)::
+
+ NPY_CASTING
+ resolve_descriptors(
+ PyArrayMethod_Context *context,
+ PyArray_Descr *given_dtypes[nin+nout],
+ PyArray_Descr *loop_dtypes[nin+nout]);
+
+ The function fills ``loop_dtypes`` based on the given ``given_dtypes``.
+ This requires filling in the descriptor of the output(s).
+ Often also the input descriptor(s) have to be found, e.g. to ensure native
+ byteorder when needed by the inner-loop.
+
+ In most cases an ``ArrayMethod`` will have non-parametric output DTypes
+ so that a default implementation can be provided.
+
+* An additional ``void *user_data`` will usually be typed to extend
+ the existing ``NpyAuxData *`` struct::
+
+ struct {
+ NpyAuxData_FreeFunc *free;
+ NpyAuxData_CloneFunc *clone;
+ /* To allow for a bit of expansion without breaking the ABI */
+ void *reserved[2];
+ } NpyAuxData;
+
+ This struct is currently mainly used for the NumPy internal casting
+ machinery and as of now both ``free`` and ``clone`` must be provided,
+ although this could be relaxed.
+
+ Unlike NumPy casts, the vast majority of ufuncs currently do not require
+ this additional scratch-space, but may need simple flagging capability
+ for example for implementing warnings (see Error and Warning Handling below).
+ To simplify this NumPy will pass a single zero initialized ``npy_intp *``
+ when ``user_data`` is not set.
+ *NOTE that it would be possible to pass this as part of ``Context``.*
+
+* The optional ``get_loop`` function will not be public initially, to avoid
+ finalizing the API which requires design choices also with casting:
+
+ .. code-block::
+
+ innerloop *
+ get_loop(
+ PyArrayMethod_Context *context,
+ /* (move_references is currently used internally for casting) */
+ int aligned, int move_references,
+ npy_intp *strides,
+ PyArray_StridedUnaryOp **out_loop,
+ NpyAuxData **userdata,
+ NPY_ARRAYMETHOD_FLAGS *flags);
+
+ The ``NPY_ARRAYMETHOD_FLAGS`` can indicate whether the Python API is required
+ and floating point errors must be checked.
+
+* The inner-loop function::
+
+ int inner_loop(PyArrayMethod_Context *context, ..., void *userdata);
+
+ Will have the identical signature to current inner-loops with the following
+ changes:
+
+ * A return value to indicate an error when returning ``-1`` instead of ``0``.
+ When returning ``-1`` a Python error must be set.
+ * The new first argument ``PyArrayMethod_Context *`` is used to pass in
+ potentially required information about the ufunc or descriptors in a
+ convenient way.
+ * The ``void *userdata`` will be the ``NpyAuxData **userdata`` as set by
+ ``get_loop``. If ``get_loop`` does not set ``userdata`` an ``npy_intp *``
+ is passed instead (see `Error Handling`_ below for the motivation).
+
+ *Note:* Since ``get_loop`` is expected to be private, the exact implementation
+ of ``userdata`` can be modified until final exposure.
+
+Creation of a new ``BoundArrayMethod`` will use a ``PyArrayMethod_FromSpec()``
+function. A shorthand will allow direct registration to a ufunc using
+``PyUFunc_AddImplementationFromSpec()``. The specification is expected
+to contain the following (this may extend in the future)::
+
+ typedef struct {
+ const char *name; /* Generic name, mainly for debugging */
+ int nin, nout;
+ NPY_CASTING casting;
+ NPY_ARRAYMETHOD_FLAGS flags;
+ PyArray_DTypeMeta **dtypes;
+ PyType_Slot *slots;
+ } PyArrayMethod_Spec;
+
+
+Discussion and alternatives
+===========================
+
+The above split into an ``ArrayMethod`` and ``Context`` and the additional
+requirement of a ``BoundArrayMethod`` is a necessary split mirroring the
+implementation of methods and bound methods in Python.
+
+One reason for this requirement is that it allows storing the ``ArrayMethod``
+object in many cases without holding references to the ``DTypes`` which may
+be important if DTypes are created (and deleted) dynamically.
+(This is a complex topic, which does not have a complete solution in current
+Python, but the approach solves the issue with respect to casting.)
+
+There seem to be no alternatives to this structure. Separating the
+DType-specific steps from the general ufunc dispatching and promotion is
+absolutely necessary to allow future extension and flexibility.
+Furthermore, it allows unifying casting and ufuncs.
+
+Since the structure of ``ArrayMethod`` and ``BoundArrayMethod`` will be
+opaque and can be extended, there are few long-term design implications aside
+from the choice of making them Python objects.
+
+
+``resolve_descriptors``
+-----------------------
+
+The ``resolve_descriptors`` method is possibly the main innovation of this
+NEP and it is central also in the implementation of casting in NEP 42.
+
+By ensuring that every ``ArrayMethod`` provides ``resolve_descriptors`` we
+define a unified, clear API for step 3 in `Steps involved in a UFunc call`_.
+This step is required to allocate output arrays and has to happen before
+casting can be prepared.
+
+While the returned casting-safety (``NPY_CASTING``) will almost always be
+"safe" for universal functions, including it has two big advantages:
+
+* Returning the casting safety is central to NEP 42 for casting and
+ allows the unmodified use of ``ArrayMethod`` there.
+* There may be a future desire to implement fast but unsafe implementations.
+ For example for ``int64 + int64 -> int32`` which is unsafe from a casting
+ perspective. Currently, this would use ``int64 + int64 -> int64`` and then
+ cast to ``int32``. An implementation that skips the cast would
+ have to signal that it effectively includes the "same-kind" cast and is
+ thus not considered "safe".
+
+
+``get_loop`` method
+-------------------
+
+Currently, NumPy ufuncs typically only provide a single strided loop, so that
+the ``get_loop`` method may seem unnecessary.
+For this reason we plan for ``get_loop`` to be a private function initially.
+
+However, ``get_loop`` is required for casting where specialized loops are
+used even beyond strided and contiguous loops.
+Thus, the ``get_loop`` function must be a full replacement for
+the internal ``PyArray_GetDTypeTransferFunction``.
+
+In the future, ``get_loop`` may be made public or a new ``setup`` function
+be exposed to allow more control, for example to allow allocating
+working memory.
+Further, we could expand ``get_loop`` and allow the ``ArrayMethod`` implementer
+to also control the outer iteration and not only the 1-D inner-loop.
+
+
+Extending the inner-loop signature
+----------------------------------
+
+Extending the inner-loop signature is another central and necessary part of
+the NEP.
+
+**Passing in the ``Context``:**
+
+Passing in the ``Context`` potentially allows for the future extension of
+the signature by adding new fields to the context struct.
+Furthermore it provides direct access to the dtype instances which
+the inner-loop operates on.
+This is necessary information for parametric dtypes since for example comparing
+two strings requires knowing the length of both strings.
+The ``Context`` can also hold potentially useful information such as the
+the original ``ufunc``, which can be helpful when reporting errors.
+
+In principle passing in Context is not necessary, as all information could be
+included in ``userdata`` and set up in the ``get_loop`` function.
+In this NEP we propose passing the struct to simplify creation of loops for
+parametric DTypes.
+
+**Passing in user data:**
+
+The current casting implementation uses the existing ``NpyAuxData *`` to pass
+in additional data as defined by ``get_loop``.
+There are certainly alternatives to the use of this structure, but it
+provides a simple solution, which is already used in NumPy and public API.
+
+``NpyAyxData *`` is a light weight, allocated structure and since it already
+exists in NumPy for this purpose, it seems a natural choice.
+To simplify some use-cases (see "Error Handling" below), we will pass a
+``npy_intp *userdata = 0`` instead when ``userdata`` is not provided.
+
+*Note: Since ``get_loop`` is expected to be private initially we can gain
+experience with ``userdata`` before exposing it as public API.*
+
+**Return value:**
+
+The return value to indicate an error is an important, but currently missing
+feature in NumPy. The error handling is further complicated by the way
+CPUs signal floating point errors.
+Both are discussed in the next section.
+
+Error Handling
+""""""""""""""
+
+We expect that future inner-loops will generally set Python errors as soon
+as an error is found. This is complicated when the inner-loop is run without
+locking the GIL. In this case the function will have to lock the GIL,
+set the Python error and return ``-1`` to indicate an error occurred::
+
+ int
+ inner_loop(PyArrayMethod_Context *context, ..., void *userdata)
+ {
+ NPY_ALLOW_C_API_DEF
+
+ for (npy_intp i = 0; i < N; i++) {
+ /* calculation */
+
+ if (error_occurred) {
+ NPY_ALLOW_C_API;
+ PyErr_SetString(PyExc_ValueError,
+ "Error occurred inside inner_loop.");
+ NPY_DISABLE_C_API
+ return -1;
+ }
+ }
+ return 0;
+ }
+
+Floating point errors are special, since they require checking the hardware
+state which is too expensive if done within the inner-loop function itself.
+Thus, NumPy will handle these if flagged by the ``ArrayMethod``.
+An ``ArrayMethod`` should never cause floating point error flags to be set
+if it flags that these should not be checked. This could interfere when
+calling multiple functions; in particular when casting is necessary.
+
+An alternative solution would be to allow setting the error only at the later
+finalization step when NumPy will also check the floating point error flags.
+
+We decided against this pattern at this time. It seems more complex and
+generally unnecessary.
+While safely grabbing the GIL in the loop may require passing in an additional
+``PyThreadState`` or ``PyInterpreterState`` in the future (for subinterpreter
+support), this is acceptable and can be anticipated.
+Setting the error at a later point would add complexity: for instance
+if an operation is paused (which can currently happen for casting in particular),
+the error check needs to run explicitly ever time this happens.
+
+We expect that setting errors immediately is the easiest and most convenient
+solution and more complex solution may be possible future extensions.
+
+Handling *warnings* is slightly more complex: A warning should be
+given exactly once for each function call (i.e. for the whole array) even
+if naively it would be given many times.
+To simplify such a use case, we will pass in ``npy_intp *userdata = 0``
+by default which can be used to store flags (or other simple persistent data).
+For instance, we could imagine an integer multiplication loop which warns
+when an overflow occurred::
+
+ int
+ integer_multiply(PyArrayMethod_Context *context, ..., npy_intp *userdata)
+ {
+ int overflow;
+ NPY_ALLOW_C_API_DEF
+
+ for (npy_intp i = 0; i < N; i++) {
+ *out = multiply_integers(*in1, *in2, &overflow);
+
+ if (overflow && !*userdata) {
+ NPY_ALLOW_C_API;
+ if (PyErr_Warn(PyExc_UserWarning,
+ "Integer overflow detected.") < 0) {
+ NPY_DISABLE_C_API
+ return -1;
+ }
+ *userdata = 1;
+ NPY_DISABLE_C_API
+ }
+ return 0;
+ }
+
+*TODO:* The idea of passing an ``npy_intp`` scratch space when ``userdata``
+is not set seems convenient, but I am uncertain about it, since I am not
+aware of any similar prior art. This "scratch space" could also be part of
+the ``context`` in principle.
+
+
+Reusing existing Loops/Implementations
+======================================
+
+For many DTypes the above definition for adding additional C-level loops will be
+sufficient and require no more than a single strided loop implementation
+and if the loop works with parametric DTypes, the
+``resolve_descriptors`` function *must* additionally be provided.
+
+However, in some use-cases it is desirable to call back to an existing implementation.
+In Python, this could be achieved by simply calling into the original ufunc.
+
+For better performance in C, and for large arrays, it is desirable to reuse
+an existing ``ArrayMethod`` as directly as possible, so that its inner-loop function
+can be used directly without additional overhead.
+We will thus allow to create a new, wrapping, ``ArrayMethod`` from an existing
+``ArrayMethod``.
+
+This wrapped ``ArrayMethod`` will have two additional methods:
+
+* ``view_inputs(Tuple[DType]: input_descr) -> Tuple[DType]`` replacing the
+ user input descriptors with descriptors matching the wrapped loop.
+ It must be possible to *view* the inputs as the output.
+ For example for ``Unit[Float64]("m") + Unit[Float32]("km")`` this will
+ return ``float64 + int32``. The original ``resolve_descriptors`` will
+ convert this to ``float64 + float64``.
+
+* ``wrap_outputs(Tuple[DType]: input_descr) -> Tuple[DType]`` replacing the
+ resolved descriptors with with the desired actual loop descriptors.
+ The original ``resolve_descriptors`` function will be called between these
+ two calls, so that the output descriptors may not be set in the first call.
+ In the above example it will use the ``float64`` as returned (which might
+ have changed the byte-order), and further resolve the physical unit making
+ the final signature::
+
+ ``Unit[Float64]("m") + Unit[Float64]("m") -> Unit[Float64]("m")``
+
+ the UFunc machinery will take care of casting the "km" input to "m".
+
+
+The ``view_inputs`` method allows passing the correct inputs into the
+original ``resolve_descriptors`` function, while ``wrap_outputs`` ensures
+the correct descriptors are used for output allocation and input buffering casts.
+
+An important use-case for this is that of an abstract Unit DType
+with subclasses for each numeric dtype (which could be dynamically created)::
+
+ Unit[Float64]("m")
+ # with Unit[Float64] being the concrete DType:
+ isinstance(Unit[Float64], Unit) # is True
+
+Such a ``Unit[Float64]("m")`` instance has a well-defined signature with
+respect to type promotion.
+The author of the ``Unit`` DType can implement most necessary logic by
+wrapping the existing math functions and using the two additional methods
+above.
+Using the *promotion* step, this will allow to create a register a single
+promoter for the abstract ``Unit`` DType with the ``ufunc``.
+The promoter can then add the wrapped concrete ``ArrayMethod`` dynamically
+at promotion time, and NumPy can cache (or store it) after the first call.
+
+**Alternative use-case:**
+
+A different use-case is that of a ``Unit(float64, "m")`` DType, where
+the numerical type is part of the DType parameter.
+This approach is possible, but will require a custom ``ArrayMethod``
+which wraps existing loops.
+It must also always require require two steps of dispatching
+(one to the ``Unit`` DType and a second one for the numerical type).
+
+Furthermore, the efficient implementation will require the ability to
+fetch and reuse the inner-loop function from another ``ArrayMethod``.
+(Which is probably necessary for users like Numba, but it is uncertain
+whether it should be a common pattern and it cannot be accessible from
+Python itself.)
+
+
+.. _promotion_and_dispatching:
+
+*************************
+Promotion and dispatching
+*************************
+
+NumPy ufuncs are multi-methods in the sense that they operate on (or with)
+multiple DTypes at once.
+While the input (and output) dtypes are attached to NumPy arrays,
+the ``ndarray`` type itself does not carry the information of which
+function to apply to the data.
+
+For example, given the input::
+
+ int_arr = np.array([1, 2, 3], dtype=np.int64)
+ float_arr = np.array([1, 2, 3], dtype=np.float64)
+ np.add(int_arr, float_arr)
+
+has to find the correct ``ArrayMethod`` to perform the operation.
+Ideally, there is an exact match defined, e.g. for ``np.add(int_arr, int_arr)``
+the ``ArrayMethod[Int64, Int64, out=Int64]`` matches exactly and can be used.
+However, for ``np.add(int_arr, float_arr)`` there is no direct match,
+requiring a promotion step.
+
+Promotion and dispatching process
+=================================
+
+In general the ``ArrayMethod`` is found by searching for an exact match of
+all input DTypes.
+The output dtypes should *not* affect calculation, but if multiple registered
+``ArrayMethod``\ s match exactly, the output DType will be used to find the
+better match.
+This will allow the current distinction for ``np.equal`` loops which define
+both ``Object, Object -> Bool`` (default) and ``Object, Object -> Object``.
+
+Initially, an ``ArrayMethod`` will be defined for *concrete* DTypes only
+and since these cannot be subclassed an exact match is guaranteed.
+In the future we expect that ``ArrayMethod``\ s can also be defined for
+*abstract* DTypes. In which case the best match is found as detailed below.
+
+**Promotion:**
+
+If a matching ``ArrayMethod`` exists, dispatching is straight forward.
+However, when it does not, require additional definitions to implement
+promotion:
+
+* By default any UFunc has a promotion which uses the common DType of all
+ inputs and dispatches a second time. This is well-defined for most
+ mathematical functions, but can be disabled or customized if necessary.
+ For instances ``int32 + float64`` tries again using ``float64 + float64``
+ which is the common DType.
+
+* Users can *register* new Promoters just as they can register a
+ new ``ArrayMethod``. These will use abstract DTypes to allow matching
+ a large variety of signatures.
+ The return value of a promotion function shall be a new ``ArrayMethod``
+ or ``NotImplemented``. It must be consistent over multiple calls with
+ the same input to allow caching of the result.
+
+The signature of a promotion function would be::
+
+ promoter(np.ufunc: ufunc, Tuple[DTypeMeta]: DTypes): -> Union[ArrayMethod, NotImplemented]
+
+Note that DTypes may include the output's DType, however, normally the
+output DType will *not* affect which ``ArrayMethod`` is chosen.
+
+In most cases, it should not be necessary to add a custom promotion function.
+An example which requires this is multiplication with a unit:
+in NumPy ``timedelta64`` can be multiplied with most integers,
+but NumPy only defines a loop (``ArrayMethod``) for ``timedelta64 * int64``
+so that multiplying with ``int32`` would fail.
+
+To allow this, the following promoter can be registered for
+``(Timedelta64, Integral, None)``::
+
+ def promote(ufunc, DTypes):
+ res = list(DTypes)
+ try:
+ res[1] = np.common_dtype(DTypes[1], Int64)
+ except TypeError:
+ return NotImplemented
+
+ # Could check that res[1] is actually Int64
+ return ufunc.resolve_impl(tuple(res))
+
+In this case, just as a ``Timedelta64 * int64`` and ``int64 * timedelta64``
+``ArrayMethod`` is necessary, a second promoter will have to be registered to
+handle the case where the integer is passed first.
+
+**Dispatching rules for ``ArrayMethod`` and Promoters:**
+
+Promoter and ``ArrayMethod`` are discovered by finding the best match as
+defined by the DType class hierarchy.
+The best match is defined if:
+
+* The signature matches for all input DTypes, so that
+ ``issubclass(input_DType, registered_DType)`` returns true.
+* No other promoter or ``ArrayMethod`` is more precise in any input:
+ ``issubclass(other_DType, this_DType)`` is true (this may include if both
+ are identical).
+* This promoter or ``ArrayMethod`` is more precise in at least one input or
+ output DType.
+
+It will be an error if ``NotImplemented`` is returned or if two
+promoters match the input equally well.
+When an existing promoter is not precise enough for new functionality, a
+new promoter has to be added.
+To ensure that this promoter takes precedence it may be necessary to define
+new abstract DTypes as more precise subclasses of existing ones.
+
+The above rules enable specialization if an output is supplied
+or the full loop is specified. This should not typically be necessary,
+but allows resolving ``np.logic_or``, etc. which have both
+``Object, Object -> Bool`` and ``Object, Object -> Object`` loops (using the
+first by default).
+
+
+Discussion and alternatives
+===========================
+
+Instead of resolving and returning a new implementation, we could also
+return a new set of DTypes to use for dispatching. This works, however,
+it has the disadvantage that it is impossible to dispatch to a loop
+defined on a different ufunc or to dynamically create a new ``ArrayMethod``.
+
+
+**Rejected Alternatives:**
+
+In the above the promoters use a multiple dispatching style type resolution
+while the current UFunc machinery uses the first
+"safe" loop (see also :ref:`NEP 40 <NEP40>`) in an ordered hierarchy.
+
+While the "safe" casting rule is not restrictive enough, we could imagine
+using a new "promote" casting rule, or the common-DType logic to find the
+best matching loop by upcasting the inputs as necessary.
+
+One downside to this approach is that upcasting alone allows upcasting the
+result beyond what is expected by users:
+Currently (which will remain supported as a fallback) any ufunc which defines
+only a float64 loop will also work for float16 and float32 by *upcasting*::
+
+ >>> from scipy.special import erf
+ >>> erf(np.array([4.], dtype=np.float16)) # float16
+ array([1.], dtype=float32)
+
+with a float32 result. It is impossible to change the ``erf`` function to
+return a float16 result without changing the result of following code.
+In general, we argue that automatic upcasting should not occur in cases
+where a less precise loop can be defined, *unless* the ufunc
+author does this intentionally using a promotion.
+
+This consideration means that upcasting has to be limited by some additional
+method.
+
+*Alternative 1:*
+
+Assuming general upcasting is not intended, a rule must be defined to
+limit upcasting the input from ``float16 -> float32`` either using generic
+logic on the DTypes or the UFunc itself (or a combination of both).
+The UFunc cannot do this easily on its own, since it cannot know all possible
+DTypes which register loops.
+Consider the two examples:
+
+First (should be rejected):
+
+* Input: ``float16 * float16``
+* Existing loop: ``float32 * float32``
+
+Second (should be accepted):
+
+* Input: ``timedelta64 * int32``
+* Existing loop: ``timedelta64 * int16``
+
+
+This requires either:
+
+1. The ``timedelta64`` to somehow signal that the ``int64`` upcast is
+ always supported if it is involved in the operation.
+2. The ``float32 * float32`` loop to reject upcasting.
+
+Implementing the first approach requires signaling that upcasts are
+acceptable in the specific context. This would require additional hooks
+and may not be simple for complex DTypes.
+
+For the second approach in most cases a simple ``np.common_dtype`` rule will
+work for initial dispatching, however, even this is only clearly the case
+for homogeneous loops.
+This option will require adding a function to check whether the input
+is a valid upcast to each loop individually, which seems problematic.
+In many cases a default could be provided (homogeneous signature).
+
+*Alternative 2:*
+
+An alternative "promotion" step is to ensure that the *output* DType matches
+with the loop after first finding the correct output DType.
+If the output DTypes are known, finding a safe loop becomes easy.
+In the majority of cases this works, the correct output dtype is just::
+
+ np.common_dtype(*input_DTypes)
+
+or some fixed DType (e.g. Bool for logical functions).
+
+However, it fails for example in the ``timedelta64 * int32`` case above since
+there is a-priori no way to know that the "expected" result type of this
+output is indeed ``timedelta64`` (``np.common_dtype(Datetime64, Int32)`` fails).
+This requires some additional knowledge of the timedelta64 precision being
+int64. Since a ufunc can have an arbitrary number of (relevant) inputs
+it would thus at least require an additional ``__promoted_dtypes__`` method
+on ``Datetime64`` (and all DTypes).
+
+A further limitation is shown by masked DTypes. Logical functions do not
+have a boolean result when masked are involved, which would thus require the
+original ufunc author to anticipate masked DTypes in this scheme.
+Similarly, some functions defined for complex values will return real numbers
+while others return complex numbers. If the original author did not anticipate
+complex numbers, the promotion may be incorrect for a later added complex loop.
+
+
+We believe that promoters, while allowing for an huge theoretical complexity,
+are the best solution:
+
+1. Promotion allows for dynamically adding new loops. E.g. it is possible
+ to define an abstract Unit DType, which dynamically creates classes to
+ wrap other existing DTypes. Using a single promoter, this DType can
+ dynamically wrap existing ``ArrayMethod`` enabling it to find the correct
+ loop in a single lookup instead of two.
+2. The promotion logic will usually err on the safe side: A newly-added
+ loop cannot be misused unless a promoter is added as well.
+3. They put the burden of carefully thinking of whether the logic is correct
+ on the programmer adding new loops to a UFunc. (Compared to Alternative 2)
+4. In case of incorrect existing promotion, writing a promoter to restrict
+ or refine a generic rule is possible. In general a promotion rule should
+ never return an *incorrect* promotion, but if it the existing promotion
+ logic fails or is incorrect for a newly-added loop, the loop can add a
+ new promoter to refine the logic.
+
+The option of having each loop verify that no upcast occured is probably
+the best alternative, but does not include the ability to dynamically
+adding new loops.
+
+The main downsides of general promoters is that they allow a possible
+very large complexity.
+A third-party library *could* add incorrect promotions to NumPy, however,
+this is already possible by adding new incorrect loops.
+In general we believe we can rely on downstream projects to use this
+power and complexity carefully and responsibly.
+
+
+***************
+User Guidelines
+***************
+
+In general adding a promoter to a UFunc must be done very carefully.
+A promoter should never affect loops which can be reasonably defined
+by other datatypes. Defining a hypothetical ``erf(UnitFloat16)`` loop
+must not lead to ``erf(float16)``.
+In general a promoter should fulfill the following requirements:
+
+* Be conservative when defining a new promotion rule. An incorrect result
+ is a much more dangerous error than an unexpected error.
+* One of the (abstract) DTypes added should typically match specifically with a
+ DType (or family of DTypes) defined by your project.
+ Never add promotion rules which go beyond normal common DType rules!
+ It is *not* reasonable to add a loop for ``int16 + uint16 -> int24`` if
+ you write an ``int24`` dtype. The result of this operation was already
+ defined previously as ``int32`` and will be used with this assumption.
+* A promoter (or loop) should never affect existing loop results.
+ This includes adding faster but less precise loops/promoters to replace
+ existing ones.
+* Try to stay within a clear, linear hierarchy for all promotion (and casting)
+ related logic. NumPy itself breaks this logic for integers and floats
+ (they are not strictly linear, since int64 cannot promote to float32).
+* Loops and promoters can be added by any project, which could be:
+
+ * The project defining the ufunc
+ * The project defining the DType
+ * A third-party project
+
+ Try to find out which is the best project to add the loop. If neither
+ the project defining the ufunc nor the project defining the DType add the
+ loop, issues with multiple definitions (which are rejected) may arise
+ and care should be taken that the loop behaviour is always more desirable
+ than an error.
+
+In some cases exceptions to these rules may make sense, however, in general
+we ask you to use extreme caution and when in doubt create a new UFunc
+instead. This clearly notifies the users of differing rules.
+When in doubt, ask on the NumPy mailing list or issue tracker!
+
+
+**************
+Implementation
+**************
+
+Implementation of this NEP will entail a large refactor and restructuring
+of the current ufunc machinery (as well as casting).
+
+The implementation unfortunately will require large maintenance of the
+UFunc machinery, since both the actual UFunc loop calls, as well as the
+the initial dispatching steps have to be modified.
+
+In general, the correct ``ArrayMethod``, also those returned by a promoter,
+will be cached (or stored) inside a hashtable for efficient lookup.
+
+
+**********
+Discussion
+**********
+
+There is a large space of possible implementations with many discussions
+in various places, as well as initial thoughts and design documents.
+These are listed in the discussion of :ref:`NEP 40 <NEP40>` and not repeated here for
+brevity.
+
+A long discussion which touches many of these points and points towards
+similar solutions can be found in
+`the github issue 12518 "What should be the calling convention for ufunc inner loop signatures?" <https://github.com/numpy/numpy/issues/12518>`_
+
+
+**********
+References
+**********
+
+Please see NEP 40 and 41 for more discussion and references.
+
+
+*********
+Copyright
+*********
+
+This document has been placed in the public domain.
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