# Source code for dask.array.gufunc

    from __future__ import annotations

    import re

    import numpy as np
    from tlz import concat, merge, unique

    from dask.array.core import Array, apply_infer_dtype, asarray, blockwise, getitem
    from dask.array.utils import meta_from_array
    from dask.core import flatten
    from dask.highlevelgraph import HighLevelGraph

    # Modified version of `numpy.lib.function_base._parse_gufunc_signature`
    # Modifications:
    #   - Allow for zero input arguments
    # See https://docs.scipy.org/doc/numpy/reference/c-api/generalized-ufuncs.html
    _DIMENSION_NAME = r"\w+"
    _CORE_DIMENSION_LIST = "(?:{0:}(?:,{0:})*,?)?".format(_DIMENSION_NAME)
    _ARGUMENT = rf"\({_CORE_DIMENSION_LIST}\)"
    _INPUT_ARGUMENTS = "(?:{0:}(?:,{0:})*,?)?".format(_ARGUMENT)
    _OUTPUT_ARGUMENTS = "{0:}(?:,{0:})*".format(
        _ARGUMENT
    )  # Use `'{0:}(?:,{0:})*,?'` if gufunc-
    # signature should be allowed for length 1 tuple returns
    _SIGNATURE = f"^{_INPUT_ARGUMENTS}->{_OUTPUT_ARGUMENTS}$"

    def _parse_gufunc_signature(signature):
        """
        Parse string signatures for a generalized universal function.

        Arguments
        ---------
        signature : string
            Generalized universal function signature, e.g., ``(m,n),(n,p)->(m,p)``
            for ``np.matmul``.

        Returns
        -------
        Tuple of input and output core dimensions parsed from the signature, each
        of the form List[Tuple[str, ...]], except for one output. For one output
        core dimension is not a list, but of the form Tuple[str, ...]
        """
        signature = re.sub(r"\s+", "", signature)
        if not re.match(_SIGNATURE, signature):
            raise ValueError(f"Not a valid gufunc signature: {signature}")
        in_txt, out_txt = signature.split("->")
        ins = [
            tuple(re.findall(_DIMENSION_NAME, arg)) for arg in re.findall(_ARGUMENT, in_txt)
        ]
        outs = [
            tuple(re.findall(_DIMENSION_NAME, arg))
            for arg in re.findall(_ARGUMENT, out_txt)
        ]
        outs = outs[0] if ((len(outs) == 1) and (out_txt[-1] != ",")) else outs
        return ins, outs

    def _validate_normalize_axes(axes, axis, keepdims, input_coredimss, output_coredimss):
        """
        Validates logic of `axes`/`axis`/`keepdims` arguments and normalize them.
        Refer to [1]_ for details

        Arguments
        ---------
        axes: List of tuples
        axis: int
        keepdims: bool
        input_coredimss: List of Tuple of dims
        output_coredimss: List of Tuple of dims

        Returns
        -------
        input_axes: List of tuple of int
        output_axes: List of tuple of int

        References
        ----------
        .. [1] https://docs.scipy.org/doc/numpy/reference/ufuncs.html#optional-keyword-arguments
        """
        nin = len(input_coredimss)
        nout = 1 if not isinstance(output_coredimss, list) else len(output_coredimss)

        if axes is not None and axis is not None:
            raise ValueError(
                "Only one of `axis` or `axes` keyword arguments should be given"
            )
        if axes and not isinstance(axes, list):
            raise ValueError("`axes` has to be of type list")

        output_coredimss = output_coredimss if nout > 1 else [output_coredimss]
        filtered_core_dims = list(filter(len, input_coredimss))
        nr_outputs_with_coredims = len([True for x in output_coredimss if len(x) > 0])

        if keepdims:
            if nr_outputs_with_coredims > 0:
                raise ValueError("`keepdims` can only be used for scalar outputs")
            output_coredimss = len(output_coredimss) * [filtered_core_dims[0]]

        core_dims = input_coredimss + output_coredimss
        if axis is not None:
            if not isinstance(axis, int):
                raise ValueError("`axis` argument has to be an integer value")
            if filtered_core_dims:
                cd0 = filtered_core_dims[0]
                if len(cd0) != 1:
                    raise ValueError(
                        "`axis` can be used only, if one core dimension is present"
                    )
                for cd in filtered_core_dims:
                    if cd0 != cd:
                        raise ValueError(
                            "To use `axis`, all core dimensions have to be equal"
                        )

        # Expand defaults or axis
        if axes is None:
            if axis is not None:
                axes = [(axis,) if cd else tuple() for cd in core_dims]
            else:
                axes = [tuple(range(-len(icd), 0)) for icd in core_dims]
        elif not isinstance(axes, list):
            raise ValueError("`axes` argument has to be a list")
        axes = [(a,) if isinstance(a, int) else a for a in axes]

        if (
            (nr_outputs_with_coredims == 0)
            and (nin != len(axes))
            and (nin + nout != len(axes))
        ) or ((nr_outputs_with_coredims > 0) and (nin + nout != len(axes))):
            raise ValueError(
                "The number of `axes` entries is not equal the number of input and output arguments"
            )

        # Treat outputs
        output_axes = axes[nin:]
        output_axes = (
            output_axes
            if output_axes
            else [tuple(range(-len(ocd), 0)) for ocd in output_coredimss]
        )
        input_axes = axes[:nin]

        # Assert we have as many axes as output core dimensions
        for idx, (iax, icd) in enumerate(zip(input_axes, input_coredimss)):
            if len(iax) != len(icd):
                raise ValueError(
                    "The number of `axes` entries for argument #{} is not equal "
                    "the number of respective input core dimensions in signature".format(
                        idx
                    )
                )
        if not keepdims:
            for idx, (oax, ocd) in enumerate(zip(output_axes, output_coredimss)):
                if len(oax) != len(ocd):
                    raise ValueError(
                        "The number of `axes` entries for argument #{} is not equal "
                        "the number of respective output core dimensions in signature".format(
                            idx
                        )
                    )
        else:
            if input_coredimss:
                icd0 = input_coredimss[0]
                for icd in input_coredimss:
                    if icd0 != icd:
                        raise ValueError(
                            "To use `keepdims`, all core dimensions have to be equal"
                        )
                iax0 = input_axes[0]
                output_axes = [iax0 for _ in output_coredimss]

        return input_axes, output_axes

    def apply_gufunc(
        func,
        signature,
        *args,
        axes=None,
        axis=None,
        keepdims=False,
        output_dtypes=None,
        output_sizes=None,
        vectorize=None,
        allow_rechunk=False,
        meta=None,
        **kwargs,
    ):
        """
        Apply a generalized ufunc or similar python function to arrays.

        ``signature`` determines if the function consumes or produces core
        dimensions. The remaining dimensions in given input arrays (``*args``)
        are considered loop dimensions and are required to broadcast
        naturally against each other.

        In other terms, this function is like ``np.vectorize``, but for
        the blocks of dask arrays. If the function itself shall also
        be vectorized use ``vectorize=True`` for convenience.

        Parameters
        ----------
        func : callable
            Function to call like ``func(*args, **kwargs)`` on input arrays
            (``*args``) that returns an array or tuple of arrays. If multiple
            arguments with non-matching dimensions are supplied, this function is
            expected to vectorize (broadcast) over axes of positional arguments in
            the style of NumPy universal functions [1]_ (if this is not the case,
            set ``vectorize=True``). If this function returns multiple outputs,
            ``output_core_dims`` has to be set as well.
        signature: string
            Specifies what core dimensions are consumed and produced by ``func``.
            According to the specification of numpy.gufunc signature [2]_
        *args : numeric
            Input arrays or scalars to the callable function.
        axes: List of tuples, optional, keyword only
            A list of tuples with indices of axes a generalized ufunc should operate on.
            For instance, for a signature of ``"(i,j),(j,k)->(i,k)"`` appropriate for
            matrix multiplication, the base elements are two-dimensional matrices
            and these are taken to be stored in the two last axes of each argument. The
            corresponding axes keyword would be ``[(-2, -1), (-2, -1), (-2, -1)]``.
            For simplicity, for generalized ufuncs that operate on 1-dimensional arrays
            (vectors), a single integer is accepted instead of a single-element tuple,
            and for generalized ufuncs for which all outputs are scalars, the output
            tuples can be omitted.
        axis: int, optional, keyword only
            A single axis over which a generalized ufunc should operate. This is a short-cut
            for ufuncs that operate over a single, shared core dimension, equivalent to passing
            in axes with entries of (axis,) for each single-core-dimension argument and ``()`` for
            all others. For instance, for a signature ``"(i),(i)->()"``, it is equivalent to passing
            in ``axes=[(axis,), (axis,), ()]``.
        keepdims: bool, optional, keyword only
            If this is set to True, axes which are reduced over will be left in the result as
            a dimension with size one, so that the result will broadcast correctly against the
            inputs. This option can only be used for generalized ufuncs that operate on inputs
            that all have the same number of core dimensions and with outputs that have no core
            dimensions , i.e., with signatures like ``"(i),(i)->()"`` or ``"(m,m)->()"``.
            If used, the location of the dimensions in the output can be controlled with axes
            and axis.
        output_dtypes : Optional, dtype or list of dtypes, keyword only
            Valid numpy dtype specification or list thereof.
            If not given, a call of ``func`` with a small set of data
            is performed in order to try to automatically determine the
            output dtypes.
        output_sizes : dict, optional, keyword only
            Optional mapping from dimension names to sizes for outputs. Only used if
            new core dimensions (not found on inputs) appear on outputs.
        vectorize: bool, keyword only
            If set to ``True``, ``np.vectorize`` is applied to ``func`` for
            convenience. Defaults to ``False``.
        allow_rechunk: Optional, bool, keyword only
            Allows rechunking, otherwise chunk sizes need to match and core
            dimensions are to consist only of one chunk.
            Warning: enabling this can increase memory usage significantly.
            Defaults to ``False``.
        meta: Optional, tuple, keyword only
            tuple of empty ndarrays describing the shape and dtype of the output of the gufunc.
            Defaults to ``None``.
        **kwargs : dict
            Extra keyword arguments to pass to `func`

        Returns
        -------
        Single dask.array.Array or tuple of dask.array.Array

        Examples
        --------
        >>> import dask.array as da
        >>> import numpy as np
        >>> def stats(x):
        ...     return np.mean(x, axis=-1), np.std(x, axis=-1)
        >>> a = da.random.normal(size=(10,20,30), chunks=(5, 10, 30))
        >>> mean, std = da.apply_gufunc(stats, "(i)->(),()", a)
        >>> mean.compute().shape
        (10, 20)

        >>> def outer_product(x, y):
        ...     return np.einsum("i,j->ij", x, y)
        >>> a = da.random.normal(size=(   20,30), chunks=(10, 30))
        >>> b = da.random.normal(size=(10, 1,40), chunks=(5, 1, 40))
        >>> c = da.apply_gufunc(outer_product, "(i),(j)->(i,j)", a, b, vectorize=True)
        >>> c.compute().shape
        (10, 20, 30, 40)

        References
        ----------
        .. [1] https://docs.scipy.org/doc/numpy/reference/ufuncs.html
        .. [2] https://docs.scipy.org/doc/numpy/reference/c-api/generalized-ufuncs.html
        """
        # Input processing:
        ## Signature
        if not isinstance(signature, str):
            raise TypeError("`signature` has to be of type string")
        # NumPy versions before https://github.com/numpy/numpy/pull/19627
        # would not ignore whitespace characters in `signature` like they
        # are supposed to. We remove the whitespace here as a workaround.
        signature = re.sub(r"\s+", "", signature)
        input_coredimss, output_coredimss = _parse_gufunc_signature(signature)

        ## Determine nout: nout = None for functions of one direct return; nout = int for return tuples
        nout = None if not isinstance(output_coredimss, list) else len(output_coredimss)

        ## Consolidate onto `meta`
        if meta is not None and output_dtypes is not None:
            raise ValueError(
                "Only one of `meta` and `output_dtypes` should be given (`meta` is preferred)."
            )
        if meta is None:
            if output_dtypes is None:
                ## Infer `output_dtypes`
                if vectorize:
                    tempfunc = np.vectorize(func, signature=signature)
                else:
                    tempfunc = func
                output_dtypes = apply_infer_dtype(
                    tempfunc, args, kwargs, "apply_gufunc", "output_dtypes", nout
                )

            ## Turn `output_dtypes` into `meta`
            if (
                nout is None
                and isinstance(output_dtypes, (tuple, list))
                and len(output_dtypes) == 1
            ):
                output_dtypes = output_dtypes[0]
            sample = args[0] if args else None
            if nout is None:
                meta = meta_from_array(sample, dtype=output_dtypes)
            else:
                meta = tuple(meta_from_array(sample, dtype=odt) for odt in output_dtypes)

        ## Normalize `meta` format
        meta = meta_from_array(meta)
        if isinstance(meta, list):
            meta = tuple(meta)

        ## Validate `meta`
        if nout is None:
            if isinstance(meta, tuple):
                if len(meta) == 1:
                    meta = meta[0]
                else:
                    raise ValueError(
                        "For a function with one output, must give a single item for `output_dtypes`/`meta`, "
                        "not a tuple or list."
                    )
        else:
            if not isinstance(meta, tuple):
                raise ValueError(
                    f"For a function with {nout} outputs, must give a tuple or list for `output_dtypes`/`meta`, "
                    "not a single item."
                )
            if len(meta) != nout:
                raise ValueError(
                    f"For a function with {nout} outputs, must give a tuple or list of {nout} items for "
                    f"`output_dtypes`/`meta`, not {len(meta)}."
                )

        ## Vectorize function, if required
        if vectorize:
            otypes = [x.dtype for x in meta] if isinstance(meta, tuple) else [meta.dtype]
            func = np.vectorize(func, signature=signature, otypes=otypes)

        ## Miscellaneous
        if output_sizes is None:
            output_sizes = {}

        ## Axes
        input_axes, output_axes = _validate_normalize_axes(
            axes, axis, keepdims, input_coredimss, output_coredimss
        )

        # Main code:
        ## Cast all input arrays to dask
        args = [asarray(a) for a in args]

        if len(input_coredimss) != len(args):
            raise ValueError(
                "According to `signature`, `func` requires %d arguments, but %s given"
                % (len(input_coredimss), len(args))
            )

        ## Axes: transpose input arguments
        transposed_args = []
        for arg, iax in zip(args, input_axes):
            shape = arg.shape
            iax = tuple(a if a < 0 else a - len(shape) for a in iax)
            tidc = tuple(i for i in range(-len(shape) + 0, 0) if i not in iax) + iax
            transposed_arg = arg.transpose(tidc)
            transposed_args.append(transposed_arg)
        args = transposed_args

        ## Assess input args for loop dims
        input_shapes = [a.shape for a in args]
        input_chunkss = [a.chunks for a in args]
        num_loopdims = [len(s) - len(cd) for s, cd in zip(input_shapes, input_coredimss)]
        max_loopdims = max(num_loopdims) if num_loopdims else None
        core_input_shapes = [
            dict(zip(icd, s[n:]))
            for s, n, icd in zip(input_shapes, num_loopdims, input_coredimss)
        ]
        core_shapes = merge(*core_input_shapes)
        core_shapes.update(output_sizes)

        loop_input_dimss = [
            tuple("__loopdim%d__" % d for d in range(max_loopdims - n, max_loopdims))
            for n in num_loopdims
        ]
        input_dimss = [l + c for l, c in zip(loop_input_dimss, input_coredimss)]

        loop_output_dims = max(loop_input_dimss, key=len) if loop_input_dimss else tuple()

        ## Assess input args for same size and chunk sizes
        ### Collect sizes and chunksizes of all dims in all arrays
        dimsizess = {}
        chunksizess = {}
        for dims, shape, chunksizes in zip(input_dimss, input_shapes, input_chunkss):
            for dim, size, chunksize in zip(dims, shape, chunksizes):
                dimsizes = dimsizess.get(dim, [])
                dimsizes.append(size)
                dimsizess[dim] = dimsizes
                chunksizes_ = chunksizess.get(dim, [])
                chunksizes_.append(chunksize)
                chunksizess[dim] = chunksizes_
        ### Assert correct partitioning, for case:
        for dim, sizes in dimsizess.items():
            #### Check that the arrays have same length for same dimensions or dimension `1`
            if set(sizes) | {1} != {1, max(sizes)}:
                raise ValueError(f"Dimension `'{dim}'` with different lengths in arrays")
            if not allow_rechunk:
                chunksizes = chunksizess[dim]
                #### Check if core dimensions consist of only one chunk
                if (dim in core_shapes) and (chunksizes[0][0] < core_shapes[dim]):
                    raise ValueError(
                        "Core dimension `'{}'` consists of multiple chunks. To fix, rechunk into a single \
    chunk along this dimension or set `allow_rechunk=True`, but beware that this may increase memory usage \
    significantly.".format(
                            dim
                        )
                    )
                #### Check if loop dimensions consist of same chunksizes, when they have sizes > 1
                relevant_chunksizes = list(
                    unique(c for s, c in zip(sizes, chunksizes) if s > 1)
                )
                if len(relevant_chunksizes) > 1:
                    raise ValueError(
                        f"Dimension `'{dim}'` with different chunksize present"
                    )

        ## Apply function - use blockwise here
        arginds = list(concat(zip(args, input_dimss)))

        ### Use existing `blockwise` but only with loopdims to enforce
        ### concatenation for coredims that appear also at the output
        ### Modifying `blockwise` could improve things here.
        tmp = blockwise(
            func, loop_output_dims, *arginds, concatenate=True, meta=meta, **kwargs
        )

        # NOTE: we likely could just use `meta` instead of `tmp._meta`,
        # but we use it and validate it anyway just to be sure nothing odd has happened.
        metas = tmp._meta
        if nout is None:
            assert not isinstance(
                metas, (list, tuple)
            ), f"meta changed from single output to multiple output during blockwise: {meta} -> {metas}"
            metas = (metas,)
        else:
            assert isinstance(
                metas, (list, tuple)
            ), f"meta changed from multiple output to single output during blockwise: {meta} -> {metas}"
            assert (
                len(metas) == nout
            ), f"Number of outputs changed from {nout} to {len(metas)} during blockwise"

        ## Prepare output shapes
        loop_output_shape = tmp.shape
        loop_output_chunks = tmp.chunks
        keys = list(flatten(tmp.__dask_keys__()))
        name, token = keys[0][0].split("-")

        ### *) Treat direct output
        if nout is None:
            output_coredimss = [output_coredimss]

        ## Split output
        leaf_arrs = []
        for i, (ocd, oax, meta) in enumerate(zip(output_coredimss, output_axes, metas)):
            core_output_shape = tuple(core_shapes[d] for d in ocd)
            core_chunkinds = len(ocd) * (0,)
            output_shape = loop_output_shape + core_output_shape
            output_chunks = loop_output_chunks + core_output_shape
            leaf_name = "%s_%d-%s" % (name, i, token)
            leaf_dsk = {
                (leaf_name,)
                + key[1:]
                + core_chunkinds: ((getitem, key, i) if nout else key)
                for key in keys
            }
            graph = HighLevelGraph.from_collections(leaf_name, leaf_dsk, dependencies=[tmp])
            meta = meta_from_array(meta, len(output_shape))
            leaf_arr = Array(
                graph, leaf_name, chunks=output_chunks, shape=output_shape, meta=meta
            )

            ### Axes:
            if keepdims:
                slices = len(leaf_arr.shape) * (slice(None),) + len(oax) * (np.newaxis,)
                leaf_arr = leaf_arr[slices]

            tidcs = [None] * len(leaf_arr.shape)
            for ii, oa in zip(range(-len(oax), 0), oax):
                tidcs[oa] = ii
            j = 0
            for ii in range(len(tidcs)):
                if tidcs[ii] is None:
                    tidcs[ii] = j
                    j += 1
            leaf_arr = leaf_arr.transpose(tidcs)
            leaf_arrs.append(leaf_arr)

        return (*leaf_arrs,) if nout else leaf_arrs[0]  # Undo *) from above

    class gufunc:
        """
        Binds `pyfunc` into ``dask.array.apply_gufunc`` when called.

        Parameters
        ----------
        pyfunc : callable
            Function to call like ``func(*args, **kwargs)`` on input arrays
            (``*args``) that returns an array or tuple of arrays. If multiple
            arguments with non-matching dimensions are supplied, this function is
            expected to vectorize (broadcast) over axes of positional arguments in
            the style of NumPy universal functions [1]_ (if this is not the case,
            set ``vectorize=True``). If this function returns multiple outputs,
            ``output_core_dims`` has to be set as well.
        signature : String, keyword only
            Specifies what core dimensions are consumed and produced by ``func``.
            According to the specification of numpy.gufunc signature [2]_
        axes: List of tuples, optional, keyword only
            A list of tuples with indices of axes a generalized ufunc should operate on.
            For instance, for a signature of ``"(i,j),(j,k)->(i,k)"`` appropriate for
            matrix multiplication, the base elements are two-dimensional matrices
            and these are taken to be stored in the two last axes of each argument. The
            corresponding axes keyword would be ``[(-2, -1), (-2, -1), (-2, -1)]``.
            For simplicity, for generalized ufuncs that operate on 1-dimensional arrays
            (vectors), a single integer is accepted instead of a single-element tuple,
            and for generalized ufuncs for which all outputs are scalars, the output
            tuples can be omitted.
        axis: int, optional, keyword only
            A single axis over which a generalized ufunc should operate. This is a short-cut
            for ufuncs that operate over a single, shared core dimension, equivalent to passing
            in axes with entries of (axis,) for each single-core-dimension argument and ``()`` for
            all others. For instance, for a signature ``"(i),(i)->()"``, it is equivalent to passing
            in ``axes=[(axis,), (axis,), ()]``.
        keepdims: bool, optional, keyword only
            If this is set to True, axes which are reduced over will be left in the result as
            a dimension with size one, so that the result will broadcast correctly against the
            inputs. This option can only be used for generalized ufuncs that operate on inputs
            that all have the same number of core dimensions and with outputs that have no core
            dimensions , i.e., with signatures like ``"(i),(i)->()"`` or ``"(m,m)->()"``.
            If used, the location of the dimensions in the output can be controlled with axes
            and axis.
        output_dtypes : Optional, dtype or list of dtypes, keyword only
            Valid numpy dtype specification or list thereof.
            If not given, a call of ``func`` with a small set of data
            is performed in order to try to automatically determine the
            output dtypes.
        output_sizes : dict, optional, keyword only
            Optional mapping from dimension names to sizes for outputs. Only used if
            new core dimensions (not found on inputs) appear on outputs.
        vectorize: bool, keyword only
            If set to ``True``, ``np.vectorize`` is applied to ``func`` for
            convenience. Defaults to ``False``.
        allow_rechunk: Optional, bool, keyword only
            Allows rechunking, otherwise chunk sizes need to match and core
            dimensions are to consist only of one chunk.
            Warning: enabling this can increase memory usage significantly.
            Defaults to ``False``.
        meta: Optional, tuple, keyword only
            tuple of empty ndarrays describing the shape and dtype of the output of the gufunc.
            Defaults to ``None``.

        Returns
        -------
        Wrapped function

        Examples
        --------
        >>> import dask.array as da
        >>> import numpy as np
        >>> a = da.random.normal(size=(10,20,30), chunks=(5, 10, 30))
        >>> def stats(x):
        ...     return np.mean(x, axis=-1), np.std(x, axis=-1)
        >>> gustats = da.gufunc(stats, signature="(i)->(),()", output_dtypes=(float, float))
        >>> mean, std = gustats(a)
        >>> mean.compute().shape
        (10, 20)

        >>> a = da.random.normal(size=(   20,30), chunks=(10, 30))
        >>> b = da.random.normal(size=(10, 1,40), chunks=(5, 1, 40))
        >>> def outer_product(x, y):
        ...     return np.einsum("i,j->ij", x, y)
        >>> guouter_product = da.gufunc(outer_product, signature="(i),(j)->(i,j)", output_dtypes=float, vectorize=True)
        >>> c = guouter_product(a, b)
        >>> c.compute().shape
        (10, 20, 30, 40)

        >>> a = da.ones((1, 5, 10), chunks=(-1, -1, -1))
        >>> def stats(x):
        ...     return np.atleast_1d(x.mean()), np.atleast_1d(x.max())
        >>> meta = (np.array((), dtype=np.float64), np.array((), dtype=np.float64))
        >>> gustats = da.gufunc(stats, signature="(i,j)->(),()", meta=meta)
        >>> result = gustats(a)
        >>> result[0].compute().shape
        (1,)
        >>> result[1].compute().shape
        (1,)

        References
        ----------
        .. [1] https://docs.scipy.org/doc/numpy/reference/ufuncs.html
        .. [2] https://docs.scipy.org/doc/numpy/reference/c-api/generalized-ufuncs.html
        """

        def __init__(
            self,
            pyfunc,
            *,
            signature=None,
            vectorize=False,
            axes=None,
            axis=None,
            keepdims=False,
            output_sizes=None,
            output_dtypes=None,
            allow_rechunk=False,
            meta=None,
        ):
            self.pyfunc = pyfunc
            self.signature = signature
            self.vectorize = vectorize
            self.axes = axes
            self.axis = axis
            self.keepdims = keepdims
            self.output_sizes = output_sizes
            self.output_dtypes = output_dtypes
            self.allow_rechunk = allow_rechunk
            self.meta = meta

            self.__doc__ = """
            Bound ``dask.array.gufunc``
            func: ``{func}``
            signature: ``'{signature}'``

            Parameters
            ----------
            *args : numpy/dask arrays or scalars
                Arrays to which to apply to ``func``. Core dimensions as specified in
                ``signature`` need to come last.
            **kwargs : dict
                Extra keyword arguments to pass to ``func``

            Returns
            -------
            Single dask.array.Array or tuple of dask.array.Array
            """.format(
                func=str(self.pyfunc), signature=self.signature
            )

        def __call__(self, *args, allow_rechunk=False, **kwargs):
            return apply_gufunc(
                self.pyfunc,
                self.signature,
                *args,
                vectorize=self.vectorize,
                axes=self.axes,
                axis=self.axis,
                keepdims=self.keepdims,
                output_sizes=self.output_sizes,
                output_dtypes=self.output_dtypes,
                allow_rechunk=self.allow_rechunk or allow_rechunk,
                meta=self.meta,
                **kwargs,
            )

    def as_gufunc(signature=None, **kwargs):
        """
        Decorator for ``dask.array.gufunc``.

        Parameters
        ----------
        signature : String
            Specifies what core dimensions are consumed and produced by ``func``.
            According to the specification of numpy.gufunc signature [2]_
        axes: List of tuples, optional, keyword only
            A list of tuples with indices of axes a generalized ufunc should operate on.
            For instance, for a signature of ``"(i,j),(j,k)->(i,k)"`` appropriate for
            matrix multiplication, the base elements are two-dimensional matrices
            and these are taken to be stored in the two last axes of each argument. The
            corresponding axes keyword would be ``[(-2, -1), (-2, -1), (-2, -1)]``.
            For simplicity, for generalized ufuncs that operate on 1-dimensional arrays
            (vectors), a single integer is accepted instead of a single-element tuple,
            and for generalized ufuncs for which all outputs are scalars, the output
            tuples can be omitted.
        axis: int, optional, keyword only
            A single axis over which a generalized ufunc should operate. This is a short-cut
            for ufuncs that operate over a single, shared core dimension, equivalent to passing
            in axes with entries of (axis,) for each single-core-dimension argument and ``()`` for
            all others. For instance, for a signature ``"(i),(i)->()"``, it is equivalent to passing
            in ``axes=[(axis,), (axis,), ()]``.
        keepdims: bool, optional, keyword only
            If this is set to True, axes which are reduced over will be left in the result as
            a dimension with size one, so that the result will broadcast correctly against the
            inputs. This option can only be used for generalized ufuncs that operate on inputs
            that all have the same number of core dimensions and with outputs that have no core
            dimensions , i.e., with signatures like ``"(i),(i)->()"`` or ``"(m,m)->()"``.
            If used, the location of the dimensions in the output can be controlled with axes
            and axis.
        output_dtypes : Optional, dtype or list of dtypes, keyword only
            Valid numpy dtype specification or list thereof.
            If not given, a call of ``func`` with a small set of data
            is performed in order to try to automatically determine the
            output dtypes.
        output_sizes : dict, optional, keyword only
            Optional mapping from dimension names to sizes for outputs. Only used if
            new core dimensions (not found on inputs) appear on outputs.
        vectorize: bool, keyword only
            If set to ``True``, ``np.vectorize`` is applied to ``func`` for
            convenience. Defaults to ``False``.
        allow_rechunk: Optional, bool, keyword only
            Allows rechunking, otherwise chunk sizes need to match and core
            dimensions are to consist only of one chunk.
            Warning: enabling this can increase memory usage significantly.
            Defaults to ``False``.
        meta: Optional, tuple, keyword only
            tuple of empty ndarrays describing the shape and dtype of the output of the gufunc.
            Defaults to ``None``.

        Returns
        -------
        Decorator for `pyfunc` that itself returns a `gufunc`.

        Examples
        --------
        >>> import dask.array as da
        >>> import numpy as np
        >>> a = da.random.normal(size=(10,20,30), chunks=(5, 10, 30))
        >>> @da.as_gufunc("(i)->(),()", output_dtypes=(float, float))
        ... def stats(x):
        ...     return np.mean(x, axis=-1), np.std(x, axis=-1)
        >>> mean, std = stats(a)
        >>> mean.compute().shape
        (10, 20)

        >>> a = da.random.normal(size=(   20,30), chunks=(10, 30))
        >>> b = da.random.normal(size=(10, 1,40), chunks=(5, 1, 40))
        >>> @da.as_gufunc("(i),(j)->(i,j)", output_dtypes=float, vectorize=True)
        ... def outer_product(x, y):
        ...     return np.einsum("i,j->ij", x, y)
        >>> c = outer_product(a, b)
        >>> c.compute().shape
        (10, 20, 30, 40)

        References
        ----------
        .. [1] https://docs.scipy.org/doc/numpy/reference/ufuncs.html
        .. [2] https://docs.scipy.org/doc/numpy/reference/c-api/generalized-ufuncs.html
        """
        _allowedkeys = {
            "vectorize",
            "axes",
            "axis",
            "keepdims",
            "output_sizes",
            "output_dtypes",
            "allow_rechunk",
            "meta",
        }
        if kwargs.keys() - _allowedkeys:
            raise TypeError("Unsupported keyword argument(s) provided")

        def _as_gufunc(pyfunc):
            return gufunc(pyfunc, signature=signature, **kwargs)

        _as_gufunc.__doc__ = """
            Decorator to make ``dask.array.gufunc``
            signature: ``'{signature}'``

            Parameters
            ----------
            pyfunc : callable
                Function matching signature ``'{signature}'``.

            Returns
            -------
            ``dask.array.gufunc``
            """.format(
            signature=signature
        )
        return _as_gufunc