text: """
  Functions for applying functions that act on arrays to xarray's labeled data.
  """
  import functools
  import itertools
  import operator
  from collections import Counter, OrderedDict
  from distutils.version import LooseVersion
  from typing import (
      AbstractSet, Any, Callable, Iterable, List, Mapping, Optional, Sequence,
      Tuple, Union)
  
  import numpy as np
  
  from . import duck_array_ops, utils
  from .alignment import deep_align
  from .merge import expand_and_merge_variables
  from .pycompat import TYPE_CHECKING, dask_array_type
  from .utils import is_dict_like
  from .variable import Variable
  
  if TYPE_CHECKING:
      from .dataset import Dataset
  
  _DEFAULT_FROZEN_SET = frozenset()  # type: frozenset
  _NO_FILL_VALUE = utils.ReprObject('<no-fill-value>')
  _DEFAULT_NAME = utils.ReprObject('<default-name>')
  _JOINS_WITHOUT_FILL_VALUES = frozenset({'inner', 'exact'})
  
  
  class _UFuncSignature:
      """Core dimensions signature for a given function.
  
      Based on the signature provided by generalized ufuncs in NumPy.
  
      Attributes
      ----------
      input_core_dims : tuple[tuple]
          Core dimension names on each input variable.
      output_core_dims : tuple[tuple]
          Core dimension names on each output variable.
      """
  
      def __init__(self, input_core_dims, output_core_dims=((),)):
          self.input_core_dims = tuple(tuple(a) for a in input_core_dims)
          self.output_core_dims = tuple(tuple(a) for a in output_core_dims)
          self._all_input_core_dims = None
          self._all_output_core_dims = None
          self._all_core_dims = None
  
      @property
      def all_input_core_dims(self):
          if self._all_input_core_dims is None:
              self._all_input_core_dims = frozenset(
                  dim for dims in self.input_core_dims for dim in dims)
          return self._all_input_core_dims
  
      @property
      def all_output_core_dims(self):
          if self._all_output_core_dims is None:
              self._all_output_core_dims = frozenset(
                  dim for dims in self.output_core_dims for dim in dims)
          return self._all_output_core_dims
  
      @property
      def all_core_dims(self):
          if self._all_core_dims is None:
              self._all_core_dims = (self.all_input_core_dims |
                                     self.all_output_core_dims)
          return self._all_core_dims
  
      @property
      def num_inputs(self):
          return len(self.input_core_dims)
  
      @property
      def num_outputs(self):
          return len(self.output_core_dims)
  
      def __eq__(self, other):
          try:
              return (self.input_core_dims == other.input_core_dims and
                      self.output_core_dims == other.output_core_dims)
          except AttributeError:
              return False
  
      def __ne__(self, other):
          return not self == other
  
      def __repr__(self):
          return ('%s(%r, %r)'
                  % (type(self).__name__,
                     list(self.input_core_dims),
                     list(self.output_core_dims)))
  
      def __str__(self):
          lhs = ','.join('({})'.format(','.join(dims))
                         for dims in self.input_core_dims)
          rhs = ','.join('({})'.format(','.join(dims))
                         for dims in self.output_core_dims)
          return '{}->{}'.format(lhs, rhs)
  
      def to_gufunc_string(self):
          """Create an equivalent signature string for a NumPy gufunc.
  
          Unlike __str__, handles dimensions that don't map to Python
          identifiers.
          """
          all_dims = self.all_core_dims
          dims_map = dict(zip(sorted(all_dims), range(len(all_dims))))
          input_core_dims = [['dim%d' % dims_map[dim] for dim in core_dims]
                             for core_dims in self.input_core_dims]
          output_core_dims = [['dim%d' % dims_map[dim] for dim in core_dims]
                              for core_dims in self.output_core_dims]
          alt_signature = type(self)(input_core_dims, output_core_dims)
          return str(alt_signature)
  
  
  def result_name(objects: list) -> Any:
      # use the same naming heuristics as pandas:
      # https://github.com/blaze/blaze/issues/458#issuecomment-51936356
      names = {getattr(obj, 'name', _DEFAULT_NAME) for obj in objects}
      names.discard(_DEFAULT_NAME)
      if len(names) == 1:
          name, = names
      else:
          name = None
      return name
  
  
  def _get_coord_variables(args):
      input_coords = []
      for arg in args:
          try:
              coords = arg.coords
          except AttributeError:
              pass  # skip this argument
          else:
              coord_vars = getattr(coords, 'variables', coords)
              input_coords.append(coord_vars)
      return input_coords
  
  
  def build_output_coords(
      args: list,
      signature: _UFuncSignature,
      exclude_dims: AbstractSet = frozenset(),
  ) -> 'List[OrderedDict[Any, Variable]]':
      """Build output coordinates for an operation.
  
      Parameters
      ----------
      args : list
          List of raw operation arguments. Any valid types for xarray operations
          are OK, e.g., scalars, Variable, DataArray, Dataset.
      signature : _UfuncSignature
          Core dimensions signature for the operation.
      exclude_dims : optional set
          Dimensions excluded from the operation. Coordinates along these
          dimensions are dropped.
  
      Returns
      -------
      OrderedDict of Variable objects with merged coordinates.
      """
      input_coords = _get_coord_variables(args)
  
      if exclude_dims:
          input_coords = [OrderedDict((k, v) for k, v in coord_vars.items()
                                      if exclude_dims.isdisjoint(v.dims))
                          for coord_vars in input_coords]
  
      if len(input_coords) == 1:
          # we can skip the expensive merge
          unpacked_input_coords, = input_coords
          merged = OrderedDict(unpacked_input_coords)
      else:
          merged = expand_and_merge_variables(input_coords)
  
      output_coords = []
      for output_dims in signature.output_core_dims:
          dropped_dims = signature.all_input_core_dims - set(output_dims)
          if dropped_dims:
              filtered = OrderedDict((k, v) for k, v in merged.items()
                                     if dropped_dims.isdisjoint(v.dims))
          else:
              filtered = merged
          output_coords.append(filtered)
  
      return output_coords
  
  
  def apply_dataarray_vfunc(
      func,
      *args,
      signature,
      join='inner',
      exclude_dims=frozenset(),
      keep_attrs=False
  ):
      """Apply a variable level function over DataArray, Variable and/or ndarray
      objects.
      """
      from .dataarray import DataArray
  
      if len(args) > 1:
          args = deep_align(args, join=join, copy=False, exclude=exclude_dims,
                            raise_on_invalid=False)
  
      if keep_attrs and hasattr(args[0], 'name'):
          name = args[0].name
      else:
          name = result_name(args)
      result_coords = build_output_coords(args, signature, exclude_dims)
  
      data_vars = [getattr(a, 'variable', a) for a in args]
      result_var = func(*data_vars)
  
      if signature.num_outputs > 1:
          out = tuple(DataArray(variable, coords, name=name, fastpath=True)
                      for variable, coords in zip(result_var, result_coords))
      else:
          coords, = result_coords
          out = DataArray(result_var, coords, name=name, fastpath=True)
  
      return out
  
  
  def ordered_set_union(all_keys: List[Iterable]) -> Iterable:
      result_dict = OrderedDict()  # type: OrderedDict[Any, None]
      for keys in all_keys:
          for key in keys:
              result_dict[key] = None
      return result_dict.keys()
  
  
  def ordered_set_intersection(all_keys: List[Iterable]) -> Iterable:
      intersection = set(all_keys[0])
      for keys in all_keys[1:]:
          intersection.intersection_update(keys)
      return [key for key in all_keys[0] if key in intersection]
  
  
  def assert_and_return_exact_match(all_keys):
      first_keys = all_keys[0]
      for keys in all_keys[1:]:
          if keys != first_keys:
              raise ValueError(
                  'exact match required for all data variable names, '
                  'but %r != %r' % (keys, first_keys))
      return first_keys
  
  
  _JOINERS = {
      'inner': ordered_set_intersection,
      'outer': ordered_set_union,
      'left': operator.itemgetter(0),
      'right': operator.itemgetter(-1),
      'exact': assert_and_return_exact_match,
  }
  
  
  def join_dict_keys(
      objects: Iterable[Union[Mapping, Any]], how: str = 'inner',
  ) -> Iterable:
      joiner = _JOINERS[how]
      all_keys = [obj.keys() for obj in objects if hasattr(obj, 'keys')]
      return joiner(all_keys)
  
  
  def collect_dict_values(
      objects: Iterable[Union[Mapping, Any]],
      keys: Iterable,
      fill_value: object = None,
  ) -> List[list]:
      return [[obj.get(key, fill_value)
               if is_dict_like(obj)
               else obj
               for obj in objects]
              for key in keys]
  
  
  def _as_variables_or_variable(arg):
      try:
          return arg.variables
      except AttributeError:
          try:
              return arg.variable
          except AttributeError:
              return arg
  
  
  def _unpack_dict_tuples(
      result_vars: Mapping[Any, Tuple[Variable]],
      num_outputs: int,
  ) -> 'Tuple[OrderedDict[Any, Variable], ...]':
      out = tuple(OrderedDict() for _ in range(num_outputs))  # type: ignore
      for name, values in result_vars.items():
          for value, results_dict in zip(values, out):
              results_dict[name] = value
      return out
  
  
  def apply_dict_of_variables_vfunc(
      func, *args, signature, join='inner', fill_value=None
  ):
      """Apply a variable level function over dicts of DataArray, DataArray,
      Variable and ndarray objects.
      """
      args = [_as_variables_or_variable(arg) for arg in args]
      names = join_dict_keys(args, how=join)
      grouped_by_name = collect_dict_values(args, names, fill_value)
  
      result_vars = OrderedDict()
      for name, variable_args in zip(names, grouped_by_name):
          result_vars[name] = func(*variable_args)
  
      if signature.num_outputs > 1:
          return _unpack_dict_tuples(result_vars, signature.num_outputs)
      else:
          return result_vars
  
  
  def _fast_dataset(
      variables: 'OrderedDict[Any, Variable]',
      coord_variables: Mapping[Any, Variable],
  ) -> 'Dataset':
      """Create a dataset as quickly as possible.
  
      Beware: the `variables` OrderedDict is modified INPLACE.
      """
      from .dataset import Dataset
      variables.update(coord_variables)
      coord_names = set(coord_variables)
      return Dataset._from_vars_and_coord_names(variables, coord_names)
  
  
  def apply_dataset_vfunc(
      func,
      *args,
      signature,
      join='inner',
      dataset_join='exact',
      fill_value=_NO_FILL_VALUE,
      exclude_dims=frozenset(),
      keep_attrs=False
  ):
      """Apply a variable level function over Dataset, dict of DataArray,
      DataArray, Variable and/or ndarray objects.
      """
      from .dataset import Dataset
      first_obj = args[0]  # we'll copy attrs from this in case keep_attrs=True
  
      if (dataset_join not in _JOINS_WITHOUT_FILL_VALUES and
              fill_value is _NO_FILL_VALUE):
          raise TypeError('to apply an operation to datasets with different '
                          'data variables with apply_ufunc, you must supply the '
                          'dataset_fill_value argument.')
  
      if len(args) > 1:
          args = deep_align(args, join=join, copy=False, exclude=exclude_dims,
                            raise_on_invalid=False)
  
      list_of_coords = build_output_coords(args, signature, exclude_dims)
      args = [getattr(arg, 'data_vars', arg) for arg in args]
  
      result_vars = apply_dict_of_variables_vfunc(
          func, *args, signature=signature, join=dataset_join,
          fill_value=fill_value)
  
      if signature.num_outputs > 1:
          out = tuple(_fast_dataset(*args)
                      for args in zip(result_vars, list_of_coords))
      else:
          coord_vars, = list_of_coords
          out = _fast_dataset(result_vars, coord_vars)
  
      if keep_attrs and isinstance(first_obj, Dataset):
          if isinstance(out, tuple):
              out = tuple(ds._copy_attrs_from(first_obj) for ds in out)
          else:
              out._copy_attrs_from(first_obj)
      return out
  
  
  def _iter_over_selections(obj, dim, values):
      """Iterate over selections of an xarray object in the provided order."""
      from .groupby import _dummy_copy
  
      dummy = None
      for value in values:
          try:
              obj_sel = obj.sel(**{dim: value})
          except (KeyError, IndexError):
              if dummy is None:
                  dummy = _dummy_copy(obj)
              obj_sel = dummy
          yield obj_sel
  
  
  def apply_groupby_func(func, *args):
      """Apply a dataset or datarray level function over GroupBy, Dataset,
      DataArray, Variable and/or ndarray objects.
      """
      from .groupby import GroupBy, peek_at
      from .variable import Variable
  
      groupbys = [arg for arg in args if isinstance(arg, GroupBy)]
      assert groupbys, 'must have at least one groupby to iterate over'
      first_groupby = groupbys[0]
      if any(not first_groupby._group.equals(gb._group) for gb in groupbys[1:]):
          raise ValueError('apply_ufunc can only perform operations over '
                           'multiple GroupBy objets at once if they are all '
                           'grouped the same way')
  
      grouped_dim = first_groupby._group.name
      unique_values = first_groupby._unique_coord.values
  
      iterators = []
      for arg in args:
          if isinstance(arg, GroupBy):
              iterator = (value for _, value in arg)
          elif hasattr(arg, 'dims') and grouped_dim in arg.dims:
              if isinstance(arg, Variable):
                  raise ValueError(
                      'groupby operations cannot be performed with '
                      'xarray.Variable objects that share a dimension with '
                      'the grouped dimension')
              iterator = _iter_over_selections(arg, grouped_dim, unique_values)
          else:
              iterator = itertools.repeat(arg)
          iterators.append(iterator)
  
      applied = (func(*zipped_args) for zipped_args in zip(*iterators))
      applied_example, applied = peek_at(applied)
      combine = first_groupby._combine
      if isinstance(applied_example, tuple):
          combined = tuple(combine(output) for output in zip(*applied))
      else:
          combined = combine(applied)
      return combined
  
  
  def unified_dim_sizes(
      variables: Iterable[Variable],
      exclude_dims: AbstractSet = frozenset()
  ) -> 'OrderedDict[Any, int]':
  
      dim_sizes = OrderedDict()  # type: OrderedDict[Any, int]
  
      for var in variables:
          if len(set(var.dims)) < len(var.dims):
              raise ValueError('broadcasting cannot handle duplicate '
                               'dimensions on a variable: %r' % list(var.dims))
          for dim, size in zip(var.dims, var.shape):
              if dim not in exclude_dims:
                  if dim not in dim_sizes:
                      dim_sizes[dim] = size
                  elif dim_sizes[dim] != size:
                      raise ValueError('operands cannot be broadcast together '
                                       'with mismatched lengths for dimension '
                                       '%r: %s vs %s'
                                       % (dim, dim_sizes[dim], size))
      return dim_sizes
  
  
  SLICE_NONE = slice(None)
  
  
  def broadcast_compat_data(variable, broadcast_dims, core_dims):
      # type: (Variable, tuple, tuple) -> Any
      data = variable.data
  
      old_dims = variable.dims
      new_dims = broadcast_dims + core_dims
  
      if new_dims == old_dims:
          # optimize for the typical case
          return data
  
      set_old_dims = set(old_dims)
      missing_core_dims = [d for d in core_dims if d not in set_old_dims]
      if missing_core_dims:
          raise ValueError(
              'operand to apply_ufunc has required core dimensions %r, but '
              'some of these are missing on the input variable:  %r'
              % (list(core_dims), missing_core_dims))
  
      set_new_dims = set(new_dims)
      unexpected_dims = [d for d in old_dims if d not in set_new_dims]
      if unexpected_dims:
          raise ValueError('operand to apply_ufunc encountered unexpected '
                           'dimensions %r on an input variable: these are core '
                           'dimensions on other input or output variables'
                           % unexpected_dims)
  
      # for consistency with numpy, keep broadcast dimensions to the left
      old_broadcast_dims = tuple(d for d in broadcast_dims if d in set_old_dims)
      reordered_dims = old_broadcast_dims + core_dims
      if reordered_dims != old_dims:
          order = tuple(old_dims.index(d) for d in reordered_dims)
          data = duck_array_ops.transpose(data, order)
  
      if new_dims != reordered_dims:
          key_parts = []
          for dim in new_dims:
              if dim in set_old_dims:
                  key_parts.append(SLICE_NONE)
              elif key_parts:
                  # no need to insert new axes at the beginning that are already
                  # handled by broadcasting
                  key_parts.append(np.newaxis)
          data = data[tuple(key_parts)]
  
      return data
  
  
  def apply_variable_ufunc(
      func,
      *args,
      signature,
      exclude_dims=frozenset(),
      dask='forbidden',
      output_dtypes=None,
      output_sizes=None,
      keep_attrs=False
  ):
      """Apply a ndarray level function over Variable and/or ndarray objects.
      """
      from .variable import Variable, as_compatible_data
  
      dim_sizes = unified_dim_sizes((a for a in args if hasattr(a, 'dims')),
                                    exclude_dims=exclude_dims)
      broadcast_dims = tuple(dim for dim in dim_sizes
                             if dim not in signature.all_core_dims)
      output_dims = [broadcast_dims + out for out in signature.output_core_dims]
  
      input_data = [broadcast_compat_data(arg, broadcast_dims, core_dims)
                    if isinstance(arg, Variable)
                    else arg
                    for arg, core_dims in zip(args, signature.input_core_dims)]
  
      if any(isinstance(array, dask_array_type) for array in input_data):
          if dask == 'forbidden':
              raise ValueError('apply_ufunc encountered a dask array on an '
                               'argument, but handling for dask arrays has not '
                               'been enabled. Either set the ``dask`` argument '
                               'or load your data into memory first with '
                               '``.load()`` or ``.compute()``')
          elif dask == 'parallelized':
              input_dims = [broadcast_dims + dims
                            for dims in signature.input_core_dims]
              numpy_func = func
  
              def func(*arrays):
                  return _apply_with_dask_atop(
                      numpy_func, arrays, input_dims, output_dims,
                      signature, output_dtypes, output_sizes)
          elif dask == 'allowed':
              pass
          else:
              raise ValueError('unknown setting for dask array handling in '
                               'apply_ufunc: {}'.format(dask))
      result_data = func(*input_data)
  
      if signature.num_outputs == 1:
          result_data = (result_data,)
      elif (not isinstance(result_data, tuple) or
              len(result_data) != signature.num_outputs):
          raise ValueError('applied function does not have the number of '
                           'outputs specified in the ufunc signature. '
                           'Result is not a tuple of {} elements: {!r}'
                           .format(signature.num_outputs, result_data))
  
      output = []
      for dims, data in zip(output_dims, result_data):
          data = as_compatible_data(data)
          if data.ndim != len(dims):
              raise ValueError(
                  'applied function returned data with unexpected '
                  'number of dimensions: {} vs {}, for dimensions {}'
                  .format(data.ndim, len(dims), dims))
  
          var = Variable(dims, data, fastpath=True)
          for dim, new_size in var.sizes.items():
              if dim in dim_sizes and new_size != dim_sizes[dim]:
                  raise ValueError(
                      'size of dimension {!r} on inputs was unexpectedly '
                      'changed by applied function from {} to {}. Only '
                      'dimensions specified in ``exclude_dims`` with '
                      'xarray.apply_ufunc are allowed to change size.'
                      .format(dim, dim_sizes[dim], new_size))
  
          if keep_attrs and isinstance(args[0], Variable):
              var.attrs.update(args[0].attrs)
          output.append(var)
  
      if signature.num_outputs == 1:
          return output[0]
      else:
          return tuple(output)
  
  
  def _apply_with_dask_atop(func, args, input_dims, output_dims, signature,
                            output_dtypes, output_sizes=None):
      import dask.array as da
  
      if signature.num_outputs > 1:
          raise NotImplementedError('multiple outputs from apply_ufunc not yet '
                                    "supported with dask='parallelized'")
  
      if output_dtypes is None:
          raise ValueError('output dtypes (output_dtypes) must be supplied to '
                           "apply_func when using dask='parallelized'")
      if not isinstance(output_dtypes, list):
          raise TypeError('output_dtypes must be a list of objects coercible to '
                          'numpy dtypes, got {}'.format(output_dtypes))
      if len(output_dtypes) != signature.num_outputs:
          raise ValueError('apply_ufunc arguments output_dtypes and '
                           'output_core_dims must have the same length: {} vs {}'
                           .format(len(output_dtypes), signature.num_outputs))
      (dtype,) = output_dtypes
  
      if output_sizes is None:
          output_sizes = {}
  
      new_dims = signature.all_output_core_dims - signature.all_input_core_dims
      if any(dim not in output_sizes for dim in new_dims):
          raise ValueError("when using dask='parallelized' with apply_ufunc, "
                           'output core dimensions not found on inputs must '
                           'have explicitly set sizes with ``output_sizes``: {}'
                           .format(new_dims))
  
      for n, (data, core_dims) in enumerate(
              zip(args, signature.input_core_dims)):
          if isinstance(data, dask_array_type):
              # core dimensions cannot span multiple chunks
              for axis, dim in enumerate(core_dims, start=-len(core_dims)):
                  if len(data.chunks[axis]) != 1:
                      raise ValueError(
                          'dimension {!r} on {}th function argument to '
                          "apply_ufunc with dask='parallelized' consists of "
                          'multiple chunks, but is also a core dimension. To '
                          'fix, rechunk into a single dask array chunk along '
                          'this dimension, i.e., ``.chunk({})``, but beware '
                          'that this may significantly increase memory usage.'
                          .format(dim, n, {dim: -1}))
  
      (out_ind,) = output_dims
  
      atop_args = []
      for arg, dims in zip(args, input_dims):
          # skip leading dimensions that are implicitly added by broadcasting
          ndim = getattr(arg, 'ndim', 0)
          trimmed_dims = dims[-ndim:] if ndim else ()
          atop_args.extend([arg, trimmed_dims])
  
      return da.atop(func, out_ind, *atop_args, dtype=dtype, concatenate=True,
                     new_axes=output_sizes)
  
  
  def apply_array_ufunc(func, *args, dask='forbidden'):
      """Apply a ndarray level function over ndarray objects."""
      if any(isinstance(arg, dask_array_type) for arg in args):
          if dask == 'forbidden':
              raise ValueError('apply_ufunc encountered a dask array on an '
                               'argument, but handling for dask arrays has not '
                               'been enabled. Either set the ``dask`` argument '
                               'or load your data into memory first with '
                               '``.load()`` or ``.compute()``')
          elif dask == 'parallelized':
              raise ValueError("cannot use dask='parallelized' for apply_ufunc "
                               'unless at least one input is an xarray object')
          elif dask == 'allowed':
              pass
          else:
              raise ValueError('unknown setting for dask array handling: {}'
                               .format(dask))
      return func(*args)
  
  
  def apply_ufunc(
      func: Callable,
      *args: Any,
      input_core_dims: Optional[Sequence[Sequence]] = None,
      output_core_dims: Optional[Sequence[Sequence]] = ((),),
      exclude_dims: AbstractSet = frozenset(),
      vectorize: bool = False,
      join: str = 'exact',
      dataset_join: str = 'exact',
      dataset_fill_value: object = _NO_FILL_VALUE,
      keep_attrs: bool = False,
      kwargs: Mapping = None,
      dask: str = 'forbidden',
      output_dtypes: Optional[Sequence] = None,
      output_sizes: Optional[Mapping[Any, int]] = None
  ) -> Any:
      """Apply a vectorized function for unlabeled arrays on xarray objects.
  
      The function will be mapped over the data variable(s) of the input
      arguments using xarray's standard rules for labeled computation, including
      alignment, broadcasting, looping over GroupBy/Dataset variables, and
      merging of coordinates.
  
      Parameters
      ----------
      func : callable
          Function to call like ``func(*args, **kwargs)`` on unlabeled arrays
          (``.data``) 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, you
          must set ``output_core_dims`` as well.
      *args : Dataset, DataArray, GroupBy, Variable, numpy/dask arrays or scalars
          Mix of labeled and/or unlabeled arrays to which to apply the function.
      input_core_dims : Sequence[Sequence], optional
          List of the same length as ``args`` giving the list of core dimensions
          on each input argument that should not be broadcast. By default, we
          assume there are no core dimensions on any input arguments.
  
          For example, ``input_core_dims=[[], ['time']]`` indicates that all
          dimensions on the first argument and all dimensions other than 'time'
          on the second argument should be broadcast.
  
          Core dimensions are automatically moved to the last axes of input
          variables before applying ``func``, which facilitates using NumPy style
          generalized ufuncs [2]_.
      output_core_dims : List[tuple], optional
          List of the same length as the number of output arguments from
          ``func``, giving the list of core dimensions on each output that were
          not broadcast on the inputs. By default, we assume that ``func``
          outputs exactly one array, with axes corresponding to each broadcast
          dimension.
  
          Core dimensions are assumed to appear as the last dimensions of each
          output in the provided order.
      exclude_dims : set, optional
          Core dimensions on the inputs to exclude from alignment and
          broadcasting entirely. Any input coordinates along these dimensions
          will be dropped. Each excluded dimension must also appear in
          ``input_core_dims`` for at least one argument. Only dimensions listed
          here are allowed to change size between input and output objects.
      vectorize : bool, optional
          If True, then assume ``func`` only takes arrays defined over core
          dimensions as input and vectorize it automatically with
          :py:func:`numpy.vectorize`. This option exists for convenience, but is
          almost always slower than supplying a pre-vectorized function.
          Using this option requires NumPy version 1.12 or newer.
      join : {'outer', 'inner', 'left', 'right', 'exact'}, optional
          Method for joining the indexes of the passed objects along each
          dimension, and the variables of Dataset objects with mismatched
          data variables:
  
          - 'outer': use the union of object indexes
          - 'inner': use the intersection of object indexes
          - 'left': use indexes from the first object with each dimension
          - 'right': use indexes from the last object with each dimension
          - 'exact': raise `ValueError` instead of aligning when indexes to be
            aligned are not equal
      dataset_join : {'outer', 'inner', 'left', 'right', 'exact'}, optional
          Method for joining variables of Dataset objects with mismatched
          data variables.
  
          - 'outer': take variables from both Dataset objects
          - 'inner': take only overlapped variables
          - 'left': take only variables from the first object
          - 'right': take only variables from the last object
          - 'exact': data variables on all Dataset objects must match exactly
      dataset_fill_value : optional
          Value used in place of missing variables on Dataset inputs when the
          datasets do not share the exact same ``data_vars``. Required if
          ``dataset_join not in {'inner', 'exact'}``, otherwise ignored.
      keep_attrs: boolean, Optional
          Whether to copy attributes from the first argument to the output.
      kwargs: dict, optional
          Optional keyword arguments passed directly on to call ``func``.
      dask: 'forbidden', 'allowed' or 'parallelized', optional
          How to handle applying to objects containing lazy data in the form of
          dask arrays:
  
          - 'forbidden' (default): raise an error if a dask array is encountered.
          - 'allowed': pass dask arrays directly on to ``func``.
          - 'parallelized': automatically parallelize ``func`` if any of the
            inputs are a dask array. If used, the ``output_dtypes`` argument must
            also be provided. Multiple output arguments are not yet supported.
      output_dtypes : list of dtypes, optional
          Optional list of output dtypes. Only used if dask='parallelized'.
      output_sizes : dict, optional
          Optional mapping from dimension names to sizes for outputs. Only used
          if dask='parallelized' and new dimensions (not found on inputs) appear
          on outputs.
  
      Returns
      -------
      Single value or tuple of Dataset, DataArray, Variable, dask.array.Array or
      numpy.ndarray, the first type on that list to appear on an input.
  
      Examples
      --------
  
      Calculate the vector magnitude of two arguments:
  
      >>> def magnitude(a, b):
      ...     func = lambda x, y: np.sqrt(x ** 2 + y ** 2)
      ...     return xr.apply_ufunc(func, a, b)
  
      You can now apply ``magnitude()`` to ``xr.DataArray`` and ``xr.Dataset``
      objects, with automatically preserved dimensions and coordinates, e.g.,
  
      >>> array = xr.DataArray([1, 2, 3], coords=[('x', [0.1, 0.2, 0.3])])
      >>> magnitude(array, -array)
      <xarray.DataArray (x: 3)>
      array([1.414214, 2.828427, 4.242641])
      Coordinates:
        * x        (x) float64 0.1 0.2 0.3
  
      Plain scalars, numpy arrays and a mix of these with xarray objects is also
      supported:
  
      >>> magnitude(4, 5)
      5.0
      >>> magnitude(3, np.array([0, 4]))
      array([3., 5.])
      >>> magnitude(array, 0)
      <xarray.DataArray (x: 3)>
      array([1., 2., 3.])
      Coordinates:
        * x        (x) float64 0.1 0.2 0.3
  
      Other examples of how you could use ``apply_ufunc`` to write functions to
      (very nearly) replicate existing xarray functionality:
  
      Compute the mean (``.mean``) over one dimension::
  
          def mean(obj, dim):
              # note: apply always moves core dimensions to the end
              return apply_ufunc(np.mean, obj,
                                 input_core_dims=[[dim]],
                                 kwargs={'axis': -1})
  
      Inner product over a specific dimension (like ``xr.dot``)::
  
          def _inner(x, y):
              result = np.matmul(x[..., np.newaxis, :], y[..., :, np.newaxis])
              return result[..., 0, 0]
  
          def inner_product(a, b, dim):
              return apply_ufunc(_inner, a, b, input_core_dims=[[dim], [dim]])
  
      Stack objects along a new dimension (like ``xr.concat``)::
  
          def stack(objects, dim, new_coord):
              # note: this version does not stack coordinates
              func = lambda *x: np.stack(x, axis=-1)
              result = apply_ufunc(func, *objects,
                                   output_core_dims=[[dim]],
                                   join='outer',
                                   dataset_fill_value=np.nan)
              result[dim] = new_coord
              return result
  
      If your function is not vectorized but can be applied only to core
      dimensions, you can use ``vectorize=True`` to turn into a vectorized
      function. This wraps :py:func:`numpy.vectorize`, so the operation isn't
      terribly fast. Here we'll use it to calculate the distance between
      empirical samples from two probability distributions, using a scipy
      function that needs to be applied to vectors::
  
          import scipy.stats
  
          def earth_mover_distance(first_samples,
                                   second_samples,
                                   dim='ensemble'):
              return apply_ufunc(scipy.stats.wasserstein_distance,
                                 first_samples, second_samples,
                                 input_core_dims=[[dim], [dim]],
                                 vectorize=True)
  
      Most of NumPy's builtin functions already broadcast their inputs
      appropriately for use in `apply`. You may find helper functions such as
      numpy.broadcast_arrays helpful in writing your function. `apply_ufunc` also
      works well with numba's vectorize and guvectorize. Further explanation with
      examples are provided in the xarray documentation [3].
  
      See also
      --------
      numpy.broadcast_arrays
      numba.vectorize
      numba.guvectorize
  
      References
      ----------
      .. [1] http://docs.scipy.org/doc/numpy/reference/ufuncs.html
      .. [2] http://docs.scipy.org/doc/numpy/reference/c-api.generalized-ufuncs.html
      .. [3] http://xarray.pydata.org/en/stable/computation.html#wrapping-custom-computation
      """  # noqa: E501  # don't error on that URL one line up
      from .groupby import GroupBy
      from .dataarray import DataArray
      from .variable import Variable
  
      if input_core_dims is None:
          input_core_dims = ((),) * (len(args))
      elif len(input_core_dims) != len(args):
          raise ValueError(
              'input_core_dims must be None or a tuple with the length same to '
              'the number of arguments. Given input_core_dims: {}, '
              'number of args: {}.'.format(input_core_dims, len(args)))
  
      if kwargs is None:
          kwargs = {}
  
      signature = _UFuncSignature(input_core_dims, output_core_dims)
  
      if exclude_dims and not exclude_dims <= signature.all_core_dims:
          raise ValueError('each dimension in `exclude_dims` must also be a '
                           'core dimension in the function signature')
  
      if kwargs:
          func = functools.partial(func, **kwargs)
  
      if vectorize:
          if signature.all_core_dims:
              # we need the signature argument
              if LooseVersion(np.__version__) < '1.12':  # pragma: no cover
                  raise NotImplementedError(
                      'numpy 1.12 or newer required when using vectorize=True '
                      'in xarray.apply_ufunc with non-scalar output core '
                      'dimensions.')
              func = np.vectorize(func,
                                  otypes=output_dtypes,
                                  signature=signature.to_gufunc_string())
          else:
              func = np.vectorize(func, otypes=output_dtypes)
  
      variables_vfunc = functools.partial(apply_variable_ufunc, func,
                                          signature=signature,
                                          exclude_dims=exclude_dims,
                                          keep_attrs=keep_attrs,
                                          dask=dask,
                                          output_dtypes=output_dtypes,
                                          output_sizes=output_sizes)
  
      if any(isinstance(a, GroupBy) for a in args):
          this_apply = functools.partial(apply_ufunc, func,
                                         input_core_dims=input_core_dims,
                                         output_core_dims=output_core_dims,
                                         exclude_dims=exclude_dims,
                                         join=join,
                                         dataset_join=dataset_join,
                                         dataset_fill_value=dataset_fill_value,
                                         keep_attrs=keep_attrs,
                                         dask=dask)
          return apply_groupby_func(this_apply, *args)
      elif any(is_dict_like(a) for a in args):
          return apply_dataset_vfunc(variables_vfunc, *args,
                                     signature=signature,
                                     join=join,
                                     exclude_dims=exclude_dims,
                                     dataset_join=dataset_join,
                                     fill_value=dataset_fill_value,
                                     keep_attrs=keep_attrs)
      elif any(isinstance(a, DataArray) for a in args):
          return apply_dataarray_vfunc(variables_vfunc, *args,
                                       signature=signature,
                                       join=join,
                                       exclude_dims=exclude_dims,
                                       keep_attrs=keep_attrs)
      elif any(isinstance(a, Variable) for a in args):
          return variables_vfunc(*args)
      else:
          return apply_array_ufunc(func, *args, dask=dask)
  
  
  def dot(*arrays, dims=None, **kwargs):
      """Generalized dot product for xarray objects. Like np.einsum, but
      provides a simpler interface based on array dimensions.
  
      Parameters
      ----------
      arrays: DataArray (or Variable) objects
          Arrays to compute.
      dims: str or tuple of strings, optional
          Which dimensions to sum over.
          If not speciified, then all the common dimensions are summed over.
      **kwargs: dict
          Additional keyword arguments passed to numpy.einsum or
          dask.array.einsum
  
      Returns
      -------
      dot: DataArray
  
      Examples
      --------
  
      >>> da_a = xr.DataArray(np.arange(3 * 4).reshape(3, 4), dims=['a', 'b'])
      >>> da_b = xr.DataArray(np.arange(3 * 4 * 5).reshape(3, 4, 5),
      >>>                     dims=['a', 'b', 'c'])
      >>> da_c = xr.DataArray(np.arange(5 * 6).reshape(5, 6), dims=['c', 'd'])
      >>>
      >>> xr.dot(da_a, da_b, dims=['a', 'b']).dims
      ('c', )
      >>> xr.dot(da_a, da_b, dims=['a']).dims
      ('b', 'c')
      >>> xr.dot(da_a, da_b, da_c, dims=['b', 'c']).dims
      ('a', 'd')
      """
      from .dataarray import DataArray
      from .variable import Variable
  
      if any(not isinstance(arr, (Variable, DataArray)) for arr in arrays):
          raise TypeError('Only xr.DataArray and xr.Variable are supported.'
                          'Given {}.'.format([type(arr) for arr in arrays]))
  
      if len(arrays) == 0:
          raise TypeError('At least one array should be given.')
  
      if isinstance(dims, str):
          dims = (dims, )
  
      common_dims = set.intersection(*[set(arr.dims) for arr in arrays])
      all_dims = []
      for arr in arrays:
          all_dims += [d for d in arr.dims if d not in all_dims]
  
      einsum_axes = 'abcdefghijklmnopqrstuvwxyz'
      dim_map = {d: einsum_axes[i] for i, d in enumerate(all_dims)}
  
      if dims is None:
          # find dimensions that occur more than one times
          dim_counts = Counter()
          for arr in arrays:
              dim_counts.update(arr.dims)
          dims = tuple(d for d, c in dim_counts.items() if c > 1)
  
      dims = tuple(dims)  # make dims a tuple
  
      # dimensions to be parallelized
      broadcast_dims = tuple(d for d in all_dims
                             if d in common_dims and d not in dims)
      input_core_dims = [[d for d in arr.dims if d not in broadcast_dims]
                         for arr in arrays]
      output_core_dims = [tuple(d for d in all_dims if d not in
                                dims + broadcast_dims)]
  
      # older dask than 0.17.4, we use tensordot if possible.
      if isinstance(arr.data, dask_array_type):
          import dask
          if LooseVersion(dask.__version__) < LooseVersion('0.17.4'):
              if len(broadcast_dims) == 0 and len(arrays) == 2:
                  axes = [[arr.get_axis_num(d) for d in arr.dims if d in dims]
                          for arr in arrays]
                  return apply_ufunc(duck_array_ops.tensordot, *arrays,
                                     dask='allowed',
                                     input_core_dims=input_core_dims,
                                     output_core_dims=output_core_dims,
                                     kwargs={'axes': axes})
  
      # construct einsum subscripts, such as '...abc,...ab->...c'
      # Note: input_core_dims are always moved to the last position
      subscripts_list = ['...' + ''.join([dim_map[d] for d in ds]) for ds
                         in input_core_dims]
      subscripts = ','.join(subscripts_list)
      subscripts += '->...' + ''.join([dim_map[d] for d in output_core_dims[0]])
  
      # subscripts should be passed to np.einsum as arg, not as kwargs. We need
      # to construct a partial function for apply_ufunc to work.
      func = functools.partial(duck_array_ops.einsum, subscripts, **kwargs)
      result = apply_ufunc(func, *arrays,
                           input_core_dims=input_core_dims,
                           output_core_dims=output_core_dims,
                           dask='allowed')
      return result.transpose(*[d for d in all_dims if d in result.dims])
  
  
  def where(cond, x, y):
      """Return elements from `x` or `y` depending on `cond`.
  
      Performs xarray-like broadcasting across input arguments.
  
      Parameters
      ----------
      cond : scalar, array, Variable, DataArray or Dataset with boolean dtype
          When True, return values from `x`, otherwise returns values from `y`.
      x, y : scalar, array, Variable, DataArray or Dataset
          Values from which to choose. All dimension coordinates on these objects
          must be aligned with each other and with `cond`.
  
      Returns
      -------
      In priority order: Dataset, DataArray, Variable or array, whichever
      type appears as an input argument.
  
      Examples
      --------
  
      >>> cond = xr.DataArray([True, False], dims=['x'])
      >>> x = xr.DataArray([1, 2], dims=['y'])
      >>> xr.where(cond, x, 0)
      <xarray.DataArray (x: 2, y: 2)>
      array([[1, 2],
             [0, 0]])
      Dimensions without coordinates: x, y
  
      See also
      --------
      numpy.where : corresponding numpy function
      Dataset.where, DataArray.where : equivalent methods
      """
      # alignment for three arguments is complicated, so don't support it yet
      return apply_ufunc(duck_array_ops.where,
                         cond, x, y,
                         join='exact',
                         dataset_join='exact',
                         dask='allowed')
  