text: """Define the ExecComp class, a component that evaluates an expression."""
  import re
  from itertools import product
  from contextlib import contextmanager
  
  import numpy as np
  from numpy import ndarray, imag, complex as npcomplex
  
  from openmdao.core.constants import INT_DTYPE
  from openmdao.core.explicitcomponent import ExplicitComponent
  from openmdao.utils.units import valid_units
  from openmdao.utils import cs_safe
  from openmdao.utils.om_warnings import issue_warning, DerivativesWarning, warn_deprecation
  
  # regex to check for variable names.
  VAR_RGX = re.compile(r'([.]*[_a-zA-Z]\w*[ ]*\(?)')
  
  # Names of metadata entries allowed for ExecComp variables.
  _allowed_meta = {'value', 'val', 'shape', 'units', 'res_units', 'desc',
                   'ref', 'ref0', 'res_ref', 'lower', 'upper', 'src_indices',
                   'flat_src_indices', 'tags', 'shape_by_conn', 'copy_shape'}
  
  # Names that are not allowed for input or output variables (keywords for options)
  _disallowed_names = {'has_diag_partials', 'units', 'shape', 'shape_by_conn', 'run_root_only'}
  
  
  def check_option(option, value):
      """
      Check option for validity.
  
      Parameters
      ----------
      option : str
          The name of the option.
      value : any
          The value of the option.
  
      Raises
      ------
      ValueError
      """
      if option == 'units' and value is not None and not valid_units(value):
          raise ValueError("The units '%s' are invalid." % value)
  
  
  def array_idx_iter(shape):
      """
      Return an iterator over the indices into a n-dimensional array.
  
      Parameters
      ----------
      shape : tuple
          Shape of the array.
  
      Yields
      ------
      int
      """
      for p in product(*[range(s) for s in shape]):
          yield p
  
  
  class ExecComp(ExplicitComponent):
      """
      A component defined by an expression string.
  
      Parameters
      ----------
      exprs : str, tuple of str or list of str
          An assignment statement or iter of them. These express how the
          outputs are calculated based on the inputs.  In addition to
          standard Python operators, a subset of numpy and scipy functions
          is supported.
  
      **kwargs : dict of named args
          Initial values of variables can be set by setting a named
          arg with the var name.  If the value is a dict it is assumed
          to contain metadata.  To set the initial value in addition to
          other metadata, assign the initial value to the 'val' entry
          of the dict.
  
      Attributes
      ----------
      _kwargs : dict of named args
          Initial values of variables.
      _exprs : list
          List of expressions.
      _codes : list
          List of code objects.
      _exprs_info : list
          List of tuples containing output and inputs for each expression.
      _has_diag_partials : bool
          If True, treat all array/array partials as diagonal if both arrays have size > 1.
          All arrays with size > 1 must have the same flattened size or an exception will be raised.
      _units : str or None
          Units to be assigned to all variables in this component.
          Default is None, which means units are provided for variables individually.
      complex_stepsize : double
          Step size used for complex step which is used for derivatives.
      _manual_decl_partials : bool
          If True, at least one partial has been declared by the user.
      _requires_fd : dict
          Contains a mapping of 'of' variables to a tuple of the form (wrts, functs) for those
          'of' variables that require finite difference to be used to compute their derivatives.
      """
  
      def __init__(self, exprs=[], **kwargs):
          r"""
          Create a <Component> using only an expression string.
  
          Given a list of assignment statements, this component creates
          input and output variables at construction time.  All variables
          appearing on the left-hand side of an assignment are outputs,
          and the rest are inputs.  Each variable is assumed to be of
          type float unless the initial value for that variable is supplied
          in \*\*kwargs.  Derivatives are calculated using complex step.
  
          The following functions are available for use in expressions:
  
          =========================  ====================================
          Function                   Description
          =========================  ====================================
          abs(x)                     Absolute value of x
          acos(x)                    Inverse cosine of x
          acosh(x)                   Inverse hyperbolic cosine of x
          arange(start, stop, step)  Array creation
          arccos(x)                  Inverse cosine of x
          arccosh(x)                 Inverse hyperbolic cosine of x
          arcsin(x)                  Inverse sine of x
          arcsinh(x)                 Inverse hyperbolic sine of x
          arctan(x)                  Inverse tangent of x
          arctan2(y, x)              4-quadrant arctangent function of y and x
          asin(x)                    Inverse sine of x
          asinh(x)                   Inverse hyperbolic sine of x
          atan(x)                    Inverse tangent of x
          cos(x)                     Cosine of x
          cosh(x)                    Hyperbolic cosine of x
          dot(x, y)                  Dot product of x and y
          e                          Euler's number
          erf(x)                     Error function
          erfc(x)                    Complementary error function
          exp(x)                     Exponential function
          expm1(x)                   exp(x) - 1
          factorial(x)               Factorial of all numbers in x
                                     (DEPRECATED, not available with SciPy >=1.5)
          fmax(x, y)                 Element-wise maximum of x and y
          fmin(x, y)                 Element-wise minimum of x and y
          inner(x, y)                Inner product of arrays x and y
          isinf(x)                   Element-wise detection of np.inf
          isnan(x)                   Element-wise detection of np.nan
          kron(x, y)                 Kronecker product of arrays x and y
          linspace(x, y, N)          Numpy linear spaced array creation
          log(x)                     Natural logarithm of x
          log10(x)                   Base-10 logarithm of x
          log1p(x)                   log(1+x)
          matmul(x, y)               Matrix multiplication of x and y
          maximum(x, y)              Element-wise maximum of x and y
          minimum(x, y)              Element-wise minimum of x and y
          ones(N)                    Create an array of ones
          outer(x, y)                Outer product of x and y
          pi                         Pi
          power(x, y)                Element-wise x**y
          prod(x)                    The product of all elements in x
          sin(x)                     Sine of x
          sinh(x)                    Hyperbolic sine of x
          sum(x)                     The sum of all elements in x
          tan(x)                     Tangent of x
          tanh(x)                    Hyperbolic tangent of x
          tensordot(x, y)            Tensor dot product of x and y
          zeros(N)                   Create an array of zeros
          =========================  ====================================
  
          Notes
          -----
          If a variable has an initial value that is anything other than 1.0,
          either because it has a different type than float or just because its
          initial value is != 1.0, you must use a keyword arg
          to set the initial value.  For example, let's say we have an
          ExecComp that takes an array 'x' as input and outputs a float variable
          'y' which is the sum of the entries in 'x'.
  
          .. code-block:: python
  
              import numpy
              import openmdao.api as om
              excomp = om.ExecComp('y=sum(x)', x=numpy.ones(10, dtype=float))
  
          In this example, 'y' would be assumed to be the default type of float
          and would be given the default initial value of 1.0, while 'x' would be
          initialized with a size 10 float array of ones.
  
          If you want to assign certain metadata for 'x' in addition to its
          initial value, you can do it as follows:
  
          .. code-block:: python
  
              excomp = ExecComp('y=sum(x)',
                                x={'val': numpy.ones(10, dtype=float),
                                   'units': 'ft'})
          """
          options = {}
          for name in _disallowed_names:
              if name in kwargs:
                  options[name] = kwargs.pop(name)
  
          super().__init__(**options)
  
          # if complex step is used for derivatives, this is the stepsize
          self.complex_stepsize = 1.e-40
  
          if isinstance(exprs, str):
              exprs = [exprs]
  
          self._exprs = exprs[:]
          self._exprs_info = []
          self._codes = []
          self._kwargs = kwargs
  
          self._manual_decl_partials = False
          self._no_check_partials = True
  
      def initialize(self):
          """
          Declare options.
          """
          self.options.declare('has_diag_partials', types=bool, default=False,
                               desc='If True, treat all array/array partials as diagonal if both '
                                    'arrays have size > 1. All arrays with size > 1 must have the '
                                    'same flattened size or an exception will be raised.')
  
          self.options.declare('units', types=str, allow_none=True, default=None,
                               desc='Units to be assigned to all variables in this component. '
                                    'Default is None, which means units may be provided for variables'
                                    ' individually.',
                               check_valid=check_option)
  
          self.options.declare('shape', types=(int, tuple, list), allow_none=True, default=None,
                               desc='Shape to be assigned to all variables in this component. '
                                    'Default is None, which means shape may be provided for variables'
                                    ' individually.')
  
          self.options.declare('shape_by_conn', types=bool, default=False,
                               desc='If True, shape all inputs and outputs based on their '
                                    'connection. Default is False.')
  
      @classmethod
      def register(cls, name, callable_obj, complex_safe):
          """
          Register a callable to be usable within ExecComp expressions.
  
          Parameters
          ----------
          name : str
              Name of the callable.
          callable_obj : callable
              The callable.
          complex_safe : bool
              If True, the given callable works correctly with complex numbers.
          """
          global _expr_dict, _not_complex_safe
  
          if not callable(callable_obj):
              raise TypeError(f"{cls.__name__}: '{name}' passed to register() of type "
                              f"'{type(callable_obj).__name__}' is not callable.")
          if name in _expr_dict:
              raise NameError(f"{cls.__name__}: '{name}' has already been registered.")
  
          if name in _disallowed_names:
              raise NameError(f"{cls.__name__}: cannot register name '{name}' because "
                              "it's a reserved keyword.")
  
          if '.' in name:
              raise NameError(f"{cls.__name__}: cannot register name '{name}' because "
                              "it contains '.'.")
          _expr_dict[name] = callable_obj
  
          if not complex_safe:
              _not_complex_safe.add(name)
  
      def setup(self):
          """
          Set up variable name and metadata lists.
          """
          if self._exprs:
              self._setup_expressions()
  
      def _setup_expressions(self):
          """
          Set up the expressions.
  
          This is called during setup_procs and after each call to "add_expr" from configure.
          """
          global _not_complex_safe
  
          exprs = self._exprs
          kwargs = self._kwargs
  
          units = self.options['units']
          shape = self.options['shape']
          shape_by_conn = self.options['shape_by_conn']
  
          warned = False
  
          if shape is not None and shape_by_conn:
              raise RuntimeError(f"{self.msginfo}: Can't set both shape and shape_by_conn.")
  
          outs = set()
          allvars = set()
  
          self._exprs_info = exprs_info = [(self._parse_for_out_vars(expr.split('=', 1)[0]),
                                            self._parse_for_names(expr)) for expr in exprs]
  
          self._requires_fd = {}
  
          # find all of the variables and which ones are outputs
          for onames, names in exprs_info:
              outs.update(onames)
              allvars.update(names[0])
              if _not_complex_safe.intersection(names[1]):
                  for o in onames:
                      self._requires_fd[o] = names
  
          if self._requires_fd:
              inps = []
              for out, (rhsvars, funcs) in self._requires_fd.items():
                  iset = rhsvars.difference(outs)
                  self._requires_fd[out] = (iset, funcs)
                  inps.extend(iset)
              self._no_check_partials = False
              self.set_check_partial_options(wrt=inps, method='fd')
  
          kwargs2 = {}
          init_vals = {}
  
          # make sure all kwargs are legit
          for arg, val in kwargs.items():
              if arg not in allvars:
                  raise RuntimeError("%s: arg '%s' in call to ExecComp() "
                                     "does not refer to any variable in the "
                                     "expressions %s" % (self.msginfo,
                                                         arg, exprs))
              if isinstance(val, dict):
                  diff = set(val.keys()) - _allowed_meta
                  if diff:
                      raise RuntimeError("%s: the following metadata names were not "
                                         "recognized for variable '%s': %s" %
                                         (self.msginfo, arg, sorted(diff)))
  
                  if 'val' in val and 'value' in val:
                      raise RuntimeError(f"{self.msginfo}: 'val' and 'value' at the same time, use "
                                         "'val'.")
                  elif 'value' in val and not warned:
                      warn_deprecation(f"{self.msginfo}: 'value' will be deprecated in 4.0. Please "
                                       "use 'val' in the future.")
  
                  if 'value' in val:
                      val['val'] = val.pop('value')
                      warned = True
  
                  kwargs2[arg] = val.copy()
  
                  if units is not None:
                      if 'units' in val and val['units'] != units:
                          raise RuntimeError("%s: units of '%s' have been specified for "
                                             "variable '%s', but units of '%s' have been "
                                             "specified for the entire component." %
                                             (self.msginfo, val['units'], arg, units))
                      else:
                          kwargs2[arg]['units'] = units
  
                  if shape is not None:
                      if 'shape' in val and val['shape'] != shape:
                          raise RuntimeError("%s: shape of %s has been specified for "
                                             "variable '%s', but shape of %s has been "
                                             "specified for the entire component." %
                                             (self.msginfo, val['shape'], arg, shape))
                      elif 'val' in val and np.atleast_1d(val['val']).shape != shape:
                          raise RuntimeError("%s: value of shape %s has been specified for "
                                             "variable '%s', but shape of %s has been "
                                             "specified for the entire component." %
                                             (self.msginfo, np.atleast_1d(val['val']).shape,
                                              arg, shape))
                      else:
                          init_vals[arg] = np.ones(shape)
  
                  if 'val' in val:
                      init_vals[arg] = val['val']
                      del kwargs2[arg]['val']
  
                  if shape_by_conn or 'shape_by_conn' in val or 'copy_shape' in val:
                      if val.get('shape') is not None or val.get('val') is not None:
                          raise RuntimeError(f"{self.msginfo}: Can't set 'shape' or 'val' for "
                                             f"variable '{arg}' along with 'copy_shape' or "
                                             "'shape_by_conn'.")
  
                  if 'shape' in val:
                      if arg not in init_vals:
                          init_vals[arg] = np.ones(val['shape'])
                      elif np.atleast_1d(init_vals[arg]).shape != val['shape']:
                          raise RuntimeError("%s: shape of %s has been specified for variable "
                                             "'%s', but a value of shape %s has been provided." %
                                             (self.msginfo, str(val['shape']), arg,
                                              str(np.atleast_1d(init_vals[arg]).shape)))
                      del kwargs2[arg]['shape']
              else:
                  init_vals[arg] = val
  
          if self._static_mode:
              var_rel2meta = self._static_var_rel2meta
          else:
              var_rel2meta = self._var_rel2meta
  
          for var in sorted(allvars):
              meta = kwargs2.get(var, {
                  'units': units,
                  'shape': shape,
                  'shape_by_conn': shape_by_conn})
  
              # if user supplied an initial value, use it, otherwise set to 1.0
              if var in init_vals:
                  val = init_vals[var]
              else:
                  val = 1.0
  
              if var in var_rel2meta:
                  # Input/Output already exists, but we may be setting defaults for the first time.
                  # Note that there is only one submitted dictionary of defaults.
                  current_meta = var_rel2meta[var]
  
                  for kname, kvalue in meta.items():
                      if kvalue is not None:
                          current_meta[kname] = kvalue
  
                  new_val = kwargs[var].get('val')
                  if new_val is not None:
                      # val is normally ensured to be a numpy array in add_input/add_output,
                      # do the same here...
                      current_meta['val'] = np.atleast_1d(new_val)
              else:
                  # new input and/or output.
                  if var in outs:
                      current_meta = self.add_output(var, val, **meta)
                  else:
                      current_meta = self.add_input(var, val, **meta)
  
              if var not in init_vals:
                  init_vals[var] = current_meta['val']
  
          self._codes = self._compile_exprs(self._exprs)
  
      def add_expr(self, expr, **kwargs):
          """
          Add an expression to the ExecComp.
  
          Parameters
          ----------
          expr : str
              An assignment statement that expresses how the outputs are calculated based on the
              inputs. In addition to standard Python operators, a subset of numpy and scipy
              functions is supported.
          **kwargs : dict of named args
              Initial values of variables can be set by setting a named arg with the var name.  If
              the value is a dict it is assumed to contain metadata.  To set the initial value in
              addition to other metadata, assign the initial value to the 'val' entry of the dict.
              Do not include for inputs whose default kwargs have been declared on previous
              expressions.
          """
          if not isinstance(expr, str):
              typ = type(expr).__name__
              msg = f"Argument 'expr' must be of type 'str', but type '{typ}' was found."
              raise TypeError(msg)
  
          self._exprs.append(expr)
          for name in kwargs:
              if name in self._kwargs:
                  raise NameError(f"Defaults for '{name}' have already been defined in a previous "
                                  "expression.")
  
          self._kwargs.update(kwargs)
  
          if not self._static_mode:
              self._setup_expressions()
  
      def _compile_exprs(self, exprs):
          compiled = []
          outputs = []
          for i, expr in enumerate(exprs):
  
              # Quick dupe check.
              lhs_name = expr.split('=', 1)[0].strip()
              if lhs_name in outputs:
                  # Can't add two equations with the same output.
                  raise RuntimeError(f"{self.msginfo}: The output '{lhs_name}' has already been "
                                     "defined by an expression.")
              else:
                  outputs.append(lhs_name)
  
              try:
                  compiled.append(compile(expr, expr, 'exec'))
              except Exception:
                  raise RuntimeError("%s: failed to compile expression '%s'." %
                                     (self.msginfo, exprs[i]))
          return compiled
  
      def _parse_for_out_vars(self, s):
          vnames = set([x.strip() for x in re.findall(VAR_RGX, s)
                        if not x.endswith('(') and not x.startswith('.')])
          for v in vnames:
              if v in _expr_dict:
                  raise NameError("%s: cannot assign to variable '%s' "
                                  "because it's already defined as an internal "
                                  "function or constant." % (self.msginfo, v))
          return vnames
  
      def _parse_for_names(self, s):
          names = [x.strip() for x in re.findall(VAR_RGX, s) if not x.startswith('.')]
          vnames = set([n for n in names if not n.endswith('(')])
          fnames = [n[:-1] for n in names if n[-1] == '(']
          to_remove = []
          for v in vnames:
              if v in _disallowed_names:
                  raise NameError("%s: cannot use variable name '%s' because "
                                  "it's a reserved keyword." % (self.msginfo, v))
              if v in _expr_dict:
                  expvar = _expr_dict[v]
                  if callable(expvar):
                      raise NameError("%s: cannot use '%s' as a variable because "
                                      "it's already defined as an internal "
                                      "function or constant." % (self.msginfo, v))
                  else:
                      to_remove.append(v)
  
          for f in fnames:
              if f not in _expr_dict:
                  raise NameError(f"{self.msginfo}: can't use '{f}' as a function because "
                                  "it hasn't been registered.")
  
          return vnames.difference(to_remove), fnames
  
      def __getstate__(self):
          """
          Return state as a dict.
  
          Returns
          -------
          dict
              State to get.
          """
          state = self.__dict__.copy()
          del state['_codes']
          return state
  
      def __setstate__(self, state):
          """
          Restore state from `state`.
  
          Parameters
          ----------
          state : dict
              State to restore.
          """
          self.__dict__.update(state)
          self._codes = self._compile_exprs(self._exprs)
  
      def declare_partials(self, *args, **kwargs):
          """
          Declare information about this component's subjacobians.
  
          Parameters
          ----------
          *args : list
              Positional args to be passed to base class version of declare_partials.
          **kwargs : dict
              Keyword args  to be passed to base class version of declare_partials.
  
          Returns
          -------
          dict
              Metadata dict for the specified partial(s).
          """
          if 'method' not in kwargs or kwargs['method'] == 'exact':
              raise RuntimeError(f"{self.msginfo}: declare_partials must be called with method='cs' "
                                 "or method='fd'.")
          if self.options['has_diag_partials']:
              raise RuntimeError(f"{self.msginfo}: declare_partials cannot be called manually if "
                                 "has_diag_partials has been set.")
  
          self._manual_decl_partials = True
          return super().declare_partials(*args, **kwargs)
  
      def _setup_partials(self):
          """
          Check that all partials are declared.
          """
          if not self._manual_decl_partials:
              meta = self._var_rel2meta
              decl_partials = super().declare_partials
              for i, (outs, tup) in enumerate(self._exprs_info):
                  vs, funcs = tup
                  ins = sorted(set(vs).difference(outs))
                  for out in sorted(outs):
                      for inp in ins:
                          if self.options['has_diag_partials']:
                              ival = meta[inp]['val']
                              iarray = isinstance(ival, ndarray) and ival.size > 1
                              oval = meta[out]['val']
                              if iarray and isinstance(oval, ndarray) and oval.size > 1:
                                  if oval.size != ival.size:
                                      raise RuntimeError(
                                          "%s: has_diag_partials is True but partial(%s, %s) "
                                          "is not square (shape=(%d, %d))." %
                                          (self.msginfo, out, inp, oval.size, ival.size))
                                  # partial will be declared as diagonal
                                  inds = np.arange(oval.size, dtype=INT_DTYPE)
                              else:
                                  inds = None
                              decl_partials(of=out, wrt=inp, rows=inds, cols=inds)
                          else:
                              decl_partials(of=out, wrt=inp)
  
          super()._setup_partials()
          if self._manual_decl_partials:
              undeclared = []
              for i, (outs, tup) in enumerate(self._exprs_info):
                  vs, funcs = tup
                  ins = sorted(set(vs).difference(outs))
                  for out in sorted(outs):
                      out = '.'.join((self.pathname, out)) if self.pathname else out
                      for inp in ins:
                          inp = '.'.join((self.pathname, inp)) if self.pathname else inp
                          if (out, inp) not in self._subjacs_info:
                              undeclared.append((out, inp))
              if undeclared:
                  idx = len(self.pathname) + 1 if self.pathname else 0
                  undeclared = ', '.join([' wrt '.join((f"'{of[idx:]}'", f"'{wrt[idx:]}'"))
                                          for of, wrt in undeclared])
                  issue_warning(f"The following partial derivatives have not been "
                                f"declared so they are assumed to be zero: [{undeclared}].",
                                prefix=self.msginfo, category=DerivativesWarning)
  
      def compute(self, inputs, outputs):
          """
          Execute this component's assignment statements.
  
          Parameters
          ----------
          inputs : `Vector`
              `Vector` containing inputs.
  
          outputs : `Vector`
              `Vector` containing outputs.
          """
          for i, expr in enumerate(self._codes):
              try:
                  exec(expr, _expr_dict, _IODict(outputs, inputs))  # nosec: limited to _expr_dict
              except Exception as err:
                  raise RuntimeError(f"{self.msginfo}: Error occurred evaluating '{self._exprs[i]}':"
                                     f"\n{err}")
  
      def _linearize(self, jac=None, sub_do_ln=False):
          """
          Compute jacobian / factorization. The model is assumed to be in a scaled state.
  
          Parameters
          ----------
          jac : Jacobian or None
              Ignored.
          sub_do_ln : bool
              Flag indicating if the children should call linearize on their linear solvers.
          """
          if self._requires_fd:
              if 'fd' in self._approx_schemes:
                  fdins = {wrt.rsplit('.', 1)[1] for wrt in self._approx_schemes['fd']._wrt_meta}
              else:
                  fdins = set()
  
              for _, (inps, funcs) in self._requires_fd.items():
                  diff = inps.difference(fdins)
                  if diff:
                      raise RuntimeError(f"{self.msginfo}: expression contains functions "
                                         f"{sorted(funcs)} that are not complex safe. To fix this, "
                                         f"call declare_partials('*', {sorted(diff)}, method='fd') "
                                         f"on this component prior to setup.")
              self._requires_fd = False  # only need to do this check the first time around
  
          super()._linearize(jac, sub_do_ln)
  
      def compute_partials(self, inputs, partials):
          """
          Use complex step method to update the given Jacobian.
  
          Parameters
          ----------
          inputs : `VecWrapper`
              `VecWrapper` containing parameters (p).
          partials : `Jacobian`
              Contains sub-jacobians.
          """
          if self._manual_decl_partials:
              return
  
          step = self.complex_stepsize * 1j
          out_names = self._var_rel_names['output']
          inv_stepsize = 1.0 / self.complex_stepsize
          has_diag_partials = self.options['has_diag_partials']
  
          for inp in inputs:
  
              pwrap = _TmpDict(inputs)
              pval = inputs[inp]
              psize = pval.size
              pwrap[inp] = np.asarray(pval, npcomplex)
  
              if has_diag_partials or psize == 1:
                  # set a complex inpup value
                  pwrap[inp] += step
  
                  uwrap = _TmpDict(self._outputs, return_complex=True)
  
                  # solve with complex input value
                  self._residuals.set_val(0.0)
                  self.compute(pwrap, uwrap)
  
                  for u in out_names:
                      if (u, inp) in self._declared_partials:
                          partials[(u, inp)] = imag(uwrap[u] * inv_stepsize).flat
  
                  # restore old input value
                  pwrap[inp] -= step
              else:
                  for i, idx in enumerate(array_idx_iter(pwrap[inp].shape)):
                      # set a complex input value
                      pwrap[inp][idx] += step
  
                      uwrap = _TmpDict(self._outputs, return_complex=True)
  
                      # solve with complex input value
                      self._residuals.set_val(0.0)
                      self.compute(pwrap, uwrap)
  
                      for u in out_names:
                          if (u, inp) in self._declared_partials:
                              # set the column in the Jacobian entry
                              partials[(u, inp)][:, i] = imag(uwrap[u] * inv_stepsize).flat
  
                      # restore old input value
                      pwrap[inp][idx] -= step
  
  
  class _TmpDict(object):
      """
      Dict wrapper that allows modification without changing the wrapped dict.
  
      It will allow getting of values
      from its inner dict unless those values get modified via
      __setitem__.  After values have been modified they are managed
      thereafter by the wrapper.  This protects the inner dict from
      modification.
  
      Attributes
      ----------
      _inner : dict-like
          The dictionary to be wrapped.
      _changed : dict-like
          The key names for the values that were changed.
      _complex : bool
          If True, return a complex version of values from __getitem__.
      """
  
      def __init__(self, inner, return_complex=False):
          """
          Construct the dictionary object.
  
          Parameters
          ----------
          inner : dict-like
              The dictionary to be wrapped.
          return_complex : bool, optional
              If True, return a complex version of values from __getitem__
          """
          self._inner = inner
          self._changed = {}
          self._complex = return_complex
  
      def __getitem__(self, name):
          if name in self._changed:
              return self._changed[name]
          elif self._complex:
              val = self._inner[name]
              if isinstance(val, ndarray):
                  self._changed[name] = np.asarray(val, dtype=npcomplex)
              else:
                  self._changed[name] = npcomplex(val)
              return self._changed[name]
          else:
              return self._inner[name]
  
      def __setitem__(self, name, value):
          self._changed[name] = value
  
      def __contains__(self, name):
          return name in self._inner or name in self._changed
  
      def __getattr__(self, name):
          return getattr(self._inner, name)
  
  
  class _IODict(object):
      """
      A dict wrapper that contains 2 different dicts.
  
      Items are first looked for in the outputs
      and then the inputs.
  
      Attributes
      ----------
      _inputs : dict-like
          The inputs object to be wrapped.
      _outputs : dict-like
          The outputs object to be wrapped.
      """
  
      def __init__(self, outputs, inputs):
          """
          Create the dict wrapper.
  
          Parameters
          ----------
          outputs : dict-like
              The outputs object to be wrapped.
  
          inputs : dict-like
              The inputs object to be wrapped.
          """
          self._outputs = outputs
          self._inputs = inputs
  
      def __getitem__(self, name):
          try:
              return self._inputs[name]
          except KeyError:
              return self._outputs[name]
  
      def __setitem__(self, name, value):
          self._outputs[name] = value
  
      def __contains__(self, name):
          return name in self._outputs or name in self._inputs
  
  
  def _import_functs(mod, dct, names=None):
      """
      Map attributes attrs from the given module into the given dict.
  
      Parameters
      ----------
      mod : object
          Module to check.
      dct : dict
          Dictionary that will contain the mapping
      names : iter of str, optional
          If supplied, only map attrs that match the given names
      """
      if names is None:
          names = dir(mod)
      for name in names:
          if isinstance(name, tuple):
              name, alias = name
          else:
              alias = name
          if not name[0] == '_':
              dct[name] = getattr(mod, name)
              dct[alias] = dct[name]
  
  
  _expr_dict = {}  # this dict will act as the local scope when we eval our expressions
  _not_complex_safe = set()  # this is the set of registered functions that are not complex safe
  
  
  _import_functs(np, _expr_dict,
                 names=['arange', 'ones', 'zeros', 'linspace',  # Array creation
                        'e', 'pi',  # Constants
                        'isinf', 'isnan',  # Logic
                        'log', 'log10', 'log1p', 'power',  # Math operations
                        'exp', 'expm1', 'fmax', 'min', 'max', 'diff',
                        'fmin', 'maximum', 'minimum',
                        'sum', 'dot', 'prod',  # Reductions
                        'tensordot', 'matmul',  # Linear algebra
                        'outer', 'inner', 'kron',
                        'sin', 'cos', 'tan', ('arcsin', 'asin'),  # Trig
                        ('arccos', 'acos'), ('arctan', 'atan'),
                        'sinh', 'cosh', 'tanh', ('arcsinh', 'asinh'),  # Hyperbolic trig
                        ('arccosh', 'acosh')])
  
  
  # if scipy is available, add some functions
  try:
      import scipy.special
  except ImportError:
      pass
  else:
      _import_functs(scipy.special, _expr_dict, names=['erf', 'erfc'])
  
      from distutils.version import LooseVersion
      if LooseVersion(scipy.__version__) >= LooseVersion("1.5.0"):
          def factorial(*args):
              """
              Raise a RuntimeError stating that the factorial function is not supported.
              """
              raise RuntimeError("The 'factorial' function is not supported for SciPy "
                                 f"versions >= 1.5, current version: {scipy.__version__}")
      else:
          def factorial(*args):
              """
              Raise a warning stating that the factorial function is deprecated.
              """
              warn_deprecation("The 'factorial' function is deprecated. "
                               "It is no longer supported for SciPy versions >= 1.5.")
              return scipy.special.factorial(*args)
  
      _expr_dict['factorial'] = factorial
  
  
  # put any functions that need custom complex-safe versions here
  
  _expr_dict['abs'] = cs_safe.abs
  _expr_dict['arctan2'] = cs_safe.arctan2
  
  
  class _NumpyMsg(object):
      """
      A class that will raise an error if an attempt is made to access any attribute/function.
      """
  
      def __init__(self, namespace):
          """
          Construct the _NumpyMsg object.
  
          Parameters
          ----------
          namespace : str
              The numpy namespace (e.g. 'numpy' or 'np).
          """
          self.namespace = namespace
  
      def __getattr__(self, name):
          """
          Attempt to access an attribute/function.
  
          Parameters
          ----------
          name : str
              The name of the attribute/function.
  
          Raises
          ------
          RuntimeError
              When an attempt is made to access any attribute/function.
          """
          raise RuntimeError('\n'.join([
              "    ExecComp supports a subset of numpy functions directly, without the '%s' prefix.",
              "    '%s' is %ssupported (See the documentation)."
          ]) % (self.namespace, name, '' if name in _expr_dict else 'not '))
  
  
  _expr_dict['np'] = _NumpyMsg('np')
  _expr_dict['numpy'] = _NumpyMsg('numpy')
  
  
  @contextmanager
  def _temporary_expr_dict():
      """
      During a test, it's useful to be able to save and restore the _expr_dict.
      """
      global _expr_dict, _not_complex_safe
  
      save = (_expr_dict.copy(), _not_complex_safe.copy())
  
      try:
          yield
      finally:
          _expr_dict, _not_complex_safe = save
  