
===============================
Making arithmetic Ops on double
===============================

Now that we have a ``double`` type, we have yet to use it to perform
computations. We'll start by defining multiplication.

.. _op_contract:

Op's contract
=============

An Op is any object which inherits from :class:`gof.Op`.  It has to
define the following methods.

.. function:: make_node(*inputs)

  This method is responsible for creating output Variables of a
  suitable symbolic Type to serve as the outputs of this Op's
  application.  The Variables found in ``*inputs`` must be operated on
  using Theano's symbolic language to compute the symbolic output
  Variables. This method should put these outputs into an Apply
  instance, and return the Apply instance.

  This method creates an Apply node representing the application of
  the Op on the inputs provided. If the Op cannot be applied to these
  inputs, it must raise an appropriate exception.

  The inputs of the Apply instance returned by this call must be
  ordered correctly: a subsequent ``self.make_node(*apply.inputs)``
  must produce something equivalent to the first ``apply``.

.. function:: perform(node, inputs, output_storage)

  This method computes the function associated to this Op. ``node`` is
  an Apply node created by the Op's ``make_node`` method. ``inputs``
  is a list of references to data to operate on using non-symbolic
  statements, (i.e., statements in Python, Numpy). ``output_storage``
  is a list of storage cells where the variables of the computation
  must be put.

  More specifically:

    - ``node``: This is a reference to an Apply node which was previously
      obtained via the ``Op``'s ``make_node`` method. It is typically not
      used in simple Ops, but it contains symbolic information that
      could be required for complex Ops.

    - ``inputs``: This is a list of data from which the values stored in ``output_storage``
      are to be computed using non-symbolic language.

    - ``output_storage``: This is a list of storage cells where the output is to be stored.
      A storage cell is a one-element list. It is forbidden to change
      the length of the list(s) contained in ``output_storage``.
      There is one storage cell for each output of the Op.

      The data put in ``output_storage`` must match the type of the
      symbolic output. This is a situation where the ``node`` argument
      can come in handy.

      A function Mode may allow ``output_storage`` elements to persist
      between evaluations, or it may reset ``output_storage`` cells to
      hold a value of ``None``.  It can also pre-allocate some memory
      for the Op to use.  This feature can allow ``perform`` to reuse
      memory between calls, for example. If there is something
      preallocated in the ``output_storage``, it will be of the good
      dtype, but can have the wrong shape and have any stride pattern.

  This method must be determined by the inputs. That is to say, if
  it is evaluated once on inputs A and returned B, then if ever
  inputs C, equal to A, are presented again, then outputs equal to
  B must be returned again.

  You must be careful about aliasing outputs to inputs, and making
  modifications to any of the inputs. See :ref:`Views and inplace
  operations <views_and_inplace>` before writing a ``perform``
  implementation that does either of these things.

Instead (or in addition to) ``perform()`` You can also provide a
:ref:`C implementation <cop>` of For more details, refer to the
documentation for :ref:`op`.

.. function:: __eq__(other)

  ``other`` is also an Op.

  Returning ``True`` here is a promise to the optimization system
  that the other Op will produce exactly the same graph effects
  (from perform) as this one, given identical inputs. This means it
  will produce the same output values, it will destroy the same
  inputs (same destroy_map), and will alias outputs to the same
  inputs (same view_map). For more details, see
  :ref:`views_and_inplace`.

   .. note::

     If you set `__props__`, this will be automatically generated.

.. function:: __hash__()

  If two Op instances compare equal, then they **must** return the
  same hash value.

  Equally important, this hash value must not change during the
  lifetime of self.  Op instances should be immutable in this
  sense.

   .. note::

     If you set `__props__`, this will be automatically generated.

.. op_optional:

Optional methods or attributes
==============================

.. attribute:: __props__

  *Default:* Undefined

  Must be a tuple.  Lists the name of the attributes which influence
  the computation performed.  This will also enable the automatic
  generation of appropriate __eq__, __hash__ and __str__ methods.
  Should be set to `()` if you have no attributes that are relevant to
  the computation to generate the methods.

  .. versionadded:: 0.7

.. attribute:: default_output

  *Default:* None

  If this member variable is an integer, then the default
  implementation of ``__call__`` will return
  ``node.outputs[self.default_output]``, where ``node`` was returned
  by ``make_node``.  Otherwise, the entire list of outputs will be
  returned, unless it is of length 1, where the single element will be
  returned by itself.

.. function:: make_thunk(node, storage_map, compute_map, no_recycling)

   This function must return a thunk, that is a zero-arguments
   function that encapsulates the computation to be performed by this
   op on the arguments of the node.

   :param node: Apply instance
     The node for which a thunk is requested.
   :param storage_map: dict of lists
     This maps variables to a one-element lists holding the variable's
     current value. The one-element list acts as pointer to the value
     and allows sharing that "pointer" with other nodes and instances.
   :param compute_map: dict of lists
     This maps variables to one-element lists holding booleans.  If
     the value is 0 then the variable has not been computed and the
     value should not be considered valid.  If the value is 1 the
     variable has been computed and the value is valid.  If the value
     is 2 the variable has been garbage-collected and is no longer
     valid, but shouldn't be required anymore for this call.
   :param no_recycling: WRITEME
     WRITEME

   The returned function must ensure that is sets the computed
   variables as computed in the `compute_map`.

   Defining this function removes the requirement for :meth:`perform`
   or C code, as you will define the thunk for the computation
   yourself.

.. function:: __call__(*inputs, **kwargs)

   By default this is a convenience function which calls
   :meth:`make_node` with the supplied arguments and returns the
   result indexed by `default_output`.  This can be overridden by
   subclasses to do anything else, but must return either a theano
   Variable or a list of Variables.

   If you feel the need to override `__call__` to change the graph
   based on the arguments, you should instead create a function that
   will use your Op and build the graphs that you want and call that
   instead of the Op instance directly.

.. function:: infer_shape(node, shapes)

   This function is needed for shape optimization. ``shapes`` is a
   list with one tuple for each input of the Apply node (which corresponds
   to the inputs of the op).  Each tuple contains as many elements as the
   number of dimensions of the corresponding input. The value of each element
   is the shape (number of items) along the corresponding dimension of that
   specific input.

   While this might sound complicated, it is nothing more than the shape
   of each input as symbolic variables (one per dimension).

   The function should return a list with one tuple for each output.
   Each tuple should contain the corresponding output's computed shape.

   Implementing this method will allow Theano to compute the output's
   shape without computing the output itself, potentially sparing you
   a costly recomputation.

.. function:: flops(inputs, outputs)

   It is only used to have more information printed by the memory
   profiler.  It makes it print the mega flops and giga flops per
   second for each apply node. It takes as inputs two lists: one for the
   inputs and one for the outputs. They contain tuples that are the
   shapes of the corresponding inputs/outputs.

.. function:: __str__()

   This allows you to specify a more informative string representation of your
   Op. If an Op has parameters, it is highly recommended to have the
   ``__str__`` method include the name of the op and the Op's parameters'
   values.

   .. note::

     If you set `__props__`, this will be automatically generated.
     You can still overide it for custom output.

.. function:: do_constant_folding(node)

   *Default:* Return True

   By default when optimizations are enabled, we remove during
   function compilation Apply nodes whose inputs are all constants.
   We replace the Apply node with a Theano constant variable.
   This way, the Apply node is not executed at each function
   call. If you want to force the execution of an op during the
   function call, make do_constant_folding return False.

   As done in the Alloc op, you can return False only in some cases by
   analyzing the graph from the node parameter.

If you want your op to work with gradient.grad() you also need to
implement the functions described below.

Gradient
========

These are the function required to work with gradient.grad().

.. function:: grad(inputs, output_gradients)

  If the Op being defined is differentiable, its gradient may be
  specified symbolically in this method. Both ``inputs`` and
  ``output_gradients`` are lists of symbolic Theano Variables and
  those must be operated on using Theano's symbolic language. The grad
  method must return a list containing one Variable for each
  input. Each returned Variable represents the gradient with respect
  to that input computed based on the symbolic gradients with respect
  to each output.

  If the output is not differentiable with respect to an input then
  this method should be defined to return a variable of type NullType
  for that input. Likewise, if you have not implemented the grad
  computation for some input, you may return a variable of type
  NullType for that input. theano.gradient contains convenience
  methods that can construct the variable for you:
  :func:`theano.gradient.grad_undefined` and
  :func:`theano.gradient.grad_not_implemented`, respectively.

  If an element of output_gradient is of type
  theano.gradient.DisconnectedType, it means that the cost is not a
  function of this output. If any of the op's inputs participate in
  the computation of only disconnected outputs, then Op.grad should
  return DisconnectedType variables for those inputs.

  If the grad method is not defined, then Theano assumes it has been
  forgotten.  Symbolic differentiation will fail on a graph that
  includes this Op.

  It must be understood that the Op's grad method is not meant to
  return the gradient of the Op's output. theano.tensor.grad computes
  gradients; Op.grad is a helper function that computes terms that
  appear in gradients.
  
  If an Op has a single vector-valued output y and a single
  vector-valued input x, then the grad method will be passed x and a
  second vector z. Define J to be the Jacobian of y with respect to
  x. The Op's grad method should return dot(J.T,z). When
  theano.tensor.grad calls the grad method, it will set z to be the
  gradient of the cost C with respect to y. If this op is the only op
  that acts on x, then dot(J.T,z) is the gradient of C with respect to
  x.  If there are other ops that act on x, theano.tensor.grad will
  have to add up the terms of x's gradient contributed by the other
  op's grad method.

  In practice, an op's input and output are rarely implemented as
  single vectors.  Even if an op's output consists of a list
  containing a scalar, a sparse matrix, and a 4D tensor, you can think
  of these objects as being formed by rearranging a vector. Likewise
  for the input. In this view, the values computed by the grad method
  still represent a Jacobian-vector product.

  In practice, it is probably not a good idea to explicitly construct
  the Jacobian, which might be very large and very sparse. However,
  the returned value should be equal to the Jacobian-vector product.

  So long as you implement this product correctly, you need not
  understand what theano.tensor.grad is doing, but for the curious the
  mathematical justification is as follows:

  In essence, the grad method must simply implement through symbolic
  Variables and operations the chain rule of differential
  calculus. The chain rule is the mathematical procedure that allows
  one to calculate the total derivative :math:`\frac{d C}{d x}` of the
  final scalar symbolic Variable C with respect to a primitive
  symbolic Variable x found in the list ``inputs``.  The grad method
  does this using ``output_gradients`` which provides the total
  derivative :math:`\frac{d C}{d f}` of C with respect to a symbolic
  Variable that is returned by the Op (this is provided in
  ``output_gradients``), as well as the knowledge of the total
  derivative :math:`\frac{d f}{d x}` of the latter with respect to the
  primitive Variable (this has to be computed).

  In mathematics, the total derivative of a scalar variable (C) with
  respect to a vector of scalar variables (x), i.e. the gradient, is
  customarily represented as the row vector of the partial
  derivatives, whereas the total derivative of a vector of scalar
  variables (f) with respect to another (x), is customarily
  represented by the matrix of the partial derivatives, i.e.the
  jacobian matrix. In this convenient setting, the chain rule
  instructs that the gradient of the final scalar variable C with
  respect to the primitive scalar variables in x through those in f is
  simply given by the matrix product: :math:`\frac{d C}{d x} = \frac{d
  C}{d f} * \frac{d f}{d x}`.

  Here, the chain rule must be implemented in a similar but slightly
  more complex setting: Theano provides in the list
  ``output_gradients`` one gradient for each of the Variables returned
  by the Op. Where f is one such particular Variable, the
  corresponding gradient found in ``output_gradients`` and
  representing :math:`\frac{d C}{d f}` is provided with a shape
  similar to f and thus not necessarily as a row vector of scalars.
  Furthermore, for each Variable x of the Op's list of input variables
  ``inputs``, the returned gradient representing :math:`\frac{d C}{d
  x}` must have a shape similar to that of Variable x.

  If the output list of the op is :math:`[f_1, ... f_n]`, then the
  list ``output_gradients`` is :math:`[grad_{f_1}(C), grad_{f_2}(C),
  ... , grad_{f_n}(C)]`.  If ``inputs`` consists of the list
  :math:`[x_1, ..., x_m]`, then Op.grad should return the list
  :math:`[grad_{x_1}(C), grad_{x_2}(C), ..., grad_{x_m}(C)]`, where
  :math:`(grad_{y}(Z))_i = \frac{\partial Z}{\partial y_i}` (and
  :math:`i` can stand for multiple dimensions).
 
  In other words, :func:`grad` does not return :math:`\frac{d f_i}{d
  x_j}`, but instead the appropriate dot product specified by the
  chain rule: :math:`\frac{d C}{d x_j} = \frac{d C}{d f_i} \cdot
  \frac{d f_i}{d x_j}`.  Both the partial differentiation and the
  multiplication have to be performed by :func:`grad`.

  Theano currently imposes the following constraints on the values
  returned by the grad method:
  
  1) They must be Variable instances.
  2) When they are types that have dtypes, they must never have an integer dtype.

  The output gradients passed *to* Op.grad will also obey these constraints.

  Integers are a tricky subject. Integers are the main reason for
  having DisconnectedType, NullType or zero gradient. When you have an
  integer as an argument to your grad method, recall the definition of
  a derivative to help you decide what value to return:

  :math:`\frac{d f}{d x} = \lim_{\epsilon \rightarrow 0} (f(x+\epsilon)-f(x))/\epsilon`.

  Suppose your function f has an integer-valued output. For most
  functions you're likely to implement in theano, this means your
  gradient should be zero, because f(x+epsilon) = f(x) for almost all
  x. (The only other option is that the gradient could be undefined,
  if your function is discontinuous everywhere, like the rational
  indicator function)

  Suppose your function f has an integer-valued input. This is a
  little trickier, because you need to think about what you mean
  mathematically when you make a variable integer-valued in
  theano. Most of the time in machine learning we mean "f is a
  function of a real-valued x, but we are only going to pass in
  integer-values of x". In this case, f(x+epsilon) exists, so the
  gradient through f should be the same whether x is an integer or a
  floating point variable. Sometimes what we mean is "f is a function
  of an integer-valued x, and f is only defined where x is an
  integer." Since f(x+epsilon) doesn't exist, the gradient is
  undefined.  Finally, many times in theano, integer valued inputs
  don't actually affect the elements of the output, only its shape.

  If your function f has both an integer-valued input and an
  integer-valued output, then both rules have to be combined:

  - If f is defined at (x+epsilon), then the input gradient is
    defined. Since f(x+epsilon) would be equal to f(x) almost
    everywhere, the gradient should be 0 (first rule).

  - If f is only defined where x is an integer, then the gradient
    is undefined, regardless of what the gradient with respect to the
    output is.

  Examples:

  1) f(x,y) = dot product between x and y. x and y are integers.
        Since the output is also an integer, f is a step function.
        Its gradient is zero almost everywhere, so Op.grad should return
        zeros in the shape of x and y.
  2) f(x,y) = dot product between x and y. x is floating point and y is an integer.
        In this case the output is floating point. It doesn't matter
        that y is an integer.  We consider f to still be defined at
        f(x,y+epsilon). The gradient is exactly the same as if y were
        floating point.
  3) f(x,y) = argmax of x along axis y.
        The gradient with respect to y is undefined, because f(x,y) is
        not defined for floating point y. How could you take an argmax
        along a fraActional axis?  The gradient with respect to x is
        0, because f(x+epsilon, y) = f(x) almost everywhere.
  4) f(x,y) = a vector with y elements, each of which taking on the value x
        The grad method should return DisconnectedType()() for y,
        because the elements of f don't depend on y. Only the shape of
        f depends on y. You probably also want to implement a
        connection_pattern method to encode this.
  5) f(x) = int(x) converts float x into an int. g(y) = float(y) converts an integer y into a float.
        If the final cost C = 0.5 * g(y) = 0.5 g(f(x)), then the
        gradient with respect to y will be 0.5, even if y is an
        integer. However, the gradient with respect to x will be 0,
        because the output of f is integer-valued.

.. function:: connection_pattern(node):

  Sometimes needed for proper operation of gradient.grad().

  Returns a list of list of bools.

  Op.connection_pattern[input_idx][output_idx] is true if the
  elements of inputs[input_idx] have an effect on the elements of
  outputs[output_idx].

  The ``node`` parameter is needed to determine the number of
  inputs. Some ops such as Subtensor take a variable number of
  inputs.

  If no connection_pattern is specified, gradient.grad will
  assume that all inputs have some elements connected to some
  elements of all outputs.

  This method conveys two pieces of information that are otherwise
  not part of the theano graph:

  1) Which of the op's inputs are truly ancestors of each of the
     op's outputs. Suppose an op has two inputs, x and y, and
     outputs f(x) and g(y). y is not really an ancestor of f, but
     it appears to be so in the theano graph.
  2) Whether the actual elements of each input/output are relevant
     to a computation.
     For example, the shape op does not read its input's elements,
     only its shape metadata. d shape(x) / dx should thus raise
     a disconnected input exception (if these exceptions are
     enabled).
     As another example, the elements of the Alloc op's outputs
     are not affected by the shape arguments to the Alloc op.

  Failing to implement this function for an op that needs it can
  result in two types of incorrect behavior:

  1) gradient.grad erroneously raising a TypeError reporting that
     a gradient is undefined.
  2) gradient.grad failing to raise a ValueError reporting that
     an input is disconnected.

  Even if connection_pattern is not implemented correctly, if
  gradient.grad returns an expression, that expression will be
  numerically correct.

.. function:: R_op(inputs, eval_points)

   Optional, to work with gradient.R_op().

   This function implements the application of the R-operator on the
   function represented by your op. Let assume that function is :math:`f`,
   with input :math:`x`, applying the R-operator means computing the 
   Jacobian of :math:`f` and right-multiplying it by :math:`v`, the evaluation 
   point, namely: :math:`\frac{\partial f}{\partial x} v`. 

   ``inputs`` are the symbolic variables corresponding to the value of 
   the input where you want to evaluate the jacobian, and ``eval_points``
   are the symbolic variables corresponding to the value you want to
   right multiply the jacobian with. 

   Same conventions as for the grad method hold. If your op is not
   differentiable, you can return None. Note that in contrast to 
   the method :func:`grad`, for :func:`R_op` you need to return the
   same number of outputs as there are ouputs of the op. You can think
   of it in the following terms. You have all your inputs concatenated
   into a single vector :math:`x`. You do the same with the evaluation 
   points (which are as many as inputs and of the shame shape) and obtain
   another vector :math:`v`. For each output, you reshape it into a vector, 
   compute the jacobian of that vector with respect to :math:`x` and 
   multiply it by :math:`v`. As a last step you reshape each of these
   vectors you obtained for each outputs (that have the same shape as 
   the outputs) back to their corresponding shapes and return them as the 
   output of the :func:`R_op` method.

Defining an Op: ``mul``
=======================

We'll define multiplication as a *binary* operation, even though a
multiplication Op could take an arbitrary number of arguments.

First, we'll instantiate a ``mul`` Op:

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_extending.test_extending_1

.. code-block:: python

   from theano import gof
   mul = gof.Op()


**make_node**

This function must take as many arguments as the operation we are
defining is supposed to take as inputs---in this example that would be
two.  This function ensures that both inputs have the ``double`` type.
Since multiplying two doubles yields a double, this function makes an
Apply node with an output Variable of type ``double``.

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_extending.test_extending_1

.. code-block:: python

   def make_node(x, y):
       if x.type != double or y.type != double:
           raise TypeError('mul only works on doubles')
       return gof.Apply(mul, [x, y], [double()])
   mul.make_node = make_node


The first two lines make sure that both inputs are Variables of the
``double`` type that we created in the previous section. We would not
want to multiply two arbitrary types, it would not make much sense
(and we'd be screwed when we implement this in C!)

The last line is the meat of the definition. There we create an Apply
node representing the application of Op ``mul`` to inputs ``x`` and
``y``, giving a Variable instance of type ``double`` as the output.

.. note::

   Theano relies on the fact that if you call the ``make_node`` method
   of Apply's first argument on the inputs passed as the Apply's
   second argument, the call will not fail and the returned Apply
   instance will be equivalent.  This is how graphs are copied.

**perform**

This code actually computes the function.
In our example, the data in ``inputs`` will be instances of Python's
built-in type ``float`` because this is the type that ``double.filter()``
will always return, per our own definition. ``output_storage`` will
contain a single storage cell for the multiplication's variable.

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_extending.test_extending_1
.. code-block:: python

   def perform(node, inputs, output_storage):
       x, y = inputs[0], inputs[1]
       z = output_storage[0]
       z[0] = x * y
   mul.perform = perform

Here, ``z`` is a list of one element. By default, ``z == [None]``.

.. note::

   It is possible that ``z`` does not contain ``None``. If it contains
   anything else, Theano guarantees that whatever it contains is what
   ``perform`` put there the last time it was called with this
   particular storage. Furthermore, Theano gives you permission to do
   whatever you want with ``z``'s contents, chiefly reusing it or the
   memory allocated for it. More information can be found in the
   :ref:`op` documentation.

.. warning::

   We gave ``z`` the Theano type ``double`` in ``make_node``, which means
   that a Python ``float`` must be put there. You should not put, say, an
   ``int`` in ``z[0]`` because Theano assumes Ops handle typing properly.


Trying out our new Op
=====================

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_extending.test_extending_1

In the following code, we use our new Op:

>>> x, y = double('x'), double('y')
>>> z = mul(x, y)
>>> f = theano.function([x, y], z)
>>> f(5, 6)
30.0
>>> f(5.6, 6.7)
37.519999999999996

Note that there is an implicit call to
``double.filter()`` on each argument, so if we give integers as inputs
they are magically cast to the right type. Now, what if we try this?

>>> x = double('x')
>>> z = mul(x, 2)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/u/breuleuo/hg/theano/theano/gof/op.py", line 207, in __call__
  File "<stdin>", line 2, in make_node
AttributeError: 'int' object has no attribute 'type'

Automatic Constant Wrapping
---------------------------

Well, OK. We'd like our Op to be a bit more flexible. This can be done
by modifying ``make_node`` to accept Python ``int`` or ``float`` as
``x`` and/or ``y``:

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_extending.test_extending_1
.. code-block:: python

   def make_node(x, y):
       if isinstance(x, (int, float)):
           x = gof.Constant(double, x)
       if isinstance(y, (int, float)):
           y = gof.Constant(double, y)
       if x.type != double or y.type != double:
           raise TypeError('mul only works on doubles')
       return gof.Apply(mul, [x, y], [double()])
   mul.make_node = make_node

Whenever we pass a Python int or float instead of a Variable as ``x`` or
``y``, ``make_node`` will convert it to :ref:`constant` for us. ``gof.Constant``
is a :ref:`variable` we statically know the value of.

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_op.test_op_1

>>> x = double('x')
>>> z = mul(x, 2)
>>> f = theano.function([x], z)
>>> f(10)
20.0
>>> f(3.4)
6.7999999999999998

Now the code works the way we want it to.

.. note::
   Most Theano Ops follow this convention of up-casting literal
   make_node arguments to Constants.
   This makes typing expressions more natural.  If you do
   not want a constant somewhere in your graph, you have to pass a Variable
   (like ``double('x')`` here).



Final version
=============

The above example is pedagogical.  When you define other basic arithmetic
operations ``add``, ``sub`` and ``div``, code for ``make_node`` can be
shared between these Ops. Here is revised implementation of these four
arithmetic operators:

.. If you modify this code, also change :
.. theano/tests/test_tutorial.py:T_extending.test_extending_1

.. code-block:: python

   from theano import gof

   class BinaryDoubleOp(gof.Op):

       __props__ = ("name", "fn")

       def __init__(self, name, fn):
           self.name = name
           self.fn = fn

       def make_node(self, x, y):
           if isinstance(x, (int, float)):
               x = gof.Constant(double, x)
           if isinstance(y, (int, float)):
               y = gof.Constant(double, y)
           if x.type != double or y.type != double:
               raise TypeError('%s only works on doubles' % self.name)
           return gof.Apply(self, [x, y], [double()])

       def perform(self, node, inp, out):
           x, y = inp
           z, = out
           z[0] = self.fn(x, y)

       def __str__(self):
           return self.name

   add = BinaryDoubleOp(name='add',
                        fn=lambda x, y: x + y)

   sub = BinaryDoubleOp(name='sub',
                        fn=lambda x, y: x - y)

   mul = BinaryDoubleOp(name='mul',
                        fn=lambda x, y: x * y)

   div = BinaryDoubleOp(name='div',
                        fn=lambda x, y: x / y)

Instead of working directly on an instance of Op, we create a subclass of
Op that we can parametrize. All the operations we define are binary. They
all work on two inputs with type ``double``. They all return a single
Variable of type ``double``. Therefore, ``make_node`` does the same thing
for all these operations, except for the Op reference ``self`` passed
as first argument to Apply.  We define ``perform`` using the function
``fn`` passed in the constructor.

This design is a flexible way to define basic operations without
duplicating code. The same way a Type subclass represents a set of
structurally similar types (see previous section), an Op subclass
represents a set of structurally similar operations: operations that
have the same input/output types, operations that only differ in one
small detail, etc. If you see common patterns in several Ops that you
want to define, it can be a good idea to abstract out what you can.
Remember that an Op is just an object which satisfies the contract
described above on this page and that you should use all the tools at
your disposal to create these objects as efficiently as possible.

**Exercise**: Make a generic DoubleOp, where the number of
arguments can also be given as a parameter.

