text: import numpy
  import six
  
  from chainer.backends import cuda
  from chainer import function
  from chainer import function_node
  from chainer.utils import type_check
  
  
  def _extract_gates(x):
      r = x.reshape((len(x), x.shape[1] // 4, 4) + x.shape[2:])
      return [r[:, :, i] for i in six.moves.range(4)]
  
  
  def _sigmoid(x, xp=numpy):
      half = x.dtype.type(0.5)
      return xp.tanh(x * half) * half + half
  
  
  def _grad_sigmoid(x):
      return x * (1 - x)
  
  
  def _grad_grad_sigmoid(x):
      return x * (1 - x) * (1 - 2 * x)
  
  
  def _grad_tanh(x):
      return 1 - x * x
  
  
  def _grad_grad_tanh(x, gx):
      return -2 * x * gx
  
  
  _preamble = '''
  template <typename T> __device__ T sigmoid(T x) {
      const T half = 0.5;
      return tanh(x * half) * half + half;
  }
  template <typename T> __device__ T grad_sigmoid(T y) { return y * (1 - y); }
  template <typename T> __device__ T grad_tanh(T y) { return 1 - y * y; }
  
  #define COMMON_ROUTINE \
      T aa = tanh(a); \
      T ai = sigmoid(i_); \
      T af = sigmoid(f); \
      T ao = sigmoid(o);
  '''
  
  
  class LSTM(function_node.FunctionNode):
  
      """Long short-term memory unit with forget gate.
  
      It has two inputs (c, x) and two outputs (c, h), where c indicates the cell
      state. x must have four times channels compared to the number of units.
  
      """
  
      def check_type_forward(self, in_types):
          type_check.expect(in_types.size() == 2)
          c_type, x_type = in_types
  
          type_check.expect(
              c_type.dtype.kind == 'f',
              x_type.dtype == c_type.dtype,
  
              c_type.ndim >= 2,
              x_type.ndim >= 2,
              c_type.ndim == x_type.ndim,
  
              x_type.shape[0] <= c_type.shape[0],
              x_type.shape[1] == 4 * c_type.shape[1],
          )
          for i in six.moves.range(2, type_check.eval(c_type.ndim)):
              type_check.expect(x_type.shape[i] == c_type.shape[i])
  
      def forward(self, inputs):
          self.retain_inputs((0, 1))
          c_prev, x = inputs
          a, i, f, o = _extract_gates(x)
          batch = len(x)
  
          if isinstance(x, numpy.ndarray):
              a = numpy.tanh(a)
              i = _sigmoid(i)
              f = _sigmoid(f)
              o = _sigmoid(o)
  
              c_next = numpy.empty_like(c_prev)
              c_next[:batch] = a * i + f * c_prev[:batch]
              h = o * numpy.tanh(c_next[:batch])
          else:
              c_next = cuda.cupy.empty_like(c_prev)
              h = cuda.cupy.empty_like(c_next[:batch])
              cuda.elementwise(
                  'T c_prev, T a, T i_, T f, T o', 'T c, T h',
                  '''
                      COMMON_ROUTINE;
                      c = aa * ai + af * c_prev;
                      h = ao * tanh(c);
                  ''',
                  'lstm_fwd', preamble=_preamble)(
                      c_prev[:batch], a, i, f, o, c_next[:batch], h)
  
          c_next[batch:] = c_prev[batch:]
          self.retain_outputs((0,))
          return c_next, h
  
      def backward(self, indexes, grads):
          grad_inputs = (
              self.get_retained_inputs() + self.get_retained_outputs() + grads)
          return LSTMGrad()(*grad_inputs)
  
  
  class LSTMGrad(function.Function):
  
      def forward(self, inputs):
          xp = cuda.get_array_module(*inputs)
          c_prev, x, c_next, gc, gh = inputs
          batch = len(x)
  
          gx = xp.empty_like(x)
          ga, gi, gf, go = _extract_gates(gx)
  
          # Consider the case that either gradient is not given
          if gc is None:
              gc_update = 0
              gc_rest = 0
          else:
              gc_update = gc[:batch]
              gc_rest = gc[batch:]
          if gh is None:
              gh = 0
  
          a, i, f, o = _extract_gates(x)
          if xp is numpy:
              tanh_a = xp.tanh(a)
              sig_i = _sigmoid(i)
              sig_f = _sigmoid(f)
              sig_o = _sigmoid(o)
  
              co = numpy.tanh(c_next[:batch])
              gc_prev = numpy.empty_like(c_prev)
              # multiply f later
              gc_prev[:batch] = gh * sig_o * _grad_tanh(co) + gc_update
              gc = gc_prev[:batch]
              ga[:] = gc * sig_i * _grad_tanh(tanh_a)
              gi[:] = gc * tanh_a * _grad_sigmoid(sig_i)
              gf[:] = gc * c_prev[:batch] * _grad_sigmoid(sig_f)
              go[:] = gh * co * _grad_sigmoid(sig_o)
              gc_prev[:batch] *= sig_f  # multiply f here
              gc_prev[batch:] = gc_rest
          else:
              gc_prev = xp.empty_like(c_prev)
              cuda.elementwise(
                  'T c_prev, T c, T gc, T gh, T a, T i_, T f, T o',
                  'T gc_prev, T ga, T gi, T gf, T go',
                  '''
                      COMMON_ROUTINE;
                      T co = tanh(c);
                      T temp = gh * ao * grad_tanh(co) + gc;
                      ga = temp * ai * grad_tanh(aa);
                      gi = temp * aa * grad_sigmoid(ai);
                      gf = temp * c_prev * grad_sigmoid(af);
                      go = gh * co * grad_sigmoid(ao);
                      gc_prev = temp * af;
                  ''',
                  'lstm_bwd', preamble=_preamble)(
                      c_prev[:batch], c_next[:batch], gc_update, gh, a, i, f, o,
                      gc_prev[:batch], ga, gi, gf, go)
              gc_prev[batch:] = gc_rest
  
          return gc_prev, gx
  
      def backward(self, inputs, grads):
          xp = cuda.get_array_module(*inputs)
  
          c_prev, x, c, gc, gh = inputs
          ggc_prev, ggx = grads
  
          gc_prev = xp.empty_like(c_prev)
          gx = xp.empty_like(x)
          gc_next = xp.empty_like(c)
          ggc = xp.empty_like(ggc_prev)
          ggh = xp.empty_like(gh)
  
          batch = len(x)
          gc_prev[batch:] = 0
          gc_next[batch:] = 0
          ggc[batch:] = ggc_prev[batch:]
          ggh[batch:] = 0
  
          c_prev = c_prev[:batch]
          c = c[:batch]
          gc = gc[:batch]
          ggc_prev = ggc_prev[:batch]
          ggx = ggx[:batch]
  
          a, i, f, o = _extract_gates(x)
          gga, ggi, ggf, ggo = _extract_gates(ggx)
          ga, gi, gf, go = _extract_gates(gx)
  
          gc_prev[:batch], ga[:], gi[:], gf[:], go[:], gc_next[:batch], \
              ggc[:batch], ggh[:batch] \
              = lstm_grad_grad(
                  c_prev, a, i, f, o, c, gc, gh, ggc_prev, gga, ggi, ggf, ggo)
          return gc_prev, gx, gc_next, ggc, ggh
  
  
  def _cupy_sigmoid(x):
      half = x.dtype.type(0.5)
      return cuda.fusion.tanh(x * half) * half + half
  
  
  @cuda.fuse()
  def lstm_grad_grad(
          c_prev, a, i, f, o, c, gc, gh, ggc_prev, gga, ggi, ggf, ggo):
      sig_o = _cupy_sigmoid(o)
      gsig_o = _grad_sigmoid(sig_o)
      ggsig_o = _grad_grad_sigmoid(sig_o)
      sig_i = _cupy_sigmoid(i)
      gsig_i = _grad_sigmoid(sig_i)
      ggsig_i = _grad_grad_sigmoid(sig_i)
      sig_f = _cupy_sigmoid(f)
      gsig_f = _grad_sigmoid(sig_f)
      ggsig_f = _grad_grad_sigmoid(sig_f)
      tanh_a = cuda.fusion.tanh(a)
      gtanh_a = _grad_tanh(tanh_a)
      ggtanh_a = _grad_grad_tanh(tanh_a, gtanh_a)
      tanh_c = cuda.fusion.tanh(c)
      gtanh_c = _grad_tanh(tanh_c)
      ggtanh_c = _grad_grad_tanh(tanh_c, gtanh_c)
  
      gc_bar = gh * sig_o * gtanh_c + gc
  
      gc_prev = ggf * gc_bar * gsig_f
      ga = (gga * sig_i * ggtanh_a +
            ggi * gtanh_a * gsig_i) * gc_bar
      gi = (gga * gtanh_a * gsig_i +
            ggi * tanh_a * ggsig_i) * gc_bar
      gf = (ggc_prev * (gh * sig_o * gtanh_c + gc) * gsig_f +
            ggf * gc_bar * c_prev * ggsig_f)
  
      ggc = (
          ggc_prev * sig_f +
          gga * sig_i * gtanh_a +
          ggi * tanh_a * gsig_i +
          ggf * c_prev * gsig_f)
  
      dgc_do = gh * gsig_o * gtanh_c
      go = ggc * dgc_do + ggo * gh * tanh_c * ggsig_o
      dgc_dc = gh * sig_o * ggtanh_c
      gc_next = ggc * dgc_dc + ggo * gh * gtanh_c * gsig_o
      ggh = ggc * sig_o * gtanh_c + ggo * tanh_c * gsig_o
      return gc_prev, ga, gi, gf, go, gc_next, ggc, ggh
  
  
  def lstm(c_prev, x):
      """Long Short-Term Memory units as an activation function.
  
      This function implements LSTM units with forget gates. Let the previous
      cell state ``c_prev`` and the input array ``x``.
  
      First, the input array ``x`` is split into four arrays
      :math:`a, i, f, o` of the same shapes along the second axis. It means that
      ``x`` 's second axis must have 4 times the ``c_prev`` 's second axis.
  
      The split input arrays are corresponding to:
  
          - :math:`a` : sources of cell input
          - :math:`i` : sources of input gate
          - :math:`f` : sources of forget gate
          - :math:`o` : sources of output gate
  
      Second, it computes the updated cell state ``c`` and the outgoing signal
      ``h`` as:
  
      .. math::
  
          c &= \\tanh(a) \\sigma(i)
             + c_{\\text{prev}} \\sigma(f), \\\\
          h &= \\tanh(c) \\sigma(o),
  
      where :math:`\\sigma` is the elementwise sigmoid function.
      These are returned as a tuple of two variables.
  
      This function supports variable length inputs. The mini-batch size of
      the current input must be equal to or smaller than that of the previous
      one. When mini-batch size of ``x`` is smaller than that of ``c``, this
      function only updates ``c[0:len(x)]`` and doesn't change the rest of ``c``,
      ``c[len(x):]``.
      So, please sort input sequences in descending order of lengths before
      applying the function.
  
      Args:
          c_prev (:class:`~chainer.Variable` or :class:`numpy.ndarray` or \
          :class:`cupy.ndarray`):
              Variable that holds the previous cell state. The cell state
              should be a zero array or the output of the previous call of LSTM.
          x (:class:`~chainer.Variable` or :class:`numpy.ndarray` or \
          :class:`cupy.ndarray`):
              Variable that holds the sources of cell input, input gate, forget
              gate and output gate. It must have the second dimension whose size
              is four times of that of the cell state.
  
      Returns:
          tuple: Two :class:`~chainer.Variable` objects ``c`` and ``h``.
          ``c`` is the updated cell state. ``h`` indicates the outgoing signal.
  
      See the original paper proposing LSTM with forget gates:
      `Long Short-Term Memory in Recurrent Neural Networks \
      <http://www.felixgers.de/papers/phd.pdf>`_.
  
      .. seealso::
          :class:`~chainer.links.LSTM`
  
      .. admonition:: Example
  
          Assuming ``y`` is the current incoming signal, ``c`` is the previous
          cell state, and ``h`` is the previous outgoing signal from an ``lstm``
          function. Each of ``y``, ``c`` and ``h`` has ``n_units`` channels.
          Most typical preparation of ``x`` is:
  
          >>> n_units = 100
          >>> y = chainer.Variable(np.zeros((1, n_units), 'f'))
          >>> h = chainer.Variable(np.zeros((1, n_units), 'f'))
          >>> c = chainer.Variable(np.zeros((1, n_units), 'f'))
          >>> model = chainer.Chain()
          >>> with model.init_scope():
          ...   model.w = L.Linear(n_units, 4 * n_units)
          ...   model.v = L.Linear(n_units, 4 * n_units)
          >>> x = model.w(y) + model.v(h)
          >>> c, h = F.lstm(c, x)
  
          It corresponds to calculate the input array ``x``, or the input
          sources :math:`a, i, f, o`, from the current incoming signal ``y`` and
          the previous outgoing signal ``h``. Different parameters are used for
          different kind of input sources.
  
      .. note::
  
          We use the naming rule below.
  
          - incoming signal
              The formal input of the formulation of LSTM (e.g. in NLP, word
              vector or output of lower RNN layer). The input of
              :class:`chainer.links.LSTM` is the *incoming signal*.
          - input array
              The array which is linear transformed from *incoming signal* and
              the previous outgoing signal. The *input array* contains four
              sources, the sources of cell input, input gate, forget gate and
              output gate. The input of :class:`chainer.functions.LSTM` is the
              *input array*.
  
      """
      return LSTM().apply((c_prev, x))
  