prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  Support optimization for rsqrt
  CUDA provides `rsqrt` function, which is faster than doing `1.0 / sqrt(x)`.
  It is better to provide custom kernel in Chainer or directly support in CuPy.
  (Note that NumPy does not provide `rsqrt`)
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `chainer/functions/math/sqrt.py`
  Content:
  ```
  1 from chainer.backends import cuda
  2 from chainer import function_node
  3 from chainer import utils
  4 from chainer.utils import type_check
  5 
  6 
  7 class Sqrt(function_node.FunctionNode):
  8 
  9     @property
  10     def label(self):
  11         return 'sqrt'
  12 
  13     def check_type_forward(self, in_types):
  14         type_check.expect(
  15             in_types.size() == 1,
  16             in_types[0].dtype.kind == 'f',
  17         )
  18 
  19     def forward(self, x):
  20         self.retain_outputs((0,))
  21         xp = cuda.get_array_module(*x)
  22         return utils.force_array(xp.sqrt(x[0], dtype=x[0].dtype)),
  23 
  24     def backward(self, indexes, grad_outputs):
  25         gx = self.get_retained_outputs()[0]
  26         gy = grad_outputs[0]
  27         return gy / (gx * 2.0),
  28 
  29 
  30 def sqrt(x):
  31     """Elementwise square root function.
  32 
  33     .. math::
  34        y_i = \\sqrt x_i.
  35 
  36     If the value of :math:`x_i` is negative, it returns ``Nan`` for :math:`y_i`
  37     respect to underlying numpy and cupy specification.
  38 
  39     Args:
  40         x (~chainer.Variable): Input variable.
  41 
  42     Returns:
  43         ~chainer.Variable: Output variable.
  44     """
  45     return Sqrt().apply((x,))[0]
  46 
  47 
  48 def rsqrt(x):
  49     """Computes elementwise reciprocal of square root of input :math:`x_i`.
  50 
  51     .. math::
  52        y_i = {1 \\over \\sqrt x_i}.
  53 
  54     Args:
  55         x (~chainer.Variable): Input variable.
  56 
  57     Returns:
  58         ~chainer.Variable: Output variable.
  59 
  60     .. seealso:: :func:`~chainer.functions.sqrt`
  61     """
  62     return 1.0 / sqrt(x)
  63
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/chainer/functions/math/sqrt.py b/chainer/functions/math/sqrt.py
  --- a/chainer/functions/math/sqrt.py
  +++ b/chainer/functions/math/sqrt.py
  @@ -1,3 +1,5 @@
  +import numpy
  +
   from chainer.backends import cuda
   from chainer import function_node
   from chainer import utils
  @@ -27,6 +29,36 @@
           return gy / (gx * 2.0),
   
   
  +class Rsqrt(function_node.FunctionNode):
  +
  +    @property
  +    def label(self):
  +        return 'rsqrt'
  +
  +    def check_type_forward(self, in_types):
  +        type_check.expect(
  +            in_types.size() == 1,
  +            in_types[0].dtype.kind == 'f',
  +        )
  +
  +    def forward(self, inputs):
  +        self.retain_inputs((0,))
  +        x, = inputs
  +        xp = cuda.get_array_module(x)
  +        dtype = x.dtype
  +        if xp is numpy:
  +            out = xp.reciprocal(xp.sqrt(x, dtype=dtype), dtype=dtype)
  +        else:
  +            # CuPy provides `rsqrt` which is faster than `1.0 / sqrt(x)`.
  +            out = cuda.cupyx.rsqrt(x, dtype=dtype)
  +        return utils.force_array(out),
  +
  +    def backward(self, indexes, grad_outputs):
  +        x, = self.get_retained_inputs()
  +        gy, = grad_outputs
  +        return gy * (x ** -1.5) * -0.5,
  +
  +
   def sqrt(x):
       """Elementwise square root function.
   
  @@ -59,4 +91,4 @@
   
       .. seealso:: :func:`~chainer.functions.sqrt`
       """
  -    return 1.0 / sqrt(x)
  +    return Rsqrt().apply((x,))[0]
  
verification_info: {"golden_diff": "diff --git a/chainer/functions/math/sqrt.py b/chainer/functions/math/sqrt.py\n--- a/chainer/functions/math/sqrt.py\n+++ b/chainer/functions/math/sqrt.py\n@@ -1,3 +1,5 @@\n+import numpy\n+\n from chainer.backends import cuda\n from chainer import function_node\n from chainer import utils\n@@ -27,6 +29,36 @@\n         return gy / (gx * 2.0),\n \n \n+class Rsqrt(function_node.FunctionNode):\n+\n+    @property\n+    def label(self):\n+        return 'rsqrt'\n+\n+    def check_type_forward(self, in_types):\n+        type_check.expect(\n+            in_types.size() == 1,\n+            in_types[0].dtype.kind == 'f',\n+        )\n+\n+    def forward(self, inputs):\n+        self.retain_inputs((0,))\n+        x, = inputs\n+        xp = cuda.get_array_module(x)\n+        dtype = x.dtype\n+        if xp is numpy:\n+            out = xp.reciprocal(xp.sqrt(x, dtype=dtype), dtype=dtype)\n+        else:\n+            # CuPy provides `rsqrt` which is faster than `1.0 / sqrt(x)`.\n+            out = cuda.cupyx.rsqrt(x, dtype=dtype)\n+        return utils.force_array(out),\n+\n+    def backward(self, indexes, grad_outputs):\n+        x, = self.get_retained_inputs()\n+        gy, = grad_outputs\n+        return gy * (x ** -1.5) * -0.5,\n+\n+\n def sqrt(x):\n     \"\"\"Elementwise square root function.\n \n@@ -59,4 +91,4 @@\n \n     .. seealso:: :func:`~chainer.functions.sqrt`\n     \"\"\"\n-    return 1.0 / sqrt(x)\n+    return Rsqrt().apply((x,))[0]\n", "issue": "Support optimization for rsqrt\nCUDA provides `rsqrt` function, which is faster than doing `1.0 / sqrt(x)`.\r\nIt is better to provide custom kernel in Chainer or directly support in CuPy.\r\n(Note that NumPy does not provide `rsqrt`)\n", "before_files": [{"content": "from chainer.backends import cuda\nfrom chainer import function_node\nfrom chainer import utils\nfrom chainer.utils import type_check\n\n\nclass Sqrt(function_node.FunctionNode):\n\n    @property\n    def label(self):\n        return 'sqrt'\n\n    def check_type_forward(self, in_types):\n        type_check.expect(\n            in_types.size() == 1,\n            in_types[0].dtype.kind == 'f',\n        )\n\n    def forward(self, x):\n        self.retain_outputs((0,))\n        xp = cuda.get_array_module(*x)\n        return utils.force_array(xp.sqrt(x[0], dtype=x[0].dtype)),\n\n    def backward(self, indexes, grad_outputs):\n        gx = self.get_retained_outputs()[0]\n        gy = grad_outputs[0]\n        return gy / (gx * 2.0),\n\n\ndef sqrt(x):\n    \"\"\"Elementwise square root function.\n\n    .. math::\n       y_i = \\\\sqrt x_i.\n\n    If the value of :math:`x_i` is negative, it returns ``Nan`` for :math:`y_i`\n    respect to underlying numpy and cupy specification.\n\n    Args:\n        x (~chainer.Variable): Input variable.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n    \"\"\"\n    return Sqrt().apply((x,))[0]\n\n\ndef rsqrt(x):\n    \"\"\"Computes elementwise reciprocal of square root of input :math:`x_i`.\n\n    .. math::\n       y_i = {1 \\\\over \\\\sqrt x_i}.\n\n    Args:\n        x (~chainer.Variable): Input variable.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n\n    .. seealso:: :func:`~chainer.functions.sqrt`\n    \"\"\"\n    return 1.0 / sqrt(x)\n", "path": "chainer/functions/math/sqrt.py"}], "after_files": [{"content": "import numpy\n\nfrom chainer.backends import cuda\nfrom chainer import function_node\nfrom chainer import utils\nfrom chainer.utils import type_check\n\n\nclass Sqrt(function_node.FunctionNode):\n\n    @property\n    def label(self):\n        return 'sqrt'\n\n    def check_type_forward(self, in_types):\n        type_check.expect(\n            in_types.size() == 1,\n            in_types[0].dtype.kind == 'f',\n        )\n\n    def forward(self, x):\n        self.retain_outputs((0,))\n        xp = cuda.get_array_module(*x)\n        return utils.force_array(xp.sqrt(x[0], dtype=x[0].dtype)),\n\n    def backward(self, indexes, grad_outputs):\n        gx = self.get_retained_outputs()[0]\n        gy = grad_outputs[0]\n        return gy / (gx * 2.0),\n\n\nclass Rsqrt(function_node.FunctionNode):\n\n    @property\n    def label(self):\n        return 'rsqrt'\n\n    def check_type_forward(self, in_types):\n        type_check.expect(\n            in_types.size() == 1,\n            in_types[0].dtype.kind == 'f',\n        )\n\n    def forward(self, inputs):\n        self.retain_inputs((0,))\n        x, = inputs\n        xp = cuda.get_array_module(x)\n        dtype = x.dtype\n        if xp is numpy:\n            out = xp.reciprocal(xp.sqrt(x, dtype=dtype), dtype=dtype)\n        else:\n            # CuPy provides `rsqrt` which is faster than `1.0 / sqrt(x)`.\n            out = cuda.cupyx.rsqrt(x, dtype=dtype)\n        return utils.force_array(out),\n\n    def backward(self, indexes, grad_outputs):\n        x, = self.get_retained_inputs()\n        gy, = grad_outputs\n        return gy * (x ** -1.5) * -0.5,\n\n\ndef sqrt(x):\n    \"\"\"Elementwise square root function.\n\n    .. math::\n       y_i = \\\\sqrt x_i.\n\n    If the value of :math:`x_i` is negative, it returns ``Nan`` for :math:`y_i`\n    respect to underlying numpy and cupy specification.\n\n    Args:\n        x (~chainer.Variable): Input variable.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n    \"\"\"\n    return Sqrt().apply((x,))[0]\n\n\ndef rsqrt(x):\n    \"\"\"Computes elementwise reciprocal of square root of input :math:`x_i`.\n\n    .. math::\n       y_i = {1 \\\\over \\\\sqrt x_i}.\n\n    Args:\n        x (~chainer.Variable): Input variable.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n\n    .. seealso:: :func:`~chainer.functions.sqrt`\n    \"\"\"\n    return Rsqrt().apply((x,))[0]\n", "path": "chainer/functions/math/sqrt.py"}]}