prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  float16 support
  Add support for `np.float16`.
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `jax/abstract_arrays.py`
  Content:
  ```
  1 # Copyright 2018 Google LLC
  2 #
  3 # Licensed under the Apache License, Version 2.0 (the "License");
  4 # you may not use this file except in compliance with the License.
  5 # You may obtain a copy of the License at
  6 #
  7 #     https://www.apache.org/licenses/LICENSE-2.0
  8 #
  9 # Unless required by applicable law or agreed to in writing, software
  10 # distributed under the License is distributed on an "AS IS" BASIS,
  11 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  12 # See the License for the specific language governing permissions and
  13 # limitations under the License.
  14 
  15 from __future__ import absolute_import
  16 from __future__ import division
  17 from __future__ import print_function
  18 
  19 import numpy as onp
  20 import six
  21 
  22 from . import core
  23 from . import ad_util
  24 from . util import prod
  25 from .lib import xla_bridge
  26 
  27 
  28 def concretization_err_msg(fun):
  29   fname = getattr(fun, "__name__", fun)
  30   msg = ("Abstract value passed to `{}`, which requires a concrete value. "
  31          "The function to be transformed can't be traced at the required level "
  32          "of abstraction. If using `jit`, try using `static_argnums` or "
  33          "applying `jit` to smaller subfunctions instead.")
  34   return msg.format(fname)
  35 
  36 def concretization_function_error(fun):
  37   def error(self, *args):
  38     raise TypeError(concretization_err_msg(fun))
  39   return error
  40 
  41 
  42 class UnshapedArray(core.AbstractValue):
  43   __slots__ = ['dtype']
  44   array_abstraction_level = 3
  45 
  46   def __init__(self, dtype):
  47     self.dtype = dtype
  48 
  49   def __eq__(self, other):
  50     return type(self) is type(other) and self.dtype == other.dtype
  51 
  52   def __hash__(self):
  53     return hash(str(self.dtype))
  54 
  55   def __repr__(self):
  56     return '{}({})'.format(self.__class__.__name__, self.str_short())
  57 
  58   _bool = _nonzero = concretization_function_error(bool)
  59   _float   = concretization_function_error(float)
  60   _int     = concretization_function_error(int)
  61   if six.PY2:
  62     _long    = concretization_function_error(long)  # noqa: F821
  63   _complex = concretization_function_error(complex)
  64   _hex     = concretization_function_error(hex)
  65   _oct     = concretization_function_error(oct)
  66 
  67   def at_least_vspace(self):
  68     return self
  69 
  70   def join(self, other):
  71     return self
  72 
  73   def str_short(self):
  74     return onp.dtype(self.dtype).name
  75 
  76 
  77 class ShapedArray(UnshapedArray):
  78   __slots__ = ['shape']
  79   array_abstraction_level = 2
  80 
  81   def __init__(self, shape, dtype):
  82     self.dtype = onp.dtype(xla_bridge.canonicalize_dtype(dtype))
  83     self.shape = shape
  84 
  85   ndim = property(lambda self: len(self.shape))
  86   size = property(lambda self: prod(self.shape))
  87 
  88   def __eq__(self, other):
  89     return (type(self) is type(other)
  90             and self.dtype == other.dtype and self.shape == other.shape)
  91 
  92   def __hash__(self):
  93     return hash((self.shape, str(self.dtype)))
  94 
  95   def at_least_vspace(self):
  96     return self
  97 
  98   def join(self, other):
  99     if self.shape == other.shape and self.dtype == other.dtype:
  100       return self
  101     elif self.dtype == other.dtype:
  102       return UnshapedArray(self.dtype)
  103     else:
  104       raise TypeError(other)
  105 
  106   def str_short(self):
  107     dtypestr = onp.dtype(self.dtype).name
  108     shapestr = ','.join(map(str, self.shape))
  109     return '{}[{}]'.format(dtypestr, shapestr)
  110 
  111   def __len__(self):
  112     try:
  113       return self.shape[0]
  114     except IndexError:
  115       raise TypeError("len() of unsized object")  # same as numpy error
  116 
  117   def _len(self, ignored_tracer):
  118     return len(self)
  119 
  120 
  121 class ConcreteArray(ShapedArray):
  122   __slots__ = ['val']
  123   array_abstraction_level = 0
  124 
  125   def __init__(self, val):
  126     self.val = val
  127     self.shape = onp.shape(val)
  128     # canonicalized self.dtype doesn't necessarily match self.val
  129     self.dtype = onp.dtype(xla_bridge.canonicalize_dtype(onp.result_type(val)))
  130     assert self.dtype != onp.dtype('O')
  131 
  132   def __eq__(self, other):
  133     return (type(self) is type(other) and self.dtype == other.dtype
  134             and self.shape == other.shape and onp.all(self.val == other.val))
  135 
  136   def __hash__(self):
  137     return id(self.val)
  138 
  139   def at_least_vspace(self):
  140     return ShapedArray(self.shape, self.dtype)
  141 
  142   def join(self, other):
  143     if self == other:
  144       return self
  145     elif self.shape == other.shape and self.dtype == other.dtype:
  146       return ShapedArray(self.shape, self.dtype)
  147     elif self.dtype == other.dtype:
  148       return UnshapedArray(self.dtype)
  149     else:
  150       raise TypeError(other)
  151 
  152   def str_short(self):
  153     return str(self.val)
  154 
  155 
  156 def make_shaped_array(x):
  157   dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))
  158   return ShapedArray(onp.shape(x), dtype)
  159 
  160 def zeros_like_array(x):
  161   dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))
  162   return onp.broadcast_to(onp.array(0, dtype), onp.shape(x))
  163 
  164 array_types = [onp.ndarray, onp.float64, onp.float32, onp.complex64,
  165                onp.complex128, onp.int64, onp.int32, onp.bool_, onp.uint64,
  166                onp.uint32, complex, float, int, bool]
  167 
  168 for t in array_types:
  169   core.pytype_aval_mappings[t] = ConcreteArray
  170   ad_util.jaxval_zeros_likers[t] = zeros_like_array
  171
  ```
  --- 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/jax/abstract_arrays.py b/jax/abstract_arrays.py
  --- a/jax/abstract_arrays.py
  +++ b/jax/abstract_arrays.py
  @@ -161,9 +161,11 @@
     dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))
     return onp.broadcast_to(onp.array(0, dtype), onp.shape(x))
   
  -array_types = [onp.ndarray, onp.float64, onp.float32, onp.complex64,
  -               onp.complex128, onp.int64, onp.int32, onp.bool_, onp.uint64,
  -               onp.uint32, complex, float, int, bool]
  +array_types = [onp.ndarray, onp.float64, onp.float32, onp.float16,
  +               onp.complex64, onp.complex128,
  +               onp.int64, onp.int32, onp.int16, onp.int8,
  +               onp.bool_, onp.uint64, onp.uint32, onp.uint16, onp.uint8,
  +               complex, float, int, bool]
   
   for t in array_types:
     core.pytype_aval_mappings[t] = ConcreteArray
  
verification_info: {"golden_diff": "diff --git a/jax/abstract_arrays.py b/jax/abstract_arrays.py\n--- a/jax/abstract_arrays.py\n+++ b/jax/abstract_arrays.py\n@@ -161,9 +161,11 @@\n   dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))\n   return onp.broadcast_to(onp.array(0, dtype), onp.shape(x))\n \n-array_types = [onp.ndarray, onp.float64, onp.float32, onp.complex64,\n-               onp.complex128, onp.int64, onp.int32, onp.bool_, onp.uint64,\n-               onp.uint32, complex, float, int, bool]\n+array_types = [onp.ndarray, onp.float64, onp.float32, onp.float16,\n+               onp.complex64, onp.complex128,\n+               onp.int64, onp.int32, onp.int16, onp.int8,\n+               onp.bool_, onp.uint64, onp.uint32, onp.uint16, onp.uint8,\n+               complex, float, int, bool]\n \n for t in array_types:\n   core.pytype_aval_mappings[t] = ConcreteArray\n", "issue": "float16 support\nAdd support for `np.float16`.\n", "before_files": [{"content": "# Copyright 2018 Google LLC\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport numpy as onp\nimport six\n\nfrom . import core\nfrom . import ad_util\nfrom . util import prod\nfrom .lib import xla_bridge\n\n\ndef concretization_err_msg(fun):\n  fname = getattr(fun, \"__name__\", fun)\n  msg = (\"Abstract value passed to `{}`, which requires a concrete value. \"\n         \"The function to be transformed can't be traced at the required level \"\n         \"of abstraction. If using `jit`, try using `static_argnums` or \"\n         \"applying `jit` to smaller subfunctions instead.\")\n  return msg.format(fname)\n\ndef concretization_function_error(fun):\n  def error(self, *args):\n    raise TypeError(concretization_err_msg(fun))\n  return error\n\n\nclass UnshapedArray(core.AbstractValue):\n  __slots__ = ['dtype']\n  array_abstraction_level = 3\n\n  def __init__(self, dtype):\n    self.dtype = dtype\n\n  def __eq__(self, other):\n    return type(self) is type(other) and self.dtype == other.dtype\n\n  def __hash__(self):\n    return hash(str(self.dtype))\n\n  def __repr__(self):\n    return '{}({})'.format(self.__class__.__name__, self.str_short())\n\n  _bool = _nonzero = concretization_function_error(bool)\n  _float   = concretization_function_error(float)\n  _int     = concretization_function_error(int)\n  if six.PY2:\n    _long    = concretization_function_error(long)  # noqa: F821\n  _complex = concretization_function_error(complex)\n  _hex     = concretization_function_error(hex)\n  _oct     = concretization_function_error(oct)\n\n  def at_least_vspace(self):\n    return self\n\n  def join(self, other):\n    return self\n\n  def str_short(self):\n    return onp.dtype(self.dtype).name\n\n\nclass ShapedArray(UnshapedArray):\n  __slots__ = ['shape']\n  array_abstraction_level = 2\n\n  def __init__(self, shape, dtype):\n    self.dtype = onp.dtype(xla_bridge.canonicalize_dtype(dtype))\n    self.shape = shape\n\n  ndim = property(lambda self: len(self.shape))\n  size = property(lambda self: prod(self.shape))\n\n  def __eq__(self, other):\n    return (type(self) is type(other)\n            and self.dtype == other.dtype and self.shape == other.shape)\n\n  def __hash__(self):\n    return hash((self.shape, str(self.dtype)))\n\n  def at_least_vspace(self):\n    return self\n\n  def join(self, other):\n    if self.shape == other.shape and self.dtype == other.dtype:\n      return self\n    elif self.dtype == other.dtype:\n      return UnshapedArray(self.dtype)\n    else:\n      raise TypeError(other)\n\n  def str_short(self):\n    dtypestr = onp.dtype(self.dtype).name\n    shapestr = ','.join(map(str, self.shape))\n    return '{}[{}]'.format(dtypestr, shapestr)\n\n  def __len__(self):\n    try:\n      return self.shape[0]\n    except IndexError:\n      raise TypeError(\"len() of unsized object\")  # same as numpy error\n\n  def _len(self, ignored_tracer):\n    return len(self)\n\n\nclass ConcreteArray(ShapedArray):\n  __slots__ = ['val']\n  array_abstraction_level = 0\n\n  def __init__(self, val):\n    self.val = val\n    self.shape = onp.shape(val)\n    # canonicalized self.dtype doesn't necessarily match self.val\n    self.dtype = onp.dtype(xla_bridge.canonicalize_dtype(onp.result_type(val)))\n    assert self.dtype != onp.dtype('O')\n\n  def __eq__(self, other):\n    return (type(self) is type(other) and self.dtype == other.dtype\n            and self.shape == other.shape and onp.all(self.val == other.val))\n\n  def __hash__(self):\n    return id(self.val)\n\n  def at_least_vspace(self):\n    return ShapedArray(self.shape, self.dtype)\n\n  def join(self, other):\n    if self == other:\n      return self\n    elif self.shape == other.shape and self.dtype == other.dtype:\n      return ShapedArray(self.shape, self.dtype)\n    elif self.dtype == other.dtype:\n      return UnshapedArray(self.dtype)\n    else:\n      raise TypeError(other)\n\n  def str_short(self):\n    return str(self.val)\n\n\ndef make_shaped_array(x):\n  dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))\n  return ShapedArray(onp.shape(x), dtype)\n\ndef zeros_like_array(x):\n  dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))\n  return onp.broadcast_to(onp.array(0, dtype), onp.shape(x))\n\narray_types = [onp.ndarray, onp.float64, onp.float32, onp.complex64,\n               onp.complex128, onp.int64, onp.int32, onp.bool_, onp.uint64,\n               onp.uint32, complex, float, int, bool]\n\nfor t in array_types:\n  core.pytype_aval_mappings[t] = ConcreteArray\n  ad_util.jaxval_zeros_likers[t] = zeros_like_array\n", "path": "jax/abstract_arrays.py"}], "after_files": [{"content": "# Copyright 2018 Google LLC\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport numpy as onp\nimport six\n\nfrom . import core\nfrom . import ad_util\nfrom . util import prod\nfrom .lib import xla_bridge\n\n\ndef concretization_err_msg(fun):\n  fname = getattr(fun, \"__name__\", fun)\n  msg = (\"Abstract value passed to `{}`, which requires a concrete value. \"\n         \"The function to be transformed can't be traced at the required level \"\n         \"of abstraction. If using `jit`, try using `static_argnums` or \"\n         \"applying `jit` to smaller subfunctions instead.\")\n  return msg.format(fname)\n\ndef concretization_function_error(fun):\n  def error(self, *args):\n    raise TypeError(concretization_err_msg(fun))\n  return error\n\n\nclass UnshapedArray(core.AbstractValue):\n  __slots__ = ['dtype']\n  array_abstraction_level = 3\n\n  def __init__(self, dtype):\n    self.dtype = dtype\n\n  def __eq__(self, other):\n    return type(self) is type(other) and self.dtype == other.dtype\n\n  def __hash__(self):\n    return hash(str(self.dtype))\n\n  def __repr__(self):\n    return '{}({})'.format(self.__class__.__name__, self.str_short())\n\n  _bool = _nonzero = concretization_function_error(bool)\n  _float   = concretization_function_error(float)\n  _int     = concretization_function_error(int)\n  if six.PY2:\n    _long    = concretization_function_error(long)  # noqa: F821\n  _complex = concretization_function_error(complex)\n  _hex     = concretization_function_error(hex)\n  _oct     = concretization_function_error(oct)\n\n  def at_least_vspace(self):\n    return self\n\n  def join(self, other):\n    return self\n\n  def str_short(self):\n    return onp.dtype(self.dtype).name\n\n\nclass ShapedArray(UnshapedArray):\n  __slots__ = ['shape']\n  array_abstraction_level = 2\n\n  def __init__(self, shape, dtype):\n    self.dtype = onp.dtype(xla_bridge.canonicalize_dtype(dtype))\n    self.shape = shape\n\n  ndim = property(lambda self: len(self.shape))\n  size = property(lambda self: prod(self.shape))\n\n  def __eq__(self, other):\n    return (type(self) is type(other)\n            and self.dtype == other.dtype and self.shape == other.shape)\n\n  def __hash__(self):\n    return hash((self.shape, str(self.dtype)))\n\n  def at_least_vspace(self):\n    return self\n\n  def join(self, other):\n    if self.shape == other.shape and self.dtype == other.dtype:\n      return self\n    elif self.dtype == other.dtype:\n      return UnshapedArray(self.dtype)\n    else:\n      raise TypeError(other)\n\n  def str_short(self):\n    dtypestr = onp.dtype(self.dtype).name\n    shapestr = ','.join(map(str, self.shape))\n    return '{}[{}]'.format(dtypestr, shapestr)\n\n  def __len__(self):\n    try:\n      return self.shape[0]\n    except IndexError:\n      raise TypeError(\"len() of unsized object\")  # same as numpy error\n\n  def _len(self, ignored_tracer):\n    return len(self)\n\n\nclass ConcreteArray(ShapedArray):\n  __slots__ = ['val']\n  array_abstraction_level = 0\n\n  def __init__(self, val):\n    self.val = val\n    self.shape = onp.shape(val)\n    # canonicalized self.dtype doesn't necessarily match self.val\n    self.dtype = onp.dtype(xla_bridge.canonicalize_dtype(onp.result_type(val)))\n    assert self.dtype != onp.dtype('O')\n\n  def __eq__(self, other):\n    return (type(self) is type(other) and self.dtype == other.dtype\n            and self.shape == other.shape and onp.all(self.val == other.val))\n\n  def __hash__(self):\n    return id(self.val)\n\n  def at_least_vspace(self):\n    return ShapedArray(self.shape, self.dtype)\n\n  def join(self, other):\n    if self == other:\n      return self\n    elif self.shape == other.shape and self.dtype == other.dtype:\n      return ShapedArray(self.shape, self.dtype)\n    elif self.dtype == other.dtype:\n      return UnshapedArray(self.dtype)\n    else:\n      raise TypeError(other)\n\n  def str_short(self):\n    return str(self.val)\n\n\ndef make_shaped_array(x):\n  dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))\n  return ShapedArray(onp.shape(x), dtype)\n\ndef zeros_like_array(x):\n  dtype = xla_bridge.canonicalize_dtype(onp.result_type(x))\n  return onp.broadcast_to(onp.array(0, dtype), onp.shape(x))\n\narray_types = [onp.ndarray, onp.float64, onp.float32, onp.float16,\n               onp.complex64, onp.complex128,\n               onp.int64, onp.int32, onp.int16, onp.int8,\n               onp.bool_, onp.uint64, onp.uint32, onp.uint16, onp.uint8,\n               complex, float, int, bool]\n\nfor t in array_types:\n  core.pytype_aval_mappings[t] = ConcreteArray\n  ad_util.jaxval_zeros_likers[t] = zeros_like_array\n", "path": "jax/abstract_arrays.py"}]}