code: def Gelu(approximate: bool = False) -> InternalLayer:
      """Gelu function.
  
      Args:
        approximate:
          only relevant for finite-width network, `apply_fn`. If `True`, computes
          an approximation via `tanh`, see "`Gaussian Error Linear Units (GELUs)
          <https://arxiv.org/abs/1606.08415>`_" and :obj:`jax.nn.gelu` for details.
  
      Returns:
        `(init_fn, apply_fn, kernel_fn)`.
      """
  
      def fn(x):
          return jax.nn.gelu(x, approximate=approximate)
  
      def kernel_fn(k: Kernel) -> Kernel:
          """Compute kernels after a `Gelu` layer.
  
          For NNGP see "`Avoiding Kernel Fixed Points: Computing with ELU and GELU
          Infinite Networks <https://arxiv.org/abs/2002.08517>`_".
          """
          cov1, nngp, cov2, ntk = k.cov1, k.nngp, k.cov2, k.ntk
  
          cov1_plus_1 = cov1 + 1
          cov2_plus_1 = None if cov2 is None else cov2 + 1
  
          (
              prod11_plus_1,
              prod12_plus_1,
              prod22_plus_1,
          ) = get_diagonal_outer_prods(
              cov1_plus_1,
              cov2_plus_1,
              k.diagonal_batch,
              k.diagonal_spatial,
              op.mul,
          )
          prod11, prod12, prod22 = get_diagonal_outer_prods(
              cov1, cov2, k.diagonal_batch, k.diagonal_spatial, op.mul
          )
  
          def nngp_ntk_fn(
              nngp: jnp.ndarray,
              prod: jnp.ndarray,
              prod_plus_1: jnp.ndarray,
              ntk: Optional[jnp.ndarray] = None,
          ) -> tuple[jnp.ndarray, Optional[jnp.ndarray]]:
              delta_squared = prod_plus_1 - nngp**2
              delta = _sqrt(delta_squared)
              angles = jnp.arctan2(nngp, delta)
              new_nngp = (nngp**2 + prod * delta_squared) / (
                  prod_plus_1 * delta
              )
              new_nngp += nngp * angles
              new_nngp /= 2 * jnp.pi
              new_nngp += 0.25 * nngp
  
              if ntk is not None:
                  second_term = 0.25 + angles / (2 * jnp.pi)
                  first_term = (
                      1 / delta_squared + (1 - prod) / prod_plus_1 + 1
                  )
                  first_term *= nngp / delta / (2.0 * jnp.pi)
                  dot_sigma = first_term + second_term
                  ntk *= dot_sigma
              return new_nngp, ntk
  
          def nngp_fn_diag(nngp: jnp.ndarray) -> jnp.ndarray:
              square_root = jnp.sqrt(1.0 + 2.0 * nngp)
              new_nngp = nngp / (
                  (nngp + 1.0) * jnp.sqrt(1.0 + 2.0 * nngp)
              )
              new_nngp += jnp.arctan2(nngp, square_root) / 2
              new_nngp /= jnp.pi
              new_nngp += 0.25
              new_nngp *= nngp
              return new_nngp
  
          nngp, ntk = nngp_ntk_fn(nngp, prod12, prod12_plus_1, ntk)
  
          if k.diagonal_batch and k.diagonal_spatial:
              cov1 = nngp_fn_diag(cov1)
              if cov2 is not None:
                  cov2 = nngp_fn_diag(cov2)
          else:
              cov1, _ = nngp_ntk_fn(cov1, prod11, prod11_plus_1)
              if cov2 is not None:
                  cov2, _ = nngp_ntk_fn(cov2, prod22, prod22_plus_1)
  
          return k.replace(cov1=cov1, nngp=nngp, cov2=cov2, ntk=ntk)
  
      return _elementwise(fn, "Gelu", kernel_fn)
  
imports: import functools
  import operator
  from typing import Callable, Optional, Sequence
  import warnings
  import jax
  from jax import custom_jvp
  from jax import grad
  from jax import numpy
  from jax import vmap
  from jax.scipy.special import erf
  import numpy
  import scipy
  from ..utils import utils
  from ..utils.kernel import Kernel
  from ..utils.typing import InternalLayer
  from ..utils.typing import LayerKernelFn
  from .requirements import Diagonal
  from .requirements import get_diagonal
  from .requirements import get_diagonal_outer_prods
  from .requirements import layer
  from .requirements import requires
  from .requirements import supports_masking
label_desc: Use image node_id 4 for calling a global function with example usage: Gelu(approximate) and returns: _elementwise, jax, k, new_nngp,  ntk, new_nngp