code: def custom_root(
      f, initial_guess, solve, tangent_solve, has_aux=False
  ):
      """Differentiably solve for a roots of a function.
  
      This is a low-level routine, mostly intended for internal use in JAX.
      Gradients of custom_root() are defined with respect to closed-over variables
      from the provided function ``f`` via the implicit function theorem:
      https://en.wikipedia.org/wiki/Implicit_function_theorem
  
      Args:
        f: function for which to find a root. Should accept a single argument,
          return a tree of arrays with the same structure as its input.
        initial_guess: initial guess for a zero of f.
        solve: function to solve for the roots of f. Should take two positional
          arguments, f and initial_guess, and return a solution with the same
          structure as initial_guess such that func(solution) = 0. In other words,
          the following is assumed to be true (but not checked)::
  
            solution = solve(f, initial_guess)
            error = f(solution)
            assert all(error == 0)
  
        tangent_solve: function to solve the tangent system. Should take two
          positional arguments, a linear function ``g`` (the function ``f``
          linearized at its root) and a tree of array(s) ``y`` with the same
          structure as initial_guess, and return a solution ``x`` such that
          ``g(x)=y``:
  
          - For scalar ``y``, use ``lambda g, y: y / g(1.0)``.
          - For vector ``y``, you could use a linear solve with the Jacobian, if
            dimensionality of ``y`` is not too large:
            ``lambda g, y: np.linalg.solve(jacobian(g)(y), y)``.
        has_aux: bool indicating whether the ``solve`` function returns
          auxiliary data like solver diagnostics as a second argument.
  
      Returns:
        The result of calling solve(f, initial_guess) with gradients defined via
        implicit differentiation assuming ``f(solve(f, initial_guess)) == 0``.
      """
      guess_flat, in_args_tree = tree_flatten((initial_guess,))
      guess_avals = tuple(_map(_abstractify, guess_flat))
      f_jaxpr, f_consts, out_tree = _initial_style_jaxpr(
          f, in_args_tree, guess_avals
      )
  
      (in_tree,) = treedef_children(in_args_tree)
      _check_tree("f", "initial_guess", out_tree, in_tree, False)
  
      solve_jaxpr, solve_consts, solution_tree = _initial_style_jaxpr(
          partial(solve, f), in_args_tree, guess_avals
      )
      _check_tree(
          "solve", "initial_guess", solution_tree, in_tree, has_aux
      )
  
      def linearize_and_solve(x, b):
          unchecked_zeros, f_jvp = jax.linearize(f, x)
          return tangent_solve(f_jvp, b)
  
      l_and_s_jaxpr, l_and_s_consts, out_tree = _initial_style_jaxpr(
          linearize_and_solve,
          treedef_tuple((in_tree,) * 2),
          guess_avals * 2,
      )
      _check_tree("tangent_solve", "x", out_tree, in_tree, False)
  
      all_consts = [f_consts, solve_consts, l_and_s_consts]
      const_lengths = _RootTuple(*_map(len, all_consts))
      jaxprs = _RootTuple(f_jaxpr, solve_jaxpr, l_and_s_jaxpr)
  
      solution_flat = _custom_root(
          const_lengths, jaxprs, *(_flatten(all_consts) + guess_flat)
      )
      return tree_unflatten(solution_tree, solution_flat)
  
imports: import collections
  from functools import partial
  import operator
  import jax
  from jax.tree_util import tree_flatten, treedef_children, tree_leaves, tree_unflatten, treedef_tuple
  from jax._src import ad_util
  from jax._src import core
  from jax._src import linear_util
  from jax._src.core import raise_to_shaped
  from jax._src.interpreters import ad
  from jax._src.interpreters import batching
  from jax._src.interpreters import mlir
  from jax._src.interpreters import xla
  from jax._src.traceback_util import api_boundary
  from jax._src.util import split_list, safe_map
  import numpy
  from jax._src.lax.control_flow.common import _abstractify, _check_tree, _initial_style_jaxpr
label_desc: Use image node_id 2 for calling a global function with example usage: custom_root(f, initial_guess, solve, tangent_solve, has_aux) and returns: tree_unflatten, tangent_solve