code: def _runge_kutta_step(func, y0, f0, t0, dt, t1, tableau):
      """Take an arbitrary Runge-Kutta step and estimate error.
      Args:
          func: Function to evaluate like `func(t, y)` to compute the time derivative of `y`.
          y0: Tensor initial value for the state.
          f0: Tensor initial value for the derivative, computed from `func(t0, y0)`.
          t0: float64 scalar Tensor giving the initial time.
          dt: float64 scalar Tensor giving the size of the desired time step.
          t1: float64 scalar Tensor giving the end time; equal to t0 + dt. This is used (rather than t0 + dt) to ensure
              floating point accuracy when needed.
          tableau: _ButcherTableau describing how to take the Runge-Kutta step.
      Returns:
          Tuple `(y1, f1, y1_error, k)` giving the estimated function value after
          the Runge-Kutta step at `t1 = t0 + dt`, the derivative of the state at `t1`,
          estimated error at `t1`, and a list of Runge-Kutta coefficients `k` used for
          calculating these terms.
      """
  
      t_dtype = y0.abs().dtype
  
      t0 = t0.to(t_dtype)
      dt = dt.to(t_dtype)
      t1 = t1.to(t_dtype)
  
      # We use an unchecked assign to put data into k without incrementing its _version counter, so that the backward
      # doesn't throw an (overzealous) error about in-place correctness. We know that it's actually correct.
      k = torch.empty(
          *f0.shape,
          len(tableau.alpha) + 1,
          dtype=y0.dtype,
          device=y0.device,
      )
      k = _UncheckedAssign.apply(k, f0, (..., 0))
      for i, (alpha_i, beta_i) in enumerate(
          zip(tableau.alpha, tableau.beta)
      ):
          if alpha_i == 1.0:
              # Always step to perturbing just before the end time, in case of discontinuities.
              ti = t1
              perturb = Perturb.PREV
          else:
              ti = t0 + alpha_i * dt
              perturb = Perturb.NONE
          yi = y0 + torch.sum(
              k[..., : i + 1] * (beta_i * dt), dim=-1
          ).view_as(f0)
          f = func(ti, yi, perturb=perturb)
          k = _UncheckedAssign.apply(k, f, (..., i + 1))
  
      if not (
          tableau.c_sol[-1] == 0
          and (tableau.c_sol[:-1] == tableau.beta[-1]).all()
      ):
          # This property (true for Dormand-Prince) lets us save a few FLOPs.
          yi = y0 + torch.sum(k * (dt * tableau.c_sol), dim=-1).view_as(
              f0
          )
  
      y1 = yi
      f1 = k[..., -1]
      y1_error = torch.sum(k * (dt * tableau.c_error), dim=-1)
      return y1, f1, y1_error, k
  
imports: import bisect
  import collections
  import torch
  from .event_handling import find_event
  from .interp import _interp_evaluate, _interp_fit
  from .misc import _compute_error_ratio, _select_initial_step, _optimal_step_size
  from .misc import Perturb
  from .solvers import AdaptiveStepsizeEventODESolver
label_desc: Use image node_id 1 for calling a global function with example usage: _runge_kutta_step(func, y0, f0, t0, dt, t1, tableau) and returns: y1,  f1,  y1_error,  k