code: def _increment_and_space_time_levy_area(self):
      try:
          return self._top._increment_and_space_time_levy_area_cache[
              self
          ]
      except KeyError:
          parent = self._parent
  
          W, H = yield parent._increment_and_space_time_levy_area()
          h_reciprocal = 1 / (parent._end - parent._start)
          left_diff = parent._midway - parent._start
          right_diff = parent._end - parent._midway
  
          if self._top._have_H:
              left_diff_squared = left_diff**2
              right_diff_squared = right_diff**2
              left_diff_cubed = left_diff * left_diff_squared
              right_diff_cubed = right_diff * right_diff_squared
  
              v = 0.5 * math.sqrt(
                  left_diff
                  * right_diff
                  / (left_diff_cubed + right_diff_cubed)
              )
  
              a = v * left_diff_squared * h_reciprocal
              b = v * right_diff_squared * h_reciprocal
              c = v * _rsqrt3
  
              X1 = parent._randn(parent._W_seed)
              X2 = parent._randn(parent._H_seed)
  
              third_coeff = (
                  2 * (a * left_diff + b * right_diff) * h_reciprocal
              )
  
              if self._is_left:
                  first_coeff = left_diff * h_reciprocal
                  second_coeff = (
                      6 * first_coeff * right_diff * h_reciprocal
                  )
                  out_W = (
                      first_coeff * W
                      + second_coeff * H
                      + third_coeff * X1
                  )
                  out_H = (
                      first_coeff**2 * H
                      - a * X1
                      + c * right_diff * X2
                  )
              else:
                  first_coeff = right_diff * h_reciprocal
                  second_coeff = (
                      6 * first_coeff * left_diff * h_reciprocal
                  )
                  out_W = (
                      first_coeff * W
                      - second_coeff * H
                      - third_coeff * X1
                  )
                  out_H = (
                      first_coeff**2 * H - b * X1 - c * left_diff * X2
                  )
          else:
              # Don't compute space-time Levy area unless we need to
  
              mean = left_diff * W * h_reciprocal
              var = left_diff * right_diff * h_reciprocal
              noise = parent._randn(parent._W_seed)
              left_W = mean + math.sqrt(var) * noise
  
              if self._is_left:
                  out_W = left_W
              else:
                  out_W = W - left_W
              out_H = None
  
          self._top._increment_and_space_time_levy_area_cache[self] = (
              out_W,
              out_H,
          )
          return out_W, out_H
  
imports: import math
  import trampoline
  import warnings
  import numpy
  import torch
  from .None import brownian_base
  from ..settings import LEVY_AREA_APPROXIMATIONS
  from ..types import Scalar, Optional, Tuple, Union, Tensor
label_desc: Use image node_id 3 for calling the _Interval obj's underlying member method code with example usage: obj._increment_and_space_time_levy_area() and returns: self, out_W,  out_H