code: def backward(ctx, grad_output):
      z, log_phi_z = ctx.saved_tensors
      log_phi_z_grad = torch.zeros_like(z)
  
      z_is_small = z.lt(-1)
      z_is_not_small = ~z_is_small
  
      if z_is_small.sum() > 0:
          log_phi_z_grad[z_is_small] = torch.abs(
              ctx.denominator.div(ctx.numerator)
          ).mul(math.sqrt(2 / math.pi))
  
      exp = (
          z[z_is_not_small]
          .pow(2)
          .div(-2)
          .sub(log_phi_z[z_is_not_small])
          .add(math.log(0.5))
      )
  
      log_phi_z_grad[z_is_not_small] = torch.exp(exp).mul(
          math.sqrt(2 / math.pi)
      )
  
      return log_phi_z_grad.mul(grad_output)
  
imports: import math
  import torch
  from torch.autograd import Function
  from torch.distributions import Normal
label_desc: Use image node_id 2 for calling the LogNormalCDF obj's underlying member method code with example usage: obj.backward(ctx, grad_output) and returns: log_phi_z_grad