text: import math
  
  import torch
  
  
  class PositionalEncoding(torch.nn.Module):
      """Positional encoding module
  
      :param int d_model: embedding dim
      :param float dropout_rate: dropout rate
      :param int max_len: maximum input length
      """
  
      def __init__(self, d_model, dropout_rate, max_len=5000):
          super(PositionalEncoding, self).__init__()
          self.dropout = torch.nn.Dropout(p=dropout_rate)
          # Compute the positional encodings once in log space.
          pe = torch.zeros(max_len, d_model)
          position = torch.arange(0, max_len, dtype=torch.float32).unsqueeze(1)
          div_term = torch.exp(torch.arange(0, d_model, 2, dtype=torch.float32) *
                               -(math.log(10000.0) / d_model))
          pe[:, 0::2] = torch.sin(position * div_term)
          pe[:, 1::2] = torch.cos(position * div_term)
          pe = pe.unsqueeze(0)
          self.max_len = max_len
          self.xscale = math.sqrt(d_model)
          self.register_buffer('pe', pe)
  
      def forward(self, x):
          x = x * self.xscale + self.pe[:, :x.size(1)]
          return self.dropout(x)
  
  
  class ScaledPositionalEncoding(PositionalEncoding):
      """Scaled positional encoding module
  
      :param int d_model: embedding dim
      :param float dropout_rate: dropout rate
      :param int max_len: maximum input length
  
      See also: Sec. 3.2  https://arxiv.org/pdf/1809.08895.pdf
      """
  
      def __init__(self, d_model, dropout_rate, max_len=5000):
          super().__init__(d_model=d_model, dropout_rate=dropout_rate, max_len=max_len)
          self.alpha = torch.nn.Parameter(torch.tensor(1.0))
  
      def reset_parameters(self):
          # NOTE: I do not know how to initialize this
          self.alpha.data = torch.tensor(1.0)
  
      def forward(self, x):
          x = x + self.alpha * self.pe[:, :x.size(1)]
          return self.dropout(x)
  