Pytorch RNN implementation

I read about RNN in pytorch:
RNN — PyTorch 1.12 documentation.

According to the document the RNN run the following function:
rnn

I looked on another RNN example (from pytorch tutorial):
NLP From Scratch: Classifying Names with a Character-Level RNN — PyTorch Tutorials 1.12.0+cu102 documentation.

And they implemented RNN as:

import torch.nn as nn

class RNN(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super(RNN, self).__init__()

        self.hidden_size = hidden_size

        self.i2h = nn.Linear(input_size + hidden_size, hidden_size)
        self.i2o = nn.Linear(input_size + hidden_size, output_size)
        self.softmax = nn.LogSoftmax(dim=1)

    def forward(self, input, hidden):
        combined = torch.cat((input, hidden), 1)
        hidden = self.i2h(combined)
        output = self.i2o(combined)
        output = self.softmax(output)
        return output, hidden

    def initHidden(self):
        return torch.zeros(1, self.hidden_size)

n_hidden = 128
rnn = RNN(n_letters, n_hidden, n_categories)
  1. Why the implemented function is different from the equation ?
    (The function doesn’t contains softmax and it does contain bias which is not shown in the code)

  2. Why the code dosn’t use tanh as shown in the equation ?