Huber and Tukey are
implemented in practice using the construction <span class="math display">\[
\tilde{\psi}_{c}(r) = r^2 (1-\sigma_{c}(r)) + c^2 \sigma_{c}(r)
\]</span> <span class="math display">\[
\tilde{l}_{c}(r) = r^2 (1-\sigma_{c}(r)) + c(2r-c) \sigma_{c}(r)
\]</span> where <span class="math inline">\(\sigma(r)\)</span> is the
sigmoid function <span class="math display">\[
\sigma_{c}(r) = \dfrac{1}{1+\exp(-5(r-c))}
\]</span> The plot below shows the actual and implemented loss functions
(almost indis