Derivation

Split input into 2 regimes
if x < -0.006679389634386454 or 0.00704380983148191 < x
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification0.0
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Taylor expanded around -inf 0.0
  \[\leadsto \color{blue}{2 \cdot \frac{1}{e^{-2 \cdot x} + 1} - 1}\]
4. Simplified0.0
  \[\leadsto \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} - 1}\]
if -0.006679389634386454 < x < 0.00704380983148191
1. Initial program 59.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification59.0
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Taylor expanded around -inf 59.0
  \[\leadsto \color{blue}{2 \cdot \frac{1}{e^{-2 \cdot x} + 1} - 1}\]
4. Simplified59.0
  \[\leadsto \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} - 1}\]
5. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006679389634386454 \lor \neg \left(x \le 0.00704380983148191\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Runtime

herbie shell --seed 2018349 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

In
Out