Derivation

Split input into 2 regimes
if x < -0.00667712876377636 or 0.007224912728897627 < 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 -0.00667712876377636 < x < 0.007224912728897627
1. Initial program 58.8
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification58.8
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
4. Using strategy rm
5. Applied associate--l+0.0
  \[\leadsto \color{blue}{x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00667712876377636 \lor \neg \left(x \le 0.007224912728897627\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018304 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

In
Out