Derivation

Split input into 2 regimes
if x < -0.006892392584284423 or 0.006700868327006777 < x
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Using strategy rm
3. Applied add-exp-log0.0
  \[\leadsto \color{blue}{e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)}} - 1\]
4. Applied expm1-def0.0
  \[\leadsto \color{blue}{(e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)} - 1)^*}\]
5. Simplified0.0
  \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]
if -0.006892392584284423 < x < 0.006700868327006777
1. Initial program 59.1
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
3. Using strategy rm
4. 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
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006892392584284423 \lor \neg \left(x \le 0.006700868327006777\right):\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018219 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

In
Out