\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.007445330190004102 \lor \neg \left(x \le 0.006102529871242619\right):\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < -0.007445330190004102 or 0.006102529871242619 < 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. Using strategy rm
4. Applied add-cbrt-cube0.0
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}\]
if -0.007445330190004102 < x < 0.006102529871242619
1. Initial program 58.9
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification58.9
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007445330190004102 \lor \neg \left(x \le 0.006102529871242619\right):\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Runtime

herbie shell --seed 2018348 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Error

Try it out

Derivation

`if x < -0.007445330190004102 or 0.006102529871242619 < x`

`if -0.007445330190004102 < x < 0.006102529871242619`

Runtime

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