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Derivation

1. Split input into 3 regimes

2. if x < -0.007379778048640885

   1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube0.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}}\]

   6. Taylor expanded around inf 0.0

      \[\leadsto \sqrt[3]{\left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{e^{-2 \cdot x} + 1} - 1\right)}}\]

   7. Simplified0.0

      \[\leadsto \sqrt[3]{\left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right) \cdot \color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}\]

   if -0.007379778048640885 < x < 0.006295141041971404

   1. Initial program 58.9

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied add-log-exp58.9

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]

   4. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

   if 0.006295141041971404 < x

   1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube0.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt0.0

      \[\leadsto \sqrt[3]{\left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\right)}}\]
   8. Using strategy rm

   9. Applied flip--0.0

      \[\leadsto \sqrt[3]{\left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}}\right)\right) \cdot \left(\left(\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007379778048640885:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\right)}\\ \mathbf{elif}\;x \le 0.006295141041971404:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\log \left(e^{\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1}{1 + \frac{2}{e^{-2 \cdot x} + 1}}}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\right) \cdot \left(\sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \left(\sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Details

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