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Derivation

1. Split input into 3 regimes

2. if x < -0.007185451780055169

   1. Initial program 0.0

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified0.0

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

   if -0.007185451780055169 < x < 0.007491583784597752

   1. Initial program 59.0

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

   2. Taylor expanded around inf 59.0

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

   3. Simplified59.0

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

   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 0.007491583784597752 < x

   1. Initial program 0.0

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

   3. Applied flip3--0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007185451780055169:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.007491583784597752:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - 1}{\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) + \frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Details

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