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Derivation

1. Split input into 2 regimes

2. if x < -0.007795368062903516 or 0.006166302486698174 < 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 -0.007795368062903516 < x < 0.006166302486698174

   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

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007795368062903516 \lor \neg \left(x \le 0.006166302486698174\right):\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Details

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