Error

Derivation

1. Split input into 2 regimes

2. if x < -0.006170612698827894 or 0.007360964456689547 < x

   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 -0.006170612698827894 < x < 0.007360964456689547

   1. Initial program 58.8

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

   2. Taylor expanded around inf 58.8

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

   3. Simplified58.8

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

   4. Taylor expanded around 0 0.0

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

   5. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006170612698827894:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.007360964456689547:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) + \frac{2}{15} \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))