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Derivation

1. Split input into 2 regimes

2. if x < -0.007505523876052934 or 0.00747462309715031 < 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. Taylor expanded around inf 0.0

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

   if -0.007505523876052934 < x < 0.00747462309715031

   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

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

   5. Applied associate--l+0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007505523876052934 \lor \neg \left(x \le 0.00747462309715031\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)\\ \end{array}\]

Runtime

Time bar (total: 9.3s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))