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Derivation

1. Split input into 2 regimes

2. if x < -0.008229152713509763 or 0.007307996409688343 < 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

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

   if -0.008229152713509763 < x < 0.007307996409688343

   1. Initial program 59.1

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

   2. Initial simplification59.1

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.008229152713509763 \lor \neg \left(x \le 0.007307996409688343\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)\\ \end{array}\]

Runtime

Time bar (total: 13.6s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))