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Derivation

1. Split input into 2 regimes

2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -5.547390397855705e-06 or 4.5698625607206315e-12 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

   1. Initial program 0.4

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

   2. Initial simplification0.4

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

   3. Taylor expanded around -inf 0.4

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

   if -5.547390397855705e-06 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 4.5698625607206315e-12

   1. Initial program 59.9

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

   2. Initial simplification59.9

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

   3. Taylor expanded around -inf 59.9

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

   4. Taylor expanded around 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 simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -5.547390397855705 \cdot 10^{-06}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 4.5698625607206315 \cdot 10^{-12}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Runtime

Time bar (total: 15.6s)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))