Error

Derivation

1. Split input into 2 regimes

2. if x < -0.007185451780055169 or 0.007491583784597752 < x

   1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied flip3--0.0

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

   5. Applied add-log-exp0.0

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

   6. Applied add-log-exp0.0

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

   7. Applied sum-log0.0

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

   8. Applied add-log-exp0.0

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

   9. Applied sum-log0.0

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

   10. Simplified0.0

       \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\log \color{blue}{\left(e^{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) + 1)_*}\right)}}\]

   if -0.007185451780055169 < x < 0.007491583784597752

   1. Initial program 59.0

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

   2. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007185451780055169 \lor \neg \left(x \le 0.007491583784597752\right):\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - 1}{\log \left(e^{(\left(\frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) + 1)_*}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Details

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