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Derivation

1. Split input into 3 regimes

2. if x < -0.006955648676525426

   1. Initial program 0.0

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

   3. Applied add-exp-log0.0

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

   4. Applied expm1-def0.0

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

   5. Simplified0.0

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]
   6. Using strategy rm

   7. Applied add-cbrt-cube0.0

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

   if -0.006955648676525426 < x < 0.006295141041971404

   1. Initial program 58.9

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

   3. Applied add-exp-log58.9

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

   4. Applied expm1-def58.9

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

   5. Simplified58.8

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]
   6. Using strategy rm

   7. Applied add-cbrt-cube58.8

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

   8. Taylor expanded around 0 0.0

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

   if 0.006295141041971404 < x

   1. Initial program 0.0

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

   3. Applied add-exp-log0.0

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

   4. Applied expm1-def0.0

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

   5. Simplified0.0

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]
   6. Using strategy rm

   7. Applied flip--0.0

      \[\leadsto (e^{\color{blue}{\frac{\log 2 \cdot \log 2 - \log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x})}{\log 2 + \log_* (1 + e^{-2 \cdot x})}}} - 1)^*\]
3. Recombined 3 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006955648676525426:\\ \;\;\;\;\sqrt[3]{(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^* \cdot \left((e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^* \cdot (e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\right)}\\ \mathbf{elif}\;x \le 0.006295141041971404:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log 2 \cdot \log 2 - \log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x})}{\log_* (1 + e^{-2 \cdot x}) + \log 2}} - 1)^*\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Details

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