Error

Derivation

1. Initial program 41.4

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

2. Initial simplification41.4

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

4. Applied expm1-log1p-u38.7

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

6. Applied add-exp-log38.7

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

8. Applied add-cube-cbrt39.6

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

10. Applied add-cube-cbrt39.5

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

11. Applied fma-neg39.5

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

12. Simplified39.6

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

13. Final simplification39.6

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

Runtime

Time bar (total: 21.4s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))