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Derivation

1. Initial program 0.0

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

3. Applied add-cbrt-cube0.1

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

5. Applied associate-*l/0.1

   \[\leadsto \sqrt[3]{\color{blue}{\frac{2 \cdot \frac{2}{e^{x} + e^{-x}}}{e^{x} + e^{-x}}} \cdot \frac{2}{e^{x} + e^{-x}}}\]

6. Applied frac-times0.1

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

7. Applied cbrt-div0.0

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

8. Simplified0.0

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{8}{e^{x} + e^{-x}}}}}{\sqrt[3]{\left(e^{x} + e^{-x}\right) \cdot \left(e^{x} + e^{-x}\right)}}\]

9. Final simplification0.0

   \[\leadsto \frac{\sqrt[3]{\frac{8}{e^{x} + e^{-x}}}}{\sqrt[3]{\left(e^{x} + e^{-x}\right) \cdot \left(e^{x} + e^{-x}\right)}}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 27.7s)Debug log