Error

Derivation

1. Split input into 2 regimes

2. if x < 58.57988842383729

   1. Initial program 39.3

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Taylor expanded around 0 1.4

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

   4. Applied add-cbrt-cube1.4

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

   6. Applied flip--1.4

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

   7. Applied flip--1.4

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

   8. Applied flip3--1.4

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

   9. Applied frac-times1.4

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

   10. Applied frac-times1.4

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

   11. Applied cbrt-div1.4

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

   13. Applied add-cube-cbrt2.9

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

   14. Applied add-cube-cbrt2.9

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

   15. Applied times-frac1.4

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

   16. Simplified1.4

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{\left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right) \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{3} \cdot {x}^{3}\right))_*}}}{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(x \cdot x\right) \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right) \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{3} \cdot {x}^{3}\right))_*}}}{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(x \cdot x\right) \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}}\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{\left(\left({\left(\frac{2}{3} \cdot {x}^{3} + 2\right)}^{3} - {\left({x}^{2}\right)}^{3}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)}}}{\sqrt[3]{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) + \left({x}^{2} \cdot {x}^{2} + \left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}\right)}}}}{2}\]

   17. Simplified1.4

       \[\leadsto \frac{\left(\frac{\sqrt[3]{\sqrt[3]{\left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right) \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{3} \cdot {x}^{3}\right))_*}}}{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(x \cdot x\right) \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right) \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{3} \cdot {x}^{3}\right))_*}}}{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(x \cdot x\right) \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{3} \cdot {x}^{3}\right))_* \cdot \left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right)}}}{\sqrt[3]{\sqrt[3]{(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}}}}{2}\]

   if 58.57988842383729 < x

   1. Initial program 0.2

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Taylor expanded around -inf 0.2

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

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 58.57988842383729:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{\sqrt[3]{\left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right) \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(-{x}^{3}\right) \cdot {x}^{3}\right))_*}}}{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot \left(x \cdot x\right)\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right) \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(-{x}^{3}\right) \cdot {x}^{3}\right))_*}}}{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot \left(x \cdot x\right)\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(\left(-{x}^{3}\right) \cdot {x}^{3}\right))_* \cdot \left((\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_* \cdot (\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*\right)}}}{\sqrt[3]{\sqrt[3]{(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right))_* \cdot \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_* \cdot (\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((x \cdot x + 2)_*\right))_*\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))