Error

Try it out

Derivation

1. Split input into 2 regimes

2. if x < 42.01964098399155

   1. Initial program 39.1

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

   2. Taylor expanded around 0 1.3

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

   4. Applied add-log-exp1.3

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

   5. Applied add-log-exp1.3

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

   6. Applied diff-log1.3

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

   7. Taylor expanded around 0 1.3

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

   9. Applied add-exp-log1.3

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

   if 42.01964098399155 < x

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt0.3

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

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 42.01964098399155:\\ \;\;\;\;\frac{\log \left(e^{\log \left(e^{2} + \left({x}^{3} \cdot e^{2}\right) \cdot \frac{2}{3}\right)} - e^{2} \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018221 
(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))