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Derivation

1. Split input into 2 regimes

2. if x < 496.97590089644154

   1. Initial program 39.7

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

   2. Taylor expanded around 0 1.2

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

   4. Applied add-exp-log1.2

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

   6. Applied unpow21.2

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

   7. Applied add-sqr-sqrt2.2

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

   8. Applied difference-of-squares2.2

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

   9. Applied log-prod1.2

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

   10. Taylor expanded around inf 1.2

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

   if 496.97590089644154 < x

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube0.1

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

4. Final simplification0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 496.97590089644154:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{2 + \frac{2}{3} \cdot {x}^{3}} - x\right) + \log \left(x + \sqrt{2 + \frac{2}{3} \cdot {x}^{3}}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right)} - e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018252 
(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))