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Derivation

1. Split input into 2 regimes

2. if x < 111.9247683550667

   1. Initial program 39.6

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

   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 flip-+1.4

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

   6. Applied *-un-lft-identity1.4

      \[\leadsto \frac{\frac{\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right) - 2 \cdot 2}{\frac{2}{3} \cdot {x}^{3} - 2} - \color{blue}{1 \cdot {x}^{2}}}{2}\]

   7. Applied *-un-lft-identity1.4

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right) - 2 \cdot 2}{\frac{2}{3} \cdot {x}^{3} - 2}} - 1 \cdot {x}^{2}}{2}\]

   8. Applied distribute-lft-out--1.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\frac{\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right) - 2 \cdot 2}{\frac{2}{3} \cdot {x}^{3} - 2} - {x}^{2}\right)}}{2}\]

   9. Simplified1.4

      \[\leadsto \frac{1 \cdot \color{blue}{\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right)\right) - x \cdot x\right)}}{2}\]

   if 111.9247683550667 < 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. Using strategy rm

   3. Applied add-cube-cbrt0.2

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

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 111.9247683550667:\\ \;\;\;\;\frac{\left(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(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))

Details

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