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Derivation

1. Split input into 2 regimes

2. if x < 310.8796801255867

   1. Initial program 39.4

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

   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 associate--l+1.4

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

   6. Applied add-cube-cbrt1.4

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

   8. Applied expm1-log1p-u1.4

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

   if 310.8796801255867 < 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-sqr-sqrt0.2

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

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 310.8796801255867:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \left((e^{\log_* (1 + \sqrt[3]{\frac{2}{3} \cdot {x}^{3}})} - 1)^* \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) + \left(2 - {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}} \cdot \sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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))

Details

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