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Derivation

1. Split input into 2 regimes

2. if x < 334.2285007287434

   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 1.2

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

   3. Taylor expanded around -inf 62.0

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

   4. Simplified1.2

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

   6. Applied flip3-+1.2

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

   if 334.2285007287434 < x

   1. Initial program 0.0

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

   3. Applied *-un-lft-identity0.0

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

   4. Applied exp-prod0.0

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

   5. Simplified0.0

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

4. Final simplification0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 334.2285007287434:\\ \;\;\;\;\frac{\frac{{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}^{3} + {\left(2 - x \cdot x\right)}^{3}}{\left(2 - x \cdot x\right) \cdot \left(2 - x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) - \left(2 - x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {e}^{\left(\left(-x\right) \cdot \left(1 - \varepsilon\right)\right)} - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018242 
(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))