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Derivation

1. Split input into 2 regimes

2. if x < 3.4647329126034587

   1. Initial program 39.2

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

   2. 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 flip--1.2

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

   6. Applied add-log-exp1.2

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

   if 3.4647329126034587 < x

   1. Initial program 0.4

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

   3. Applied add-cbrt-cube0.4

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.4647329126034587:\\ \;\;\;\;\frac{\frac{\log \left(e^{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}\right)}{{x}^{2} + \left(2 + \frac{2}{3} \cdot {x}^{3}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018355 
(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))