Error

Derivation

1. Split input into 2 regimes

2. if x < 354.78477434902067

   1. Initial program 39.1

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

   2. Taylor expanded around 0 1.5

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

   4. Applied add-sqr-sqrt1.5

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

   5. Applied *-un-lft-identity1.5

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

   6. Applied prod-diff1.5

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

   7. Simplified1.5

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

   8. Simplified1.5

      \[\leadsto \frac{(\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_* + \color{blue}{0}}{2}\]
   9. Using strategy rm

   10. Applied add-exp-log1.5

       \[\leadsto \frac{\color{blue}{e^{\log \left((\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_*\right)}} + 0}{2}\]

   11. Taylor expanded around inf 1.5

       \[\leadsto \frac{e^{\log \left((\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{2}{3} \cdot x - 1\right)} + 2)_*\right)} + 0}{2}\]

   12. Simplified1.5

       \[\leadsto \frac{e^{\log \left((\left(x \cdot x\right) \cdot \color{blue}{\left((\frac{2}{3} \cdot x + -1)_*\right)} + 2)_*\right)} + 0}{2}\]

   if 354.78477434902067 < 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 prod-diff0.0

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

   4. Simplified0.0

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

   5. Simplified0.0

      \[\leadsto \frac{(\left(e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1 + \frac{-1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}\right))_* + \color{blue}{0}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 354.78477434902067:\\ \;\;\;\;\frac{e^{\log \left((\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_*\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{\left(-1 + \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) + \left(\frac{\frac{-1}{\varepsilon} + 1}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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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