Error

Derivation

1. Split input into 2 regimes

2. if x < 27.535461960897777

   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 1.2

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

   4. Applied add-cbrt-cube1.2

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

   6. Applied flip--1.2

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

   7. Applied associate-*l/1.2

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

   8. Applied associate-*l/1.2

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

   9. Applied cbrt-div1.2

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

   10. Simplified1.2

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

   11. Taylor expanded around 0 2.2

       \[\leadsto \frac{\frac{\sqrt[3]{{\left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*\right)}^{3} \cdot (\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) + \left(2 + x \cdot x\right))_*}}{\color{blue}{\frac{1}{6} \cdot \left({x}^{2} \cdot {2}^{\frac{1}{3}}\right) + \left(\frac{1}{9} \cdot \left({x}^{3} \cdot {2}^{\frac{1}{3}}\right) + {2}^{\frac{1}{3}}\right)}}}{2}\]

   12. Simplified1.2

       \[\leadsto \frac{\frac{\sqrt[3]{{\left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*\right)}^{3} \cdot (\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) + \left(2 + x \cdot x\right))_*}}{\color{blue}{(\left(\sqrt[3]{2} \cdot x\right) \cdot \left((\frac{1}{9} \cdot x + \frac{1}{6})_* \cdot x\right) + \left(\sqrt[3]{2}\right))_*}}}{2}\]

   if 27.535461960897777 < 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 prod-diff0.2

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

   4. Simplified0.2

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

   5. Simplified0.2

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 27.535461960897777:\\ \;\;\;\;\frac{\frac{\sqrt[3]{(\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) + \left(x \cdot x + 2\right))_* \cdot {\left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*\right)}^{3}}}{(\left(\sqrt[3]{2} \cdot x\right) \cdot \left(x \cdot (\frac{1}{9} \cdot x + \frac{1}{6})_*\right) + \left(\sqrt[3]{2}\right))_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{x \cdot \left(\varepsilon + -1\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{\frac{-1}{\varepsilon} + 1}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \end{array}\]

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed 2018351 +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))