Error

Derivation

1. Split input into 2 regimes

2. if x < 10.239778539899774

   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 add-log-exp1.2

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

   5. Applied add-log-exp1.2

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

   6. Applied diff-log1.2

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

   7. Simplified1.2

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

   if 10.239778539899774 < x

   1. Initial program 0.3

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

   3. Applied add-cbrt-cube0.3

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 10.239778539899774:\\ \;\;\;\;\frac{\log \left(\frac{{\left(e^{\frac{2}{3}}\right)}^{\left({x}^{3}\right)}}{e^{(x \cdot x + -2)_*}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018355 +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))