\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 41.3 s
Input Error: 22.8
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}\)
  1. Started with
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    22.8
  2. Using strategy rm
    22.8
  3. Applied add-cube-cbrt to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2}\]
    22.9
  4. Applied add-cube-cbrt to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot {\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}^3}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{{\left(\sqrt[3]{\frac{1}{\varepsilon} - 1}\right)}^3} \cdot {\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}^3}{2}\]
    23.4
  5. Applied cube-unprod to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{{\left(\sqrt[3]{\frac{1}{\varepsilon} - 1}\right)}^3 \cdot {\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{{\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2}\]
    23.4
  6. Using strategy rm
    23.4
  7. Applied exp-neg to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}\right)}^3}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}\right)}^3}{2}\]
    23.4
  8. Applied cbrt-div to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \color{red}{\sqrt[3]{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}\right)}^3}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}}\right)}^3}{2}\]
    23.4
  9. Applied flip3-- to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt[3]{\color{red}{\frac{1}{\varepsilon} - 1}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}^3}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt[3]{\color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}^3}{2}\]
    23.4
  10. Applied cbrt-div to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\color{red}{\sqrt[3]{\frac{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}^3}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\color{blue}{\frac{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}^3}{2}\]
    23.4
  11. Applied frac-times to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{red}{\left(\frac{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}}^3}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{blue}{\left(\frac{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}}{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}}^3}{2}\]
    23.4
  12. Applied cube-div to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{{\left(\frac{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}}{\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}^3}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}}{2}\]
    23.4
  13. Applied exp-neg to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2}\]
    23.4
  14. Applied un-div-inv to get
    \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2}\]
    23.4
  15. Applied frac-sub to get
    \[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{{\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3 - e^{\left(1 - \varepsilon\right) \cdot x} \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}}{2}\]
    23.4
  16. Applied simplify to get
    \[\frac{\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3 - e^{\left(1 - \varepsilon\right) \cdot x} \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}} \cdot \sqrt[3]{1}\right)}^3}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2} \leadsto \frac{\frac{\color{blue}{\left(\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left(\frac{1}{\varepsilon} + 1\right)\right) - e^{x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}{2}\]
    23.6
  17. Applied simplify to get
    \[\frac{\frac{\left(\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left(\frac{1}{\varepsilon} + 1\right)\right) - e^{x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1\right)}{\color{red}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot {\left(\sqrt[3]{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}^3}}}{2} \leadsto \frac{\frac{\left(\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left(\frac{1}{\varepsilon} + 1\right)\right) - e^{x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1\right)}{\color{blue}{e^{x \cdot \left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)} \cdot \left(\frac{1}{\varepsilon} \cdot \frac{1}{\varepsilon} + \left(1 + \frac{1}{\varepsilon}\right)\right)}}}{2}\]
    23.2
  18. Applied taylor to get
    \[\frac{\frac{\left(\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left(\frac{1}{\varepsilon} + 1\right)\right) - e^{x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1\right)}{e^{x \cdot \left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)} \cdot \left(\frac{1}{\varepsilon} \cdot \frac{1}{\varepsilon} + \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
    0.1
  19. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
    0.1
  20. Applied simplify to get
    \[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
    0.2
  21. Applied final simplification
  22. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}}\]
    0.2

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))