\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 46.5 s
Input Error: 22.8
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\frac{(\left(x \cdot \frac{2}{3}\right) * \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\)
  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 exp-neg to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
    22.8
  4. Applied flip-- to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
    23.0
  5. Applied frac-times to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
    23.3
  6. Applied flip-+ to get
    \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
    23.2
  7. Applied associate-*l/ to get
    \[\frac{\color{red}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
    22.8
  8. Applied frac-sub to get
    \[\frac{\color{red}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
    23.2
  9. Applied taylor to get
    \[\frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
    0.2
  10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
    0.2
  11. 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
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{(\left(x \cdot \frac{2}{3}\right) * \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}}\]
    0.1

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))