\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 35.0 s
Input Error: 18.4
Output Error: 3.2
Log: ⚲
Profile: 🕒
\(\frac{2 - \frac{(x * \left(\frac{2}{\varepsilon}\right) + \left(\frac{2}{\varepsilon}\right))_*}{\varepsilon}}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}\)
  1. Started with
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    18.4
  2. Using strategy rm
    18.4
  3. Applied exp-neg to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
    18.4
  4. Applied flip-- to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
    20.6
  5. Applied frac-times to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
    21.1
  6. Applied flip-+ to get
    \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
    20.9
  7. Applied associate-*l/ to get
    \[\frac{\color{red}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
    20.4
  8. Applied frac-sub to get
    \[\frac{\color{red}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
    22.9
  9. Applied taylor to get
    \[\frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\frac{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]
    10.0
  10. Taylor expanded around 0 to get
    \[\frac{\frac{\color{red}{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\frac{\color{blue}{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]
    10.0
  11. Applied simplify to get
    \[\frac{\frac{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{2 - \frac{2}{\varepsilon} \cdot \left(\frac{1}{\varepsilon} + \frac{x}{\varepsilon}\right)}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}\]
    10.7
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{2 - \frac{2}{\varepsilon} \cdot \left(\frac{1}{\varepsilon} + \frac{x}{\varepsilon}\right)}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}} \leadsto \color{blue}{\frac{2 - \frac{(x * \left(\frac{2}{\varepsilon}\right) + \left(\frac{2}{\varepsilon}\right))_*}{\varepsilon}}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}}\]
    3.2

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))