\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 33.4 s
Input Error: 18.4
Output Error: 5.6
Log: ⚲
Profile: 🕒
\(\frac{\frac{\left(2 + \frac{2}{\varepsilon}\right) + \frac{x \cdot 2}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}{2 + \frac{2}{\varepsilon}}\)
  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 flip-- to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    21.4
  4. Applied associate-*l/ to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2}\]
    21.7
  5. Applied exp-neg to get
    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    21.7
  6. Applied un-div-inv to get
    \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    21.7
  7. Applied frac-sub to get
    \[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}}{2}\]
    24.3
  8. Applied simplify to get
    \[\frac{\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}{{\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}} \cdot \left(\frac{1}{\varepsilon \cdot \varepsilon} - 1\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
    21.8
  9. Applied taylor to get
    \[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}{{\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}} \cdot \left(\frac{1}{\varepsilon \cdot \varepsilon} - 1\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
    3.0
  10. Taylor expanded around 0 to get
    \[\frac{\frac{\color{red}{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\color{blue}{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
    3.0
  11. Applied simplify to get
    \[\frac{\frac{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\left(2 + \frac{2}{\varepsilon}\right) + \frac{x \cdot 2}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}{2 + \frac{2}{\varepsilon}}\]
    5.6
  12. Applied final simplification

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))