\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 34.0 s
Input Error: 18.4
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2} & \text{when } x \le 10.704295f0 \\ \frac{\left(\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}}\right) + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} & \text{otherwise} \end{cases}\)

    if x < 10.704295f0

    1. Started with
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
      22.0
    2. Using strategy rm
      22.0
    3. Applied exp-neg to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      22.0
    4. Applied un-div-inv to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      22.0
    5. Using strategy rm
      22.0
    6. Applied div-sub to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}} - \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\]
      21.4
    7. Applied associate--r- to get
      \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}} - \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}}{2} \leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}\right) + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      20.1
    8. Applied simplify to get
      \[\frac{\color{red}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)} + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\left(\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}}\right)} + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
      19.4
    9. Applied taylor to get
      \[\frac{\left(\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}}\right) + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
      0.1
    10. Taylor expanded around 0 to get
      \[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
      0.1
    11. Applied simplify to get
      \[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
      0.1
    12. Applied final simplification
    13. Applied simplify to get
      \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}}\]
      0.1

    if 10.704295f0 < x

    1. Started with
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
      1.0
    2. Using strategy rm
      1.0
    3. Applied exp-neg to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      1.0
    4. Applied un-div-inv to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      1.0
    5. Using strategy rm
      1.0
    6. Applied div-sub to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}} - \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\]
      1.0
    7. Applied associate--r- to get
      \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}} - \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}}{2} \leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}\right) + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      1.0
    8. Applied simplify to get
      \[\frac{\color{red}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)} + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\left(\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}}\right)} + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
      0.9
  1. Removed slow pow expressions

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))