\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 37.0 s
Input Error: 17.0
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{(\left({x}^2\right) * 3 + \left((x * 4 + 2)_*\right))_*}{e^{(x * 1 + x)_*} \cdot 2} & \text{when } x \le 30.93942f0 \\ \left(\frac{1 + \frac{1}{\varepsilon}}{2} \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \frac{e^{-(\varepsilon * x + x)_*}}{2 \cdot \varepsilon}\right) + \frac{e^{-(\varepsilon * x + x)_*}}{2} & \text{otherwise} \end{cases}\)

    if x < 30.93942f0

    1. Started with
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
      20.7
    2. Using strategy rm
      20.7
    3. Applied exp-neg to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      20.6
    4. Applied un-div-inv to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      20.6
    5. Applied exp-neg to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
      20.6
    6. Applied un-div-inv to get
      \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
      20.7
    7. Applied frac-sub to get
      \[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
      20.7
    8. Applied simplify to get
      \[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{red}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{blue}{e^{(x * \left(1 + 0\right) + x)_*}}}}{2}\]
      20.7
    9. Applied taylor to get
      \[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{(x * \left(1 + 0\right) + x)_*}}}{2} \leadsto \frac{\frac{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}{e^{(x * \left(1 + 0\right) + x)_*}}}{2}\]
      0.1
    10. Taylor expanded around 0 to get
      \[\frac{\frac{\color{red}{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}}{e^{(x * \left(1 + 0\right) + x)_*}}}{2} \leadsto \frac{\frac{\color{blue}{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}}{e^{(x * \left(1 + 0\right) + x)_*}}}{2}\]
      0.1
    11. Applied simplify to get
      \[\frac{\frac{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}{e^{(x * \left(1 + 0\right) + x)_*}}}{2} \leadsto \frac{(\left(x \cdot x\right) * 3 + \left((x * 4 + 2)_*\right))_*}{2 \cdot e^{(x * 1 + x)_*}}\]
      0.1
    12. Applied final simplification
    13. Applied simplify to get
      \[\color{red}{\frac{(\left(x \cdot x\right) * 3 + \left((x * 4 + 2)_*\right))_*}{2 \cdot e^{(x * 1 + x)_*}}} \leadsto \color{blue}{\frac{(\left({x}^2\right) * 3 + \left((x * 4 + 2)_*\right))_*}{e^{(x * 1 + x)_*} \cdot 2}}\]
      0.1

    if 30.93942f0 < x

    1. Started with
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
      0.4
    2. Using strategy rm
      0.4
    3. Applied add-sqr-sqrt to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{{\left(\sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}^2}}{2}\]
      0.5
    4. Applied simplify to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{red}{\left(\sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}}^2}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{blue}{\left(\sqrt{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}}\right)}}^2}{2}\]
      0.5
    5. Applied taylor to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}}\right)}^2}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt{\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}}\right)}^2}{2}\]
      0.5
    6. Taylor expanded around 0 to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt{\color{red}{\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}}}\right)}^2}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt{\color{blue}{\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}}}\right)}^2}{2}\]
      0.5
    7. Applied simplify to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\sqrt{\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}}\right)}^2}{2} \leadsto \left(\frac{1 + \frac{1}{\varepsilon}}{2} \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \frac{e^{-(\varepsilon * x + x)_*}}{2 \cdot \varepsilon}\right) + \frac{e^{-(\varepsilon * x + x)_*}}{2}\]
      0.4
    8. Applied final simplification
  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))