\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Test:
NMSE Section 6.1 mentioned, A
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 30.9 s
Input Error: 20.2
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} 1 & \text{when } x \le 0.07415683f0 \\ \frac{\log \left(e^{\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}}\right)}{2} & \text{otherwise} \end{cases}\)

    if x < 0.07415683f0

    1. Started with
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
      23.4
    2. Applied taylor to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{2}{2}\]
      0
    3. Taylor expanded around 0 to get
      \[\frac{\color{red}{2}}{2} \leadsto \frac{\color{blue}{2}}{2}\]
      0
    4. Applied simplify to get
      \[\color{red}{\frac{2}{2}} \leadsto \color{blue}{1}\]
      0

    if 0.07415683f0 < x

    1. Started with
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
      1.9
    2. Using strategy rm
      1.9
    3. Applied add-log-exp to get
      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\log \left(e^{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\]
      2.2
    4. Applied add-log-exp to get
      \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \log \left(e^{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}{2} \leadsto \frac{\color{blue}{\log \left(e^{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}\right)} - \log \left(e^{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\]
      2.2
    5. Applied diff-log to get
      \[\frac{\color{red}{\log \left(e^{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}\right) - \log \left(e^{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}}{2} \leadsto \frac{\color{blue}{\log \left(\frac{e^{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}}{e^{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}\right)}}{2}\]
      2.1
    6. Applied simplify to get
      \[\frac{\log \color{red}{\left(\frac{e^{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}}{e^{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}\right)}}{2} \leadsto \frac{\log \color{blue}{\left(e^{\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}}\right)}}{2}\]
      2.0
  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))