Average Error: 29.2 → 1.1
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 264.3767262197937:\\ \;\;\;\;\frac{\log \left(e^{2} - \left(1 + \frac{-2}{3} \cdot x\right) \cdot \left(e^{2} \cdot \left(x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 264.3767262197937

    1. Initial program 38.7

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.4

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]
    3. Using strategy rm

    4. Applied add-log-exp1.4

      \[\leadsto \frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - \color{blue}{\log \left(e^{{x}^{2}}\right)}}{2}\]

    5. Applied add-log-exp1.4

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{2}{3} \cdot {x}^{3} + 2}\right)} - \log \left(e^{{x}^{2}}\right)}{2}\]

    6. Applied diff-log1.4

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\frac{2}{3} \cdot {x}^{3} + 2}}{e^{{x}^{2}}}\right)}}{2}\]

    7. Taylor expanded around 0 1.4

      \[\leadsto \frac{\log \color{blue}{\left(\left(\frac{2}{3} \cdot \left(e^{2} \cdot {x}^{3}\right) + e^{2}\right) - e^{2} \cdot {x}^{2}\right)}}{2}\]

    8. Simplified1.4

      \[\leadsto \frac{\log \color{blue}{\left(e^{2} - \left(x \cdot \frac{-2}{3} + 1\right) \cdot \left(e^{2} \cdot \left(x \cdot x\right)\right)\right)}}{2}\]

    if 264.3767262197937 < x

    1. Initial program 0.2

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 264.3767262197937:\\ \;\;\;\;\frac{\log \left(e^{2} - \left(1 + \frac{-2}{3} \cdot x\right) \cdot \left(e^{2} \cdot \left(x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))