Average Error: 29.5 → 1.0
Time: 2.4m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 395.70848709210344:\\ \;\;\;\;\frac{\left(2 + {e}^{\left(\log \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right)\right)} \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {e}^{\left(-\left(1 - \varepsilon\right) \cdot x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

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 < 395.70848709210344

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cube-cbrt1.3

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

    5. Applied associate-*r*1.3

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

    6. Applied simplify1.3

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

    8. Applied add-exp-log1.3

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

    10. Applied pow11.3

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

    11. Applied log-pow1.3

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

    12. Applied exp-prod1.3

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

    13. Applied simplify1.3

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

    if 395.70848709210344 < x

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied exp-prod0.1

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

    5. Applied simplify0.1

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018198 
(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))