Average Error: 28.7 → 1.1
Time: 6.3m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 13.028531706040434:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}{e^{{x}^{2}}}\right) + \log \left(\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}} \cdot \left(\frac{1}{\varepsilon} + 1\right)\right) \cdot \sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}\\ \end{array}\]

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 < 13.028531706040434

    1. Initial program 38.4

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

    2. Taylor expanded around 0 1.2

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

    4. Applied add-log-exp1.2

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

    5. Applied add-log-exp1.2

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

    6. Applied diff-log1.2

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

    8. Applied *-un-lft-identity1.2

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

    9. Applied add-cube-cbrt1.2

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

    10. Applied times-frac1.2

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

    11. Applied log-prod1.2

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

    if 13.028531706040434 < x

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt0.6

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

    4. Applied associate-*r*0.6

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

    5. Applied simplify0.6

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

  4. Applied simplify1.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 13.028531706040434:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}{e^{{x}^{2}}}\right) + \log \left(\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}} \cdot \left(\frac{1}{\varepsilon} + 1\right)\right) \cdot \sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed 2018167 
(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))