Average Error: 28.7 → 1.1
Time: 2.0m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 247.73968402852063:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}\right) + \left(\log \left(\sqrt[3]{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}} \cdot \sqrt[3]{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}\right) + \log \left(\sqrt[3]{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}{2}\\ \end{array}\]

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 < 247.73968402852063

    1. Initial program 38.6

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-sqr-sqrt2.3

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

    6. Applied add-exp-log2.3

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

    7. Applied add-exp-log2.3

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

    8. Applied prod-exp1.4

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

    10. Applied add-cube-cbrt1.4

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

    11. Applied log-prod1.4

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

    if 247.73968402852063 < 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 add-cbrt-cube0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 247.73968402852063:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}\right) + \left(\log \left(\sqrt[3]{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}} \cdot \sqrt[3]{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}\right) + \log \left(\sqrt[3]{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018220 
(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))