Average Error: 29.5 → 1.0
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 156.96717607719006:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} + \left|x\right|} \cdot \sqrt[3]{\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} + \left|x\right|}\right) \cdot \left(\sqrt[3]{\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} + \left|x\right|} \cdot \left(\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} - \left|x\right|\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{x \cdot \varepsilon - x} + e^{-\left(x + x \cdot \varepsilon\right)}\right)\right) - \frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon}}{2}\\ \end{array}\]

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 < 156.96717607719006

    1. Initial program 39.2

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

    2. Taylor expanded around 0 1.2

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

    4. Applied add-log-exp1.2

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

    6. Applied add-sqr-sqrt1.2

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

    7. Applied add-sqr-sqrt2.2

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

    8. Applied difference-of-squares2.2

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

    9. Simplified2.2

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

    10. Simplified2.2

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

    12. Applied add-cube-cbrt2.2

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

    13. Applied associate-*l*1.3

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

    if 156.96717607719006 < 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. Taylor expanded around -inf 0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 156.96717607719006:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} + \left|x\right|} \cdot \sqrt[3]{\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} + \left|x\right|}\right) \cdot \left(\sqrt[3]{\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} + \left|x\right|} \cdot \left(\sqrt{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + 2} - \left|x\right|\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{x \cdot \varepsilon - x} + e^{-\left(x + x \cdot \varepsilon\right)}\right)\right) - \frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018278 
(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))