Average Error: 40.2 → 0.3
Time: 2.0m
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot {\left(\frac{1}{2} \cdot x\right)}^{4} + \left(\left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) - \left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot \frac{1}{6}\right) + \left(1 - \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot \frac{1}{6}\right) + \left(1 - \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \frac{1}{6}\right) + \left(1 - \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right) - \left(x \cdot \left(x \cdot \frac{1}{4}\right)\right) \cdot \left(\left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) - \left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)\right)} \cdot \frac{\sqrt[3]{\left({\left({x}^{2} \cdot \frac{1}{6} + 1\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right)\right) \cdot \left({\left({x}^{2} \cdot \frac{1}{6} + 1\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}\right)\right)}}{\sqrt[3]{\left(\left({\left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3}\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right)\right)\right) \cdot \left({\left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3}\right)}}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 0.9877278429083478:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 1.753990269224867:\\ \;\;\;\;\frac{1}{2} \cdot x + \left({x}^{2} \cdot \frac{1}{6} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{e^{x} + -1}{x} \cdot \frac{e^{x} + -1}{x}\right) \cdot \frac{e^{x} + -1}{x}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.2
Target39.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (- (exp x) 1) x) < 0.0

    1. Initial program 62.0

      \[\frac{e^{x} - 1}{x}\]

    2. Initial simplification62.0

      \[\leadsto \frac{-1 + e^{x}}{x}\]

    3. Taylor expanded around 0 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right) \cdot \left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}\]
    6. Using strategy rm

    7. Applied flip3-+0

      \[\leadsto \sqrt[3]{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right) \cdot \color{blue}{\frac{{\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}}\]

    8. Applied flip3-+0

      \[\leadsto \sqrt[3]{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{\frac{{\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}\right) \cdot \frac{{\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}\]

    9. Applied flip-+0

      \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}{\frac{1}{2} \cdot x - \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}} \cdot \frac{{\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}\right) \cdot \frac{{\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}\]

    10. Applied frac-times0

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}{\left(\frac{1}{2} \cdot x - \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}} \cdot \frac{{\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}}{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}\]

    11. Applied frac-times0

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}{\left(\left(\frac{1}{2} \cdot x - \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}}\]

    12. Applied cbrt-div0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\left(\left(\frac{1}{2} \cdot x - \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}}\]
    13. Using strategy rm

    14. Applied flip3-+0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\left(\left(\frac{1}{2} \cdot x - \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) + \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right) \cdot \color{blue}{\frac{{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}}}\]

    15. Applied flip3-+0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\left(\left(\frac{1}{2} \cdot x - \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \color{blue}{\frac{{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}\right) \cdot \frac{{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}}\]

    16. Applied flip--0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\left(\color{blue}{\frac{\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}} \cdot \frac{{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}\right) \cdot \frac{{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}}\]

    17. Applied frac-times0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\color{blue}{\frac{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)}{\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)}} \cdot \frac{{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)}}}\]

    18. Applied frac-times0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\color{blue}{\frac{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)}{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)}}}}\]

    19. Applied cbrt-div0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\color{blue}{\frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)}}{\sqrt[3]{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)}}}}\]

    20. Applied associate-/r/0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)}} \cdot \sqrt[3]{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) + \left(\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) - \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right)\right)}}\]

    21. Simplified0

      \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{3} + {\left(\frac{1}{6} \cdot {x}^{2} + 1\right)}^{3}\right)}}{\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)\right) \cdot \left({\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{6} \cdot {x}^{2} + 1\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}^{3}\right)}} \cdot \color{blue}{\sqrt[3]{\left(\left(\left(\left(1 - \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right) \cdot \left(\left(\left(1 - \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right) - \left(x \cdot \left(x \cdot \frac{1}{4}\right)\right) \cdot \left(\left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right)\right) \cdot \left({\left(\frac{1}{2} \cdot x\right)}^{4} \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right) + \left(\left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right) - \left(\frac{1}{2} \cdot x\right) \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right) \cdot \left(\left(\left(\left(1 - \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right) \cdot \left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{6} \cdot x\right)\right)\right)\right)}}\]

    if 0.0 < (/ (- (exp x) 1) x) < 0.9877278429083478

    1. Initial program 0.3

      \[\frac{e^{x} - 1}{x}\]

    2. Initial simplification0.3

      \[\leadsto \frac{-1 + e^{x}}{x}\]

    3. Taylor expanded around inf 0.3

      \[\leadsto \frac{\color{blue}{e^{x} - 1}}{x}\]

    if 0.9877278429083478 < (/ (- (exp x) 1) x) < 1.753990269224867

    1. Initial program 23.3

      \[\frac{e^{x} - 1}{x}\]

    2. Initial simplification23.3

      \[\leadsto \frac{-1 + e^{x}}{x}\]

    3. Taylor expanded around 0 6.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]

    if 1.753990269224867 < (/ (- (exp x) 1) x)

    1. Initial program 0.3

      \[\frac{e^{x} - 1}{x}\]

    2. Initial simplification0.3

      \[\leadsto \frac{-1 + e^{x}}{x}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube12.5

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{-1 + e^{x}}{x} \cdot \frac{-1 + e^{x}}{x}\right) \cdot \frac{-1 + e^{x}}{x}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot {\left(\frac{1}{2} \cdot x\right)}^{4} + \left(\left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) - \left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot \frac{1}{6}\right) + \left(1 - \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot \frac{1}{6}\right) + \left(1 - \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \frac{1}{6}\right) + \left(1 - \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right)\right) - \left(x \cdot \left(x \cdot \frac{1}{4}\right)\right) \cdot \left(\left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \frac{1}{6}\right)\right) - \left(\left(\frac{1}{2} \cdot x + 1\right) + x \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)\right)} \cdot \frac{\sqrt[3]{\left({\left({x}^{2} \cdot \frac{1}{6} + 1\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}\right) \cdot \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right)\right) \cdot \left({\left({x}^{2} \cdot \frac{1}{6} + 1\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}\right)\right)}}{\sqrt[3]{\left(\left({\left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3}\right) \cdot \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right)\right)\right) \cdot \left({\left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left({x}^{2} \cdot \frac{1}{6} + 1\right) - \left({x}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3} + {\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}^{3}\right)}}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 0.9877278429083478:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 1.753990269224867:\\ \;\;\;\;\frac{1}{2} \cdot x + \left({x}^{2} \cdot \frac{1}{6} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{e^{x} + -1}{x} \cdot \frac{e^{x} + -1}{x}\right) \cdot \frac{e^{x} + -1}{x}}\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))