Average Error: 30.2 → 0.5
Time: 3.3m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0159564017437261:\\ \;\;\;\;\frac{\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\left(\sqrt[3]{\frac{1}{9} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)} \cdot x\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{81}}{x \cdot x} + \left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right)} \cdot \sqrt[3]{\frac{\frac{1}{81}}{x \cdot x} + \left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right)}\right)}\\ \mathbf{elif}\;x \le 3844.8098023854254:\\ \;\;\;\;{\left(1 + x\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{x}{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\frac{1}{729} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) + \left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\right) \cdot \left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{\frac{1}{729}}{x}}{x \cdot x} + \frac{1}{27}\right)\right)}{\left(\left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) \cdot \left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right)\right) \cdot \left(\left(\frac{1}{9} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x\right)}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0159564017437261

    1. Initial program 59.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

    2. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(\frac{5}{81} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{3}} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}}\]

    3. Simplified1.1

      \[\leadsto \color{blue}{\left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}\]
    4. Using strategy rm

    5. Applied flip3-+1.6

      \[\leadsto \left(\color{blue}{\frac{{\frac{1}{3}}^{3} + {\left(\frac{\frac{-1}{9}}{x}\right)}^{3}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)}} + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}\]

    6. Applied frac-add1.5

      \[\leadsto \color{blue}{\frac{\left({\frac{1}{3}}^{3} + {\left(\frac{\frac{-1}{9}}{x}\right)}^{3}\right) \cdot x + \left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot \frac{\frac{5}{81}}{x}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}} \cdot \frac{\sqrt[3]{x}}{x}\]

    7. Applied associate-*l/1.5

      \[\leadsto \color{blue}{\frac{\left(\left({\frac{1}{3}}^{3} + {\left(\frac{\frac{-1}{9}}{x}\right)}^{3}\right) \cdot x + \left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot \frac{\frac{5}{81}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}}\]

    8. Simplified1.2

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt1.2

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\color{blue}{\left(\left(\sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)} \cdot \sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)}\right) \cdot \sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)}\right)} \cdot x}\]

    11. Applied associate-*l*1.3

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\color{blue}{\left(\sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)} \cdot \sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)}\right) \cdot \left(\sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)} \cdot x\right)}}\]

    12. Simplified1.3

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\color{blue}{\left(\sqrt[3]{\frac{\frac{1}{81}}{x \cdot x} + \left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right)} \cdot \sqrt[3]{\frac{\frac{1}{81}}{x \cdot x} + \left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right)}\right)} \cdot \left(\sqrt[3]{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)} \cdot x\right)}\]

    if -1.0159564017437261 < x < 3844.8098023854254

    1. Initial program 0.1

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
    2. Using strategy rm

    3. Applied pow1/30.1

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

    if 3844.8098023854254 < x

    1. Initial program 60.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

    2. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(\frac{5}{81} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{3}} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}\]
    4. Using strategy rm

    5. Applied flip3-+1.1

      \[\leadsto \left(\color{blue}{\frac{{\frac{1}{3}}^{3} + {\left(\frac{\frac{-1}{9}}{x}\right)}^{3}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)}} + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}\]

    6. Applied frac-add1.0

      \[\leadsto \color{blue}{\frac{\left({\frac{1}{3}}^{3} + {\left(\frac{\frac{-1}{9}}{x}\right)}^{3}\right) \cdot x + \left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot \frac{\frac{5}{81}}{x}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}} \cdot \frac{\sqrt[3]{x}}{x}\]

    7. Applied associate-*l/1.0

      \[\leadsto \color{blue}{\frac{\left(\left({\frac{1}{3}}^{3} + {\left(\frac{\frac{-1}{9}}{x}\right)}^{3}\right) \cdot x + \left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot \frac{\frac{5}{81}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}}\]

    8. Simplified0.7

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]
    9. Using strategy rm

    10. Applied flip-+0.7

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \color{blue}{\frac{\frac{1}{27} \cdot \frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}}{\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}}}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]

    11. Applied associate-*r/0.8

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \color{blue}{\frac{\left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} \cdot \frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}}}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]

    12. Applied flip-+0.8

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) \cdot \left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x} \cdot \frac{\frac{\frac{1}{81}}{x}}{x}}{\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}}} \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \frac{\left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} \cdot \frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]

    13. Applied associate-*l/0.8

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) \cdot \left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x} \cdot \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)}{\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}}} + \frac{\left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} \cdot \frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]

    14. Applied frac-add0.8

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) \cdot \left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x} \cdot \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\right) \cdot \left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) + \left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} \cdot \frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)\right)}{\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}}}{\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x}\]

    15. Applied associate-/l/0.8

      \[\leadsto \color{blue}{\frac{\left(\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) \cdot \left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x} \cdot \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\right) \cdot \left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) + \left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} \cdot \frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)\right)}{\left(\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x\right) \cdot \left(\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)\right)}}\]

    16. Simplified0.8

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{27} + \frac{\frac{\frac{1}{729}}{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{\frac{1}{81}}{x}}{x} + \left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right)\right)\right) \cdot \left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\right) + \left(\frac{1}{729} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) \cdot \left(\left(\frac{x}{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right)\right)}}{\left(\left(\frac{1}{3} \cdot \frac{1}{3} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x\right) \cdot \left(\left(\left(\frac{\frac{1}{27}}{x} + \frac{1}{9}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0159564017437261:\\ \;\;\;\;\frac{\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(x \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{1}{27} + \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right)}{\left(\sqrt[3]{\frac{1}{9} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)} \cdot x\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{81}}{x \cdot x} + \left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right)} \cdot \sqrt[3]{\frac{\frac{1}{81}}{x \cdot x} + \left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right)}\right)}\\ \mathbf{elif}\;x \le 3844.8098023854254:\\ \;\;\;\;{\left(1 + x\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{x}{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\frac{1}{729} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x} \cdot \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) + \left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\right) \cdot \left(\left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) + \frac{\frac{\frac{1}{81}}{x}}{x}\right) \cdot \left(\frac{\frac{\frac{1}{729}}{x}}{x \cdot x} + \frac{1}{27}\right)\right)}{\left(\left(\frac{1}{27} - \frac{\frac{\frac{-1}{729}}{x}}{x \cdot x}\right) \cdot \left(\left(\frac{1}{9} + \frac{\frac{1}{27}}{x}\right) - \frac{\frac{\frac{1}{81}}{x}}{x}\right)\right) \cdot \left(\left(\frac{1}{9} + \left(\frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x} - \frac{1}{3} \cdot \frac{\frac{-1}{9}}{x}\right)\right) \cdot x\right)}\\ \end{array}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018254 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))