Average Error: 29.9 → 0.8
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \le 0.20405828448484847:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x}}{\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)} \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{x}\\ \mathbf{elif}\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \le 1.031334552932388:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot \left(\left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right)\right)}}{x}\\ \end{array}\]

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 (- (cbrt (+ 1 x)) (* (* (cbrt (cbrt x)) (cbrt (cbrt x))) (cbrt (cbrt x)))) < 0.20405828448484847

    1. Initial program 59.0

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

    2. Initial simplification59.0

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

    3. Taylor expanded around -inf 62.4

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

    4. Simplified1.1

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

    6. Applied associate-*r/1.1

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

    8. Applied flip-+1.1

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

    9. Applied associate-*l/1.1

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

    11. Applied *-un-lft-identity1.1

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

    12. Applied *-un-lft-identity1.1

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

    13. Applied distribute-lft-out--1.1

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

    14. Applied times-frac1.1

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

    15. Simplified1.1

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

    if 0.20405828448484847 < (- (cbrt (+ 1 x)) (* (* (cbrt (cbrt x)) (cbrt (cbrt x))) (cbrt (cbrt x)))) < 1.031334552932388

    1. Initial program 0.4

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

    2. Initial simplification0.4

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

    4. Applied add-cube-cbrt0.4

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

    if 1.031334552932388 < (- (cbrt (+ 1 x)) (* (* (cbrt (cbrt x)) (cbrt (cbrt x))) (cbrt (cbrt x))))

    1. Initial program 60.2

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

    2. Initial simplification60.2

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

    3. Taylor expanded around -inf 62.5

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

    4. Simplified1.5

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

    6. Applied associate-*r/1.5

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

    8. Applied add-cbrt-cube1.5

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

    9. Applied cbrt-unprod1.5

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \le 0.20405828448484847:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x}}{\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)} \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{x}\\ \mathbf{elif}\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \le 1.031334552932388:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot \left(\left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right)\right)}}{x}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))