Average Error: 29.9 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4095.8758493053733:\\ \;\;\;\;\frac{\left(\left(\frac{1}{3} \cdot \frac{1}{3} - \frac{\frac{1}{9}}{x} \cdot \frac{\frac{1}{9}}{x}\right) \cdot x + \left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\frac{5}{81}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}{\frac{1}{9} + \frac{1}{3} \cdot x}\\ \mathbf{elif}\;x \le 4622.81409987891:\\ \;\;\;\;\frac{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right) \cdot \left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right)\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x}\right) - x \cdot \left(\left(\frac{1}{3} \cdot \frac{1}{3} - \frac{\frac{1}{9}}{x} \cdot \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{3} - \frac{\frac{1}{9}}{x} \cdot \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \frac{\sqrt[3]{x}}{x}}{\left(\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(x \cdot \left(\left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right) \cdot \left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right)\right)\right)}\\ \end{array}\]

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 < -4095.8758493053733

    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

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

    5. Applied flip--0.6

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

    6. Applied frac-add0.7

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

    7. Applied associate-*r/0.7

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

    8. Simplified0.7

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

    if -4095.8758493053733 < x < 4622.81409987891

    1. Initial program 0.1

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

    3. Applied flip--0.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}}\]

    if 4622.81409987891 < x

    1. Initial program 60.3

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

    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

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

    5. Applied flip-+0.6

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

    6. Applied associate-*r/0.7

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

    8. Applied flip--0.7

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

    9. Applied flip--0.7

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

    10. Applied frac-times0.7

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

    11. Applied associate-*r/0.7

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

    12. Applied frac-sub0.8

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

    13. Applied associate-*r/0.8

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

    14. Applied associate-/l/0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4095.8758493053733:\\ \;\;\;\;\frac{\left(\left(\frac{1}{3} \cdot \frac{1}{3} - \frac{\frac{1}{9}}{x} \cdot \frac{\frac{1}{9}}{x}\right) \cdot x + \left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\frac{5}{81}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}{\frac{1}{9} + \frac{1}{3} \cdot x}\\ \mathbf{elif}\;x \le 4622.81409987891:\\ \;\;\;\;\frac{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right) \cdot \left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right)\right) \cdot \left(\frac{\frac{5}{81}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x}\right) - x \cdot \left(\left(\frac{1}{3} \cdot \frac{1}{3} - \frac{\frac{1}{9}}{x} \cdot \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{3} - \frac{\frac{1}{9}}{x} \cdot \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \frac{\sqrt[3]{x}}{x}}{\left(\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(x \cdot \left(\left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right) \cdot \left(\frac{\frac{1}{9}}{x} + \frac{1}{3}\right)\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018215 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))