Average Error: 29.8 → 2.2
Time: 22.0s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4218.046012760297:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\\ \mathbf{elif}\;x \le 1.3310074091904414 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{1}{3} \cdot \log x - \frac{\frac{1}{3}}{x}\right) + \left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right)}\\ \end{array}\]

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 < 4218.046012760297

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-sqr-sqrt0.1

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

    if 4218.046012760297 < x < 1.3310074091904414e+154

    1. Initial program 59.7

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

    2. Initial simplification59.7

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

    4. Applied add-cube-cbrt59.3

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

    5. Applied cbrt-prod59.0

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

    7. Applied add-cube-cbrt58.9

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

    8. Applied cbrt-prod59.1

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

    9. Taylor expanded around inf 5.2

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    10. Simplified0.9

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{x \cdot x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}}\]

    if 1.3310074091904414e+154 < x

    1. Initial program 61.0

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

    2. Initial simplification61.0

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

    4. Applied add-cube-cbrt61.3

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

    5. Applied cbrt-prod61.5

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

    7. Applied add-cube-cbrt61.5

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

    8. Applied cbrt-prod61.4

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

    10. Applied add-exp-log61.4

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

    11. Taylor expanded around inf 6.9

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \left(\log \left(\frac{1}{x}\right) + \frac{7}{54} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{3} \cdot \frac{1}{x}}}\]

    12. Simplified7.6

      \[\leadsto e^{\color{blue}{\left(\log x \cdot \frac{1}{3} - \frac{\frac{1}{3}}{x}\right) + \left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4218.046012760297:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\\ \mathbf{elif}\;x \le 1.3310074091904414 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{1}{3} \cdot \log x - \frac{\frac{1}{3}}{x}\right) + \left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right)}\\ \end{array}\]

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed 2018277 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))