Average Error: 29.3 → 3.4
Time: 18.5s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.8176831382692217:\\ \;\;\;\;x \cdot \left(\frac{1}{3} - \frac{1}{9} \cdot x\right) - \left(-1 + \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right) + \left(\log x \cdot \frac{1}{3} - \frac{\frac{1}{3}}{x}\right)}\\ \end{array}\]

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 2 regimes

  2. if x < 0.8176831382692217

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied cbrt-prod0.1

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

    6. Taylor expanded around 0 0.4

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

    7. Simplified0.4

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

    if 0.8176831382692217 < x

    1. Initial program 59.5

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

    2. Initial simplification59.5

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

    4. Applied add-cube-cbrt59.5

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

    5. Applied cbrt-prod59.5

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

    7. Applied add-cube-cbrt59.5

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

    9. Applied add-exp-log59.4

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

    10. Taylor expanded around inf 6.3

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

    11. Simplified6.6

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.8176831382692217:\\ \;\;\;\;x \cdot \left(\frac{1}{3} - \frac{1}{9} \cdot x\right) - \left(-1 + \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right) + \left(\log x \cdot \frac{1}{3} - \frac{\frac{1}{3}}{x}\right)}\\ \end{array}\]

Runtime

Time bar (total: 18.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.43.42.227.395.3%
herbie shell --seed 2018354 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))