Average Error: 30.1 → 0.4
Time: 24.1s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4915.556106227908 \lor \neg \left(x \le 3851.744190566523\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{{x}^{3} + 1}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}}\right)\\ \end{array}\]

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 < -4915.556106227908 or 3851.744190566523 < x

    1. Initial program 60.0

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

    3. Applied add-sqr-sqrt61.3

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

    4. Taylor expanded around -inf 62.4

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

    5. Simplified0.6

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

    if -4915.556106227908 < x < 3851.744190566523

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)}\]
    4. Using strategy rm

    5. Applied flip3-+0.1

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

    6. Applied cbrt-div0.1

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

    7. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4915.556106227908 \lor \neg \left(x \le 3851.744190566523\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{{x}^{3} + 1}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}}\right)\\ \end{array}\]

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))