Average Error: 29.7 → 0.5
Time: 20.5s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -44948342240.46133 \lor \neg \left(x \le 13955.840503997135\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{1 + x}\right)}^{3} - x}{\left(\sqrt[3]{x} \cdot \sqrt[3]{1 + x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\\ \end{array}\]

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 2 regimes

  2. if x < -44948342240.46133 or 13955.840503997135 < x

    1. Initial program 60.7

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

    2. Initial simplification60.7

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

    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. 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 -44948342240.46133 < x < 13955.840503997135

    1. Initial program 0.5

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

    2. Initial simplification0.5

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

    4. Applied flip3--0.5

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

    6. Applied rem-cube-cbrt0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -44948342240.46133 \lor \neg \left(x \le 13955.840503997135\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{1 + x}\right)}^{3} - x}{\left(\sqrt[3]{x} \cdot \sqrt[3]{1 + x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\\ \end{array}\]

Runtime

Time bar (total: 20.5s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))