Average Error: 30.2 → 0.6
Time: 11.1s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7506.02202390481 \lor \neg \left(x \le 3485.8013167965632\right):\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{-1}{9} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \frac{-4}{81} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right)}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -7506.02202390481 or 3485.8013167965632 < x

    1. Initial program 60.2

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

    2. Initial simplification60.2

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

    4. Applied flip--60.2

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

    5. Taylor expanded around 0 60.9

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

    6. Simplified60.2

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

    7. Taylor expanded around inf 33.4

      \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\]

    8. Simplified1.0

      \[\leadsto \frac{\color{blue}{\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{4}}} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{-4}{81}\right)}}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\]

    if -7506.02202390481 < x < 3485.8013167965632

    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-cbrt-cube0.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7506.02202390481 \lor \neg \left(x \le 3485.8013167965632\right):\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{-1}{9} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \frac{-4}{81} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right)}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\\ \end{array}\]

Runtime

Time bar (total: 11.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes30.20.60.529.899.6%
herbie shell --seed 2018286 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))