Average Error: 29.4 → 0.6
Time: 14.2s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3617.7226240680843:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{\sqrt[3]{1 + x}} \cdot \sqrt{\sqrt[3]{1 + x}}\right) \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{-4}{81}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{4}}}}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \end{array}\]

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 < 3617.7226240680843

    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-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. Using strategy rm

    7. Applied add-sqr-sqrt0.1

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

    if 3617.7226240680843 < x

    1. Initial program 60.0

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

    2. Initial simplification60.0

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

    4. Applied add-cbrt-cube60.0

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

    6. Applied flip--60.0

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

    7. Taylor expanded around inf 5.1

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

    8. Simplified1.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3617.7226240680843:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{\sqrt[3]{1 + x}} \cdot \sqrt{\sqrt[3]{1 + x}}\right) \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{-4}{81}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{4}}}}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 14.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.40.60.528.999.6%
herbie shell --seed 2018355 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))