Average Error: 28.8 → 0.7
Time: 18.6s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(\left(\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right)}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.8

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

  2. Initial simplification28.8

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

  4. Applied add-cube-cbrt28.8

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

  5. Applied cbrt-prod28.8

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

  7. Applied flip3--28.8

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

  8. Taylor expanded around 0 0.7

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

  10. Applied add-sqr-sqrt0.7

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

  11. Final simplification0.7

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

Runtime

Time bar (total: 18.6s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))