Average Error: 30.1 → 0.6
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{x} \cdot \left(\sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right) + \left(\sqrt[3]{x} \cdot \sqrt{\sqrt[3]{x}}\right) \cdot \sqrt{\sqrt[3]{x}}\right) + \left(\sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right)}\]

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.1

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

  3. Applied add-sqr-sqrt30.2

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

  5. Applied flip3--30.1

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

  6. Taylor expanded around inf 0.6

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

  8. Applied add-sqr-sqrt0.6

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

  9. Applied associate-*l*0.6

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

  10. Final simplification0.6

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

Runtime

Time bar (total: 18.8s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))