Average Error: 29.9 → 0.6
Time: 2.3m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 29.9

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

  3. Applied add-cbrt-cube29.9

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

  5. Applied add-cube-cbrt30.0

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

  7. Applied flip3--29.9

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

  8. Taylor expanded around inf 0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))