Average Error: 29.8 → 0.5
Time: 30.3s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{(\left(\sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \left(\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right)\right))_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 29.8

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

  3. Applied flip3--29.8

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

  4. Applied simplify0.5

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

  5. Applied simplify0.5

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied cbrt-prod0.5

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

Runtime

Time bar (total: 30.3s)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))