Average Error: 29.3 → 0.6
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3003.6159602180965:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \mathbf{elif}\;x \le 3921.2107125484745:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3003.6159602180965 or 3921.2107125484745 < x

    1. Initial program 60.2

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

    3. Applied add-cbrt-cube60.3

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

    5. Applied flip--60.3

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

    6. Taylor expanded around inf 34.4

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

    7. Simplified1.1

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

    if -3003.6159602180965 < x < 3921.2107125484745

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    5. Applied flip--0.2

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

    7. Applied add-cube-cbrt0.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3003.6159602180965:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \mathbf{elif}\;x \le 3921.2107125484745:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))