Average Error: 29.5 → 0.4
Time: 14.9s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -87243.74833790654:\\ \;\;\;\;\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{elif}\;x \le 79636.39027823936:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{x \cdot x - 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}{\sqrt[3]{\sqrt[3]{x - 1}}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -87243.74833790654 or 79636.39027823936 < x

    1. Initial program 60.3

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

    2. Taylor expanded around -inf 62.4

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

    3. Simplified0.7

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

    if -87243.74833790654 < x < 79636.39027823936

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    5. Applied flip-+0.2

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

    6. Applied cbrt-div0.2

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

    7. Applied associate-*r/0.2

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

    8. Applied cbrt-div0.2

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

    9. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -87243.74833790654:\\ \;\;\;\;\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{elif}\;x \le 79636.39027823936:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{x \cdot x - 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}{\sqrt[3]{\sqrt[3]{x - 1}}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))