Average Error: 29.3 → 15.2
Time: 27.3s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3702.482272871828:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 4010.44456416471:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3702.482272871828 or 4010.44456416471 < 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 add-cube-cbrt60.2

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

    6. Applied cbrt-prod60.3

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

    7. Applied associate-*l*60.3

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

    9. Applied add-cube-cbrt60.5

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

    10. Taylor expanded around inf 39.2

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    11. Simplified31.2

      \[\leadsto \color{blue}{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*}\]

    if -3702.482272871828 < x < 4010.44456416471

    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 cbrt-prod0.1

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]

    6. Applied fma-neg0.1

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3702.482272871828:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 4010.44456416471:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))