Average Error: 29.7 → 15.6
Time: 29.6s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3091.2076265776873:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 3746.908207933926:\\ \;\;\;\;(\left(\sqrt[3]{\frac{\sqrt[3]{(x \cdot x + -1)_*}}{\sqrt[3]{x - 1}} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3091.2076265776873 or 3746.908207933926 < x

    1. Initial program 59.9

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

    2. Taylor expanded around inf 39.6

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

    3. Simplified31.3

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

    if -3091.2076265776873 < x < 3746.908207933926

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    5. Applied fma-neg0.1

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

    7. Applied flip-+0.1

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

    8. Applied cbrt-div0.1

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

    9. Simplified0.1

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

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3091.2076265776873:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 3746.908207933926:\\ \;\;\;\;(\left(\sqrt[3]{\frac{\sqrt[3]{(x \cdot x + -1)_*}}{\sqrt[3]{x - 1}} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))