Average Error: 29.3 → 0.6
Time: 16.8s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3035.211292938797 \lor \neg \left(x \le 3365.594421354743\right):\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 - x \cdot x}}{\sqrt[3]{1 - x}} - \sqrt[3]{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3035.211292938797 or 3365.594421354743 < x

    1. Initial program 60.1

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

    2. Initial simplification60.1

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

    4. Applied add-cbrt-cube60.3

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

    6. Applied flip--60.2

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

    7. Taylor expanded around inf 33.6

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

    8. Simplified1.1

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

    if -3035.211292938797 < x < 3365.594421354743

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied flip-+0.1

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

    5. Applied cbrt-div0.1

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

    6. Simplified0.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3035.211292938797 \lor \neg \left(x \le 3365.594421354743\right):\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 - x \cdot x}}{\sqrt[3]{1 - x}} - \sqrt[3]{x}\\ \end{array}\]

Runtime

Time bar (total: 16.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.40.60.528.999.6%
herbie shell --seed 2018290 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))