Average Error: 29.7 → 0.6
Time: 21.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3115.380144796638:\\ \;\;\;\;\frac{\sqrt[3]{(x \cdot \left(x \cdot x\right) + 1)_*}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\frac{2}{3} \cdot \sqrt[3]{\frac{1}{x}}\right))_*\right))_*}{\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}} + \sqrt[3]{x + 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 3115.380144796638

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied cbrt-div0.1

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

    5. Simplified0.1

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

    if 3115.380144796638 < x

    1. Initial program 60.3

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

    3. Applied flip--60.3

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

    4. Taylor expanded around inf 5.1

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

    5. Simplified1.0

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

    7. Applied add-sqr-sqrt1.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3115.380144796638:\\ \;\;\;\;\frac{\sqrt[3]{(x \cdot \left(x \cdot x\right) + 1)_*}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\frac{2}{3} \cdot \sqrt[3]{\frac{1}{x}}\right))_*\right))_*}{\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}} + \sqrt[3]{x + 1}}\\ \end{array}\]

Runtime

Time bar (total: 21.8s)Debug logProfile

herbie shell --seed 2018278 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))