Average Error: 29.8 → 15.4
Time: 25.1s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6270.688683920158:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 5370.558225894513:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -6270.688683920158 or 5370.558225894513 < 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. Taylor expanded around inf 38.5

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

    5. Simplified31.1

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

    if -6270.688683920158 < x < 5370.558225894513

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6270.688683920158:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 5370.558225894513:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))