Average Error: 29.8 → 0.9
Time: 15.7s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.7610578009688743:\\ \;\;\;\;x \cdot \left(\frac{1}{3} - \frac{1}{9} \cdot x\right) - \left(-1 + \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{4}}} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \frac{-4}{81} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right)}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 0.7610578009688743

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    3. Taylor expanded around 0 0.4

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

    4. Simplified0.4

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

    if 0.7610578009688743 < x

    1. Initial program 59.5

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

    2. Initial simplification59.5

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

    4. Applied add-cbrt-cube59.5

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

    6. Applied flip--59.5

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

    7. Taylor expanded around inf 5.3

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

    8. Simplified1.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.7610578009688743:\\ \;\;\;\;x \cdot \left(\frac{1}{3} - \frac{1}{9} \cdot x\right) - \left(-1 + \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{4}}} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3} - \frac{-4}{81} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right)}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 15.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.80.90.529.498.5%
herbie shell --seed 2018351 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))