Average Error: 29.9 → 16.0
Time: 23.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 8.420868482517108 \cdot 10^{-06}:\\ \;\;\;\;\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{1}{9}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (cbrt (+ x 1)) (cbrt x)) < 8.420868482517108e-06

    1. Initial program 60.6

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

    3. Applied add-cube-cbrt60.5

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

    4. Applied cbrt-prod60.5

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

    6. Applied add-cube-cbrt60.5

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

    7. Applied cbrt-prod60.4

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

    8. Taylor expanded around inf 34.2

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

    9. Simplified32.1

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

    if 8.420868482517108e-06 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.4

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 8.420868482517108 \cdot 10^{-06}:\\ \;\;\;\;\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{1}{9}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

Runtime

Time bar (total: 23.8s)Debug logProfile

herbie shell --seed 2018349 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))