Average Error: 29.4 → 0.4
Time: 2.0m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 2.3940835703228913 \cdot 10^{-05}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot \left(\left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array}\]

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)) < 2.3940835703228913e-05

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 62.4

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

    3. Simplified0.6

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

    4. Taylor expanded around 0 33.5

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

    5. Simplified0.6

      \[\leadsto \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \frac{\color{blue}{\sqrt[3]{x}}}{x}\]
    6. Using strategy rm

    7. Applied associate-*r/0.6

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

    9. Applied add-cbrt-cube0.6

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

    10. Applied cbrt-unprod0.6

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

    if 2.3940835703228913e-05 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.2

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

    2. Taylor expanded around 0 31.0

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

    3. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 2.3940835703228913 \cdot 10^{-05}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot \left(\left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018251 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))