Average Error: 30.1 → 0.4
Time: 22.4s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4118.30332369231:\\ \;\;\;\;\frac{\frac{\frac{5}{81}}{x \cdot x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{elif}\;x \le 3814.846118973352:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4118.30332369231

    1. Initial program 60.2

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

    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

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

    5. Applied clear-num0.7

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

    7. Applied pow10.7

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

    8. Applied pow10.7

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

    9. Applied pow-prod-down0.7

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

    10. Simplified0.6

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

    if -4118.30332369231 < x < 3814.846118973352

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    if 3814.846118973352 < x

    1. Initial program 60.2

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

    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

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

    5. Applied div-inv0.7

      \[\leadsto \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4118.30332369231:\\ \;\;\;\;\frac{\frac{\frac{5}{81}}{x \cdot x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{elif}\;x \le 3814.846118973352:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed 2018219 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))