Average Error: 30.0 → 0.4
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5029.268298876852:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\\ \mathbf{elif}\;x \le 7250.960014455874:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{\frac{5}{81}}{{x}^{3} \cdot \frac{x}{\frac{5}{81}}} + \left(\frac{\frac{1}{9}}{x} - \frac{1}{3}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{\frac{\frac{5}{81}}{x} + \left(\frac{1}{9} - x \cdot \frac{1}{3}\right)}\\ \end{array}\]

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 < -5029.268298876852

    1. Initial program 60.1

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

    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

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

    if -5029.268298876852 < x < 7250.960014455874

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    if 7250.960014455874 < x

    1. Initial program 60.3

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

    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 flip-+0.6

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

    6. Applied frac-times0.7

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

    7. Simplified0.7

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

    8. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5029.268298876852:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\\ \mathbf{elif}\;x \le 7250.960014455874:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{\frac{5}{81}}{{x}^{3} \cdot \frac{x}{\frac{5}{81}}} + \left(\frac{\frac{1}{9}}{x} - \frac{1}{3}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{\frac{\frac{5}{81}}{x} + \left(\frac{1}{9} - x \cdot \frac{1}{3}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018221 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))