Average Error: 29.5 → 0.4
Time: 20.6s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4205.236796500165:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 4932.918022075646:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3}}{x} \cdot \sqrt[3]{x} + \left(\frac{\frac{5}{27}}{x} - \frac{1}{3}\right) \cdot \left(\frac{\frac{1}{3}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\\ \end{array}\]

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 < -4205.236796500165

    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

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

    if -4205.236796500165 < x < 4932.918022075646

    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

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

    6. Applied add-cube-cbrt0.1

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

    if 4932.918022075646 < x

    1. Initial program 60.1

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

    3. Applied add-exp-log60.1

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

    4. Taylor expanded around inf 5.9

      \[\leadsto \color{blue}{\left(\frac{5}{27} \cdot \frac{e^{\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \log \left(\frac{1}{x}\right)}}{{x}^{2}} + e^{\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \log \left(\frac{1}{x}\right)}\right) - \frac{1}{3} \cdot \frac{e^{\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \log \left(\frac{1}{x}\right)}}{x}}\]

    5. Simplified0.6

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{3}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right) \cdot \left(\frac{\frac{5}{27}}{x} - \frac{1}{3}\right) + \frac{\frac{1}{3}}{x} \cdot \sqrt[3]{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4205.236796500165:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 4932.918022075646:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3}}{x} \cdot \sqrt[3]{x} + \left(\frac{\frac{5}{27}}{x} - \frac{1}{3}\right) \cdot \left(\frac{\frac{1}{3}}{x} \cdot \frac{\sqrt[3]{x}}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 20.6s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))