Average Error: 30.5 → 0.4
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4156.062619082609:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)\right) \cdot x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 4082.7772446652702:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{\frac{\frac{5}{81}}{x} + \left(\frac{1}{9} - x \cdot \frac{1}{3}\right)}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4156.062619082609

    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

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

    6. Applied associate-*r*0.7

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

    8. Applied add-cbrt-cube0.7

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

    9. Applied cbrt-unprod0.6

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

    if -4156.062619082609 < x < 4082.7772446652702

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied cbrt-div0.1

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

    5. Simplified0.1

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

    if 4082.7772446652702 < 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 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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4156.062619082609:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)\right) \cdot x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 4082.7772446652702:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{\frac{\frac{5}{81}}{x} + \left(\frac{1}{9} - x \cdot \frac{1}{3}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018217 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))