Average Error: 30.2 → 0.5
Time: 17.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \left({\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \left({\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}\right)}
double f(double x) {
        double r35527 = x;
        double r35528 = 1.0;
        double r35529 = r35527 + r35528;
        double r35530 = cbrt(r35529);
        double r35531 = cbrt(r35527);
        double r35532 = r35530 - r35531;
        return r35532;
}
↓
double f(double x) {
        double r35533 = 1.0;
        double r35534 = x;
        double r35535 = r35534 + r35533;
        double r35536 = cbrt(r35535);
        double r35537 = cbrt(r35534);
        double r35538 = r35536 + r35537;
        double r35539 = r35536 * r35538;
        double r35540 = fabs(r35537);
        double r35541 = sqrt(r35540);
        double r35542 = 0.75;
        double r35543 = pow(r35540, r35542);
        double r35544 = r35543 * r35543;
        double r35545 = r35541 * r35544;
        double r35546 = r35539 + r35545;
        double r35547 = r35533 / r35546;
        return r35547;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.2

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

  3. Applied flip3--30.2

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

  4. Simplified0.5

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

  5. Simplified33.7

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt33.7

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\sqrt{{x}^{\frac{2}{3}}} \cdot \sqrt{{x}^{\frac{2}{3}}}}}\]

  8. Simplified33.4

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\left|\sqrt[3]{x}\right|} \cdot \sqrt{{x}^{\frac{2}{3}}}}\]

  9. Simplified0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left|\sqrt[3]{x}\right| \cdot \color{blue}{\left|\sqrt[3]{x}\right|}}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt0.6

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

  12. Applied associate-*l*0.6

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

  13. Simplified0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \color{blue}{{\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{2}}}}\]
  14. Using strategy rm

  15. Applied sqr-pow0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \color{blue}{\left({\left(\left|\sqrt[3]{x}\right|\right)}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)}}\]

  16. Simplified0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \left(\color{blue}{{\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)}\]

  17. Simplified0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \left({\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}} \cdot \color{blue}{{\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}}\right)}\]

  18. Final simplification0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt{\left|\sqrt[3]{x}\right|} \cdot \left({\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}\right)}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))