Average Error: 30.2 → 0.5
Time: 17.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left({\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}\right) \cdot \sqrt{\left|\sqrt[3]{x}\right|}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left({\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{\frac{3}{4}}\right) \cdot \sqrt{\left|\sqrt[3]{x}\right|}}
double f(double x) {
        double r35553 = x;
        double r35554 = 1.0;
        double r35555 = r35553 + r35554;
        double r35556 = cbrt(r35555);
        double r35557 = cbrt(r35553);
        double r35558 = r35556 - r35557;
        return r35558;
}
↓
double f(double x) {
        double r35559 = 1.0;
        double r35560 = x;
        double r35561 = r35560 + r35559;
        double r35562 = cbrt(r35561);
        double r35563 = cbrt(r35560);
        double r35564 = r35562 + r35563;
        double r35565 = r35562 * r35564;
        double r35566 = fabs(r35563);
        double r35567 = 0.75;
        double r35568 = pow(r35566, r35567);
        double r35569 = r35568 * r35568;
        double r35570 = sqrt(r35566);
        double r35571 = r35569 * r35570;
        double r35572 = r35565 + r35571;
        double r35573 = r35559 / r35572;
        return r35573;
}

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

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

  12. Applied associate-*r*0.6

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

  13. Simplified0.5

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

  15. Applied sqr-pow0.5

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

  16. Simplified0.5

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

  17. Simplified0.5

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

  18. Final simplification0.5

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

Reproduce

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