Average Error: 29.3 → 0.5
Time: 11.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{2}}
double f(double x) {
        double r80866 = x;
        double r80867 = 1.0;
        double r80868 = r80866 + r80867;
        double r80869 = cbrt(r80868);
        double r80870 = cbrt(r80866);
        double r80871 = r80869 - r80870;
        return r80871;
}
↓
double f(double x) {
        double r80872 = 1.0;
        double r80873 = x;
        double r80874 = r80873 + r80872;
        double r80875 = cbrt(r80874);
        double r80876 = -1.0;
        double r80877 = cbrt(r80876);
        double r80878 = -r80873;
        double r80879 = cbrt(r80878);
        double r80880 = r80877 * r80879;
        double r80881 = r80875 + r80880;
        double r80882 = r80875 * r80881;
        double r80883 = r80879 * r80879;
        double r80884 = 2.0;
        double r80885 = pow(r80877, r80884);
        double r80886 = r80883 * r80885;
        double r80887 = r80882 + r80886;
        double r80888 = r80872 / r80887;
        return r80888;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.3

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

  2. Taylor expanded around -inf 47.0

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

  3. Simplified29.3

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

  5. Applied flip3--29.2

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

  6. Simplified0.5

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\left(\sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\right)}\]

  7. Simplified32.9

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + {\left(-x\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{-1}\right)}^{2}}}\]
  8. Using strategy rm

  9. Applied sqr-pow32.9

    \[\leadsto \frac{1 + 0}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \color{blue}{\left({\left(-x\right)}^{\left(\frac{\frac{2}{3}}{2}\right)} \cdot {\left(-x\right)}^{\left(\frac{\frac{2}{3}}{2}\right)}\right)} \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\]

  10. Simplified32.6

    \[\leadsto \frac{1 + 0}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\color{blue}{\sqrt[3]{-x}} \cdot {\left(-x\right)}^{\left(\frac{\frac{2}{3}}{2}\right)}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\]

  11. Simplified0.5

    \[\leadsto \frac{1 + 0}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\sqrt[3]{-x} \cdot \color{blue}{\sqrt[3]{-x}}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{2}}\]

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019350 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))