Average Error: 30.0 → 0.7
Time: 4.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\frac{1 \cdot \left(x + \left(x + 1\right)\right)}{\left(x + 1\right) + x}}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{\frac{1 \cdot \left(x + \left(x + 1\right)\right)}{\left(x + 1\right) + x}}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}
double f(double x) {
        double r76881 = x;
        double r76882 = 1.0;
        double r76883 = r76881 + r76882;
        double r76884 = cbrt(r76883);
        double r76885 = cbrt(r76881);
        double r76886 = r76884 - r76885;
        return r76886;
}
↓
double f(double x) {
        double r76887 = 1.0;
        double r76888 = x;
        double r76889 = r76888 + r76887;
        double r76890 = r76888 + r76889;
        double r76891 = r76887 * r76890;
        double r76892 = r76889 + r76888;
        double r76893 = r76891 / r76892;
        double r76894 = cbrt(r76888);
        double r76895 = r76894 * r76894;
        double r76896 = cbrt(r76895);
        double r76897 = cbrt(r76894);
        double r76898 = r76896 * r76897;
        double r76899 = cbrt(r76889);
        double r76900 = r76898 + r76899;
        double r76901 = r76898 * r76900;
        double r76902 = r76899 * r76899;
        double r76903 = r76901 + r76902;
        double r76904 = r76893 / r76903;
        return r76904;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.0

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

  3. Applied add-cube-cbrt30.1

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

  4. Applied cbrt-prod30.1

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

  6. Applied flip3--30.1

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

  7. Simplified29.4

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

  8. Simplified29.4

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

  10. Applied flip--30.7

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))