Average Error: 29.8 → 0.5
Time: 18.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r1924834 = x;
        double r1924835 = 1.0;
        double r1924836 = r1924834 + r1924835;
        double r1924837 = cbrt(r1924836);
        double r1924838 = cbrt(r1924834);
        double r1924839 = r1924837 - r1924838;
        return r1924839;
}
↓
double f(double x) {
        double r1924840 = 1.0;
        double r1924841 = x;
        double r1924842 = r1924841 + r1924840;
        double r1924843 = cbrt(r1924842);
        double r1924844 = cbrt(r1924841);
        double r1924845 = r1924843 + r1924844;
        double r1924846 = r1924844 * r1924845;
        double r1924847 = fma(r1924843, r1924843, r1924846);
        double r1924848 = r1924840 / r1924847;
        return r1924848;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.8

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

  3. Applied flip3--29.7

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

  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. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))