Average Error: 29.7 → 0.6
Time: 14.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right), \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right), \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}
double f(double x) {
        double r1139960 = x;
        double r1139961 = 1.0;
        double r1139962 = r1139960 + r1139961;
        double r1139963 = cbrt(r1139962);
        double r1139964 = cbrt(r1139960);
        double r1139965 = r1139963 - r1139964;
        return r1139965;
}
↓
double f(double x) {
        double r1139966 = 1.0;
        double r1139967 = x;
        double r1139968 = r1139967 + r1139966;
        double r1139969 = cbrt(r1139968);
        double r1139970 = cbrt(r1139969);
        double r1139971 = r1139970 * r1139970;
        double r1139972 = r1139970 * r1139971;
        double r1139973 = cbrt(r1139967);
        double r1139974 = r1139969 + r1139973;
        double r1139975 = r1139973 * r1139974;
        double r1139976 = fma(r1139969, r1139972, r1139975);
        double r1139977 = r1139966 / r1139976;
        return r1139977;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.7

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

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

  7. Applied add-cube-cbrt0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))