Average Error: 29.6 → 0.6
Time: 8.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}
double f(double x) {
        double r102712 = x;
        double r102713 = 1.0;
        double r102714 = r102712 + r102713;
        double r102715 = cbrt(r102714);
        double r102716 = cbrt(r102712);
        double r102717 = r102715 - r102716;
        return r102717;
}
↓
double f(double x) {
        double r102718 = 1.0;
        double r102719 = x;
        double r102720 = r102719 + r102718;
        double r102721 = cbrt(r102720);
        double r102722 = cbrt(r102719);
        double r102723 = r102722 * r102722;
        double r102724 = cbrt(r102723);
        double r102725 = cbrt(r102722);
        double r102726 = r102721 + r102722;
        double r102727 = r102725 * r102726;
        double r102728 = r102724 * r102727;
        double r102729 = fma(r102721, r102721, r102728);
        double r102730 = r102718 / r102729;
        return r102730;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.6

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

  3. Applied flip3--29.6

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

  4. Simplified29.0

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

  5. Simplified29.0

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

  6. Taylor expanded around 0 0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied cbrt-prod0.6

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

  10. Applied associate-*l*0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))