Average Error: 29.6 → 15.4
Time: 6.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{3}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{3}}
double f(double x) {
        double r135821 = x;
        double r135822 = 1.0;
        double r135823 = r135821 + r135822;
        double r135824 = cbrt(r135823);
        double r135825 = cbrt(r135821);
        double r135826 = r135824 - r135825;
        return r135826;
}
↓
double f(double x) {
        double r135827 = 1.0;
        double r135828 = x;
        double r135829 = r135828 + r135827;
        double r135830 = cbrt(r135829);
        double r135831 = cbrt(r135828);
        double r135832 = cbrt(r135831);
        double r135833 = r135831 * r135831;
        double r135834 = cbrt(r135833);
        double r135835 = fma(r135832, r135834, r135830);
        double r135836 = r135834 * r135835;
        double r135837 = r135832 * r135836;
        double r135838 = fma(r135830, r135830, r135837);
        double r135839 = r135827 / r135838;
        double r135840 = -r135832;
        double r135841 = r135840 + r135832;
        double r135842 = r135834 * r135841;
        double r135843 = r135839 + r135842;
        double r135844 = 3.0;
        double r135845 = pow(r135843, r135844);
        double r135846 = cbrt(r135845);
        return r135846;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.6

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

  3. Applied add-cbrt-cube29.6

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

  4. Simplified29.6

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

  6. Applied add-cube-cbrt29.7

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

  7. Applied cbrt-prod29.7

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

  8. Applied *-un-lft-identity29.7

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

  9. Applied cbrt-prod29.7

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

  10. Applied prod-diff29.8

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

  11. Simplified29.8

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

  12. Simplified29.7

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

  14. Applied flip3--29.7

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

  15. Simplified28.9

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

  16. Simplified28.9

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

  17. Taylor expanded around 0 15.4

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

  18. Final simplification15.4

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))