Average Error: 30.2 → 0.6
Time: 15.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}}\right)\right), \left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\right)}\]
\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 \sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}}\right)\right), \left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r633810 = x;
        double r633811 = 1.0;
        double r633812 = r633810 + r633811;
        double r633813 = cbrt(r633812);
        double r633814 = cbrt(r633810);
        double r633815 = r633813 - r633814;
        return r633815;
}
↓
double f(double x) {
        double r633816 = 1.0;
        double r633817 = x;
        double r633818 = r633817 + r633816;
        double r633819 = cbrt(r633818);
        double r633820 = r633819 * r633819;
        double r633821 = cbrt(r633820);
        double r633822 = cbrt(r633819);
        double r633823 = cbrt(r633822);
        double r633824 = r633822 * r633822;
        double r633825 = cbrt(r633824);
        double r633826 = r633823 * r633825;
        double r633827 = r633821 * r633826;
        double r633828 = cbrt(r633817);
        double r633829 = r633819 + r633828;
        double r633830 = r633829 * r633828;
        double r633831 = fma(r633819, r633827, r633830);
        double r633832 = r633816 / r633831;
        return r633832;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.2

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

  3. Applied flip3--30.1

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

  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-cbrt0.5

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

  8. Applied cbrt-prod0.6

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

  10. Applied add-cube-cbrt0.6

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

  11. Applied cbrt-prod0.6

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

  12. Applied cbrt-prod0.6

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

  14. Applied cbrt-prod0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))