Average Error: 30.2 → 0.5
Time: 19.7s
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 r104233 = x;
        double r104234 = 1.0;
        double r104235 = r104233 + r104234;
        double r104236 = cbrt(r104235);
        double r104237 = cbrt(r104233);
        double r104238 = r104236 - r104237;
        return r104238;
}
↓
double f(double x) {
        double r104239 = 1.0;
        double r104240 = x;
        double r104241 = r104240 + r104239;
        double r104242 = cbrt(r104241);
        double r104243 = cbrt(r104240);
        double r104244 = r104242 + r104243;
        double r104245 = r104243 * r104244;
        double r104246 = fma(r104242, r104242, r104245);
        double r104247 = r104239 / r104246;
        return r104247;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.2

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

  3. Applied add-cbrt-cube30.3

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

  4. Simplified30.3

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

  6. Applied flip3--30.2

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

  7. Applied cube-div30.2

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

  8. Simplified29.7

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

  9. Simplified29.7

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

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

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

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

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

  13. Applied distribute-lft-out--29.7

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

  14. Simplified16.0

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

  16. Applied cbrt-div16.0

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

  17. Simplified16.0

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

  18. Simplified0.5

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

  19. 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 2019196 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))