Average Error: 29.5 → 0.5
Time: 17.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{1}}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{1}}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)\right)}
double f(double x) {
        double r92339 = x;
        double r92340 = 1.0;
        double r92341 = r92339 + r92340;
        double r92342 = cbrt(r92341);
        double r92343 = cbrt(r92339);
        double r92344 = r92342 - r92343;
        return r92344;
}
↓
double f(double x) {
        double r92345 = 1.0;
        double r92346 = cbrt(r92345);
        double r92347 = r92346 * r92346;
        double r92348 = x;
        double r92349 = r92348 + r92345;
        double r92350 = cbrt(r92349);
        double r92351 = cbrt(r92348);
        double r92352 = r92351 + r92350;
        double r92353 = r92351 * r92352;
        double r92354 = fma(r92350, r92350, r92353);
        double r92355 = r92346 / r92354;
        double r92356 = r92347 * r92355;
        return r92356;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.5

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

  3. Applied add-exp-log29.5

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

  5. Applied flip3--29.5

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

  6. Applied log-div29.5

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

  7. Simplified28.8

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

  8. Simplified28.8

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

  9. Taylor expanded around 0 2.6

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

  11. Applied add-cube-cbrt2.6

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

  12. Applied log-prod2.6

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

  13. Applied associate--l+2.6

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

  14. Applied exp-sum2.6

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

  15. Simplified2.6

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

  16. Simplified0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))