Average Error: 30.6 → 0.6
Time: 12.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\log \left(\frac{e^{1}}{1}\right)}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}, \sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{\log \left(\frac{e^{1}}{1}\right)}{\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 \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}, \sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)}
double f(double x) {
        double r290 = x;
        double r291 = 1.0;
        double r292 = r290 + r291;
        double r293 = cbrt(r292);
        double r294 = cbrt(r290);
        double r295 = r293 - r294;
        return r295;
}
↓
double f(double x) {
        double r296 = 1.0;
        double r297 = exp(r296);
        double r298 = 1.0;
        double r299 = r297 / r298;
        double r300 = log(r299);
        double r301 = x;
        double r302 = r301 + r296;
        double r303 = cbrt(r302);
        double r304 = cbrt(r301);
        double r305 = r304 * r304;
        double r306 = cbrt(r305);
        double r307 = cbrt(r304);
        double r308 = r306 * r307;
        double r309 = r304 * r308;
        double r310 = cbrt(r309);
        double r311 = fma(r310, r307, r303);
        double r312 = r307 * r311;
        double r313 = r306 * r312;
        double r314 = fma(r303, r303, r313);
        double r315 = r300 / r314;
        return r315;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.6

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

  3. Applied add-cube-cbrt30.6

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

  4. Applied cbrt-prod30.7

    \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}\]
  5. Using strategy rm

  6. Applied flip3--30.7

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

  7. Simplified29.8

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

  8. Simplified29.8

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

  10. Applied add-log-exp32.7

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

  11. Applied add-log-exp32.7

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

  12. Applied add-log-exp32.7

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

  13. Applied sum-log32.7

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

  14. Applied diff-log32.7

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

  15. Simplified0.6

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

  17. Applied add-cube-cbrt0.6

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

  18. Applied cbrt-prod0.6

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

  19. Final simplification0.6

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))