Average Error: 29.8 → 0.6
Time: 5.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\log \left(\frac{e^{1}}{1}\right)}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\]
\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 \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)
double f(double x) {
        double r100065 = x;
        double r100066 = 1.0;
        double r100067 = r100065 + r100066;
        double r100068 = cbrt(r100067);
        double r100069 = cbrt(r100065);
        double r100070 = r100068 - r100069;
        return r100070;
}
↓
double f(double x) {
        double r100071 = 1.0;
        double r100072 = exp(r100071);
        double r100073 = 1.0;
        double r100074 = r100072 / r100073;
        double r100075 = log(r100074);
        double r100076 = x;
        double r100077 = r100076 + r100071;
        double r100078 = cbrt(r100077);
        double r100079 = cbrt(r100076);
        double r100080 = cbrt(r100079);
        double r100081 = r100079 * r100079;
        double r100082 = cbrt(r100081);
        double r100083 = fma(r100080, r100082, r100078);
        double r100084 = r100082 * r100083;
        double r100085 = r100080 * r100084;
        double r100086 = fma(r100078, r100078, r100085);
        double r100087 = r100075 / r100086;
        double r100088 = -r100080;
        double r100089 = r100088 + r100080;
        double r100090 = r100082 * r100089;
        double r100091 = r100087 + r100090;
        return r100091;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.8

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

  3. Applied add-cube-cbrt29.9

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

  4. Applied cbrt-prod29.9

    \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}\]

  5. Applied *-un-lft-identity29.9

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

  6. Applied cbrt-prod29.9

    \[\leadsto \color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{x + 1}} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\]

  7. Applied prod-diff30.0

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

  8. Simplified30.0

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

  9. Simplified29.9

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

  11. Applied flip3--29.9

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

  12. Simplified29.2

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

  13. Simplified29.2

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

  15. Applied add-log-exp31.7

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

  16. Applied add-log-exp31.7

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

  17. Applied add-log-exp31.7

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

  18. Applied sum-log31.7

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

  19. Applied diff-log31.7

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

  20. Simplified0.6

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

  21. Final simplification0.6

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))