Average Error: 29.7 → 0.7
Time: 5.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\frac{\left(0 + 1\right) \cdot \left(\left(x + 1\right) + x\right)}{\mathsf{fma}\left(x, 2, 1\right)}}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{\frac{\left(0 + 1\right) \cdot \left(\left(x + 1\right) + x\right)}{\mathsf{fma}\left(x, 2, 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 r53091 = x;
        double r53092 = 1.0;
        double r53093 = r53091 + r53092;
        double r53094 = cbrt(r53093);
        double r53095 = cbrt(r53091);
        double r53096 = r53094 - r53095;
        return r53096;
}
↓
double f(double x) {
        double r53097 = 0.0;
        double r53098 = 1.0;
        double r53099 = r53097 + r53098;
        double r53100 = x;
        double r53101 = r53100 + r53098;
        double r53102 = r53101 + r53100;
        double r53103 = r53099 * r53102;
        double r53104 = 2.0;
        double r53105 = fma(r53100, r53104, r53098);
        double r53106 = r53103 / r53105;
        double r53107 = cbrt(r53101);
        double r53108 = cbrt(r53100);
        double r53109 = cbrt(r53108);
        double r53110 = r53108 * r53108;
        double r53111 = cbrt(r53110);
        double r53112 = fma(r53109, r53111, r53107);
        double r53113 = r53111 * r53112;
        double r53114 = r53109 * r53113;
        double r53115 = fma(r53107, r53107, r53114);
        double r53116 = r53106 / r53115;
        double r53117 = -r53109;
        double r53118 = r53117 + r53109;
        double r53119 = r53111 * r53118;
        double r53120 = r53116 + r53119;
        return r53120;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.7

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

  3. Applied add-cube-cbrt29.7

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

  4. Applied cbrt-prod29.8

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

  5. Applied *-un-lft-identity29.8

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

  6. Applied cbrt-prod29.8

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

  7. Applied prod-diff29.9

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

  8. Simplified29.9

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

  9. Simplified29.8

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

  11. Applied flip3--29.8

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

  12. Simplified29.1

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

  13. Simplified29.1

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

  15. Applied flip--30.4

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

  16. Simplified0.7

    \[\leadsto \frac{\frac{\color{blue}{\left(0 + 1\right) \cdot \left(\left(x + 1\right) + x\right)}}{\left(x + 1\right) + x}}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\]

  17. Simplified0.7

    \[\leadsto \frac{\frac{\left(0 + 1\right) \cdot \left(\left(x + 1\right) + x\right)}{\color{blue}{\mathsf{fma}\left(x, 2, 1\right)}}}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\]

  18. Final simplification0.7

    \[\leadsto \frac{\frac{\left(0 + 1\right) \cdot \left(\left(x + 1\right) + x\right)}{\mathsf{fma}\left(x, 2, 1\right)}}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt[3]{x + 1}\right)\right)\right)} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))