Average Error: 29.5 → 0.6
Time: 19.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\right)}\]
\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]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\right)}
double f(double x) {
        double r2102496 = x;
        double r2102497 = 1.0;
        double r2102498 = r2102496 + r2102497;
        double r2102499 = cbrt(r2102498);
        double r2102500 = cbrt(r2102496);
        double r2102501 = r2102499 - r2102500;
        return r2102501;
}
↓
double f(double x) {
        double r2102502 = 1.0;
        double r2102503 = x;
        double r2102504 = r2102503 + r2102502;
        double r2102505 = cbrt(r2102504);
        double r2102506 = cbrt(r2102503);
        double r2102507 = r2102506 * r2102506;
        double r2102508 = cbrt(r2102507);
        double r2102509 = cbrt(r2102506);
        double r2102510 = r2102508 * r2102509;
        double r2102511 = cbrt(r2102510);
        double r2102512 = r2102509 * r2102509;
        double r2102513 = r2102511 * r2102512;
        double r2102514 = r2102505 + r2102513;
        double r2102515 = r2102506 * r2102514;
        double r2102516 = fma(r2102505, r2102505, r2102515);
        double r2102517 = r2102502 / r2102516;
        return r2102517;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.5

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

  3. Applied flip3--29.5

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

  4. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied add-cube-cbrt0.6

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

  9. Applied add-cube-cbrt0.6

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

  10. Applied cbrt-prod0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))