Average Error: 29.6 → 0.6
Time: 8.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}
double f(double x) {
        double r97515 = x;
        double r97516 = 1.0;
        double r97517 = r97515 + r97516;
        double r97518 = cbrt(r97517);
        double r97519 = cbrt(r97515);
        double r97520 = r97518 - r97519;
        return r97520;
}
↓
double f(double x) {
        double r97521 = 1.0;
        double r97522 = x;
        double r97523 = r97522 + r97521;
        double r97524 = cbrt(r97523);
        double r97525 = cbrt(r97522);
        double r97526 = r97525 * r97525;
        double r97527 = cbrt(r97526);
        double r97528 = cbrt(r97525);
        double r97529 = r97524 + r97525;
        double r97530 = r97528 * r97529;
        double r97531 = r97527 * r97530;
        double r97532 = fma(r97524, r97524, r97531);
        double r97533 = r97521 / r97532;
        return r97533;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.6

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

  3. Applied flip3--29.6

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

  4. Simplified29.0

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

  5. Simplified29.0

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

  6. Taylor expanded around 0 0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied cbrt-prod0.6

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

  10. Applied associate-*l*0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))