Average Error: 30.3 → 0.6
Time: 19.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \sqrt[3]{x}\right)\right)}\]
\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]{\sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r1510478 = x;
        double r1510479 = 1.0;
        double r1510480 = r1510478 + r1510479;
        double r1510481 = cbrt(r1510480);
        double r1510482 = cbrt(r1510478);
        double r1510483 = r1510481 - r1510482;
        return r1510483;
}
↓
double f(double x) {
        double r1510484 = 1.0;
        double r1510485 = x;
        double r1510486 = r1510485 + r1510484;
        double r1510487 = cbrt(r1510486);
        double r1510488 = cbrt(r1510485);
        double r1510489 = cbrt(r1510487);
        double r1510490 = r1510489 * r1510489;
        double r1510491 = r1510489 * r1510490;
        double r1510492 = r1510491 + r1510488;
        double r1510493 = r1510488 * r1510492;
        double r1510494 = fma(r1510487, r1510487, r1510493);
        double r1510495 = r1510484 / r1510494;
        return r1510495;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.3

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

  3. Applied flip3--30.3

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

  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

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

  7. Applied add-cube-cbrt0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))