Average Error: 29.5 → 0.5
Time: 29.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}
double f(double x) {
        double r1752358 = x;
        double r1752359 = 1.0;
        double r1752360 = r1752358 + r1752359;
        double r1752361 = cbrt(r1752360);
        double r1752362 = cbrt(r1752358);
        double r1752363 = r1752361 - r1752362;
        return r1752363;
}
↓
double f(double x) {
        double r1752364 = 1.0;
        double r1752365 = x;
        double r1752366 = r1752365 + r1752364;
        double r1752367 = cbrt(r1752366);
        double r1752368 = cbrt(r1752365);
        double r1752369 = r1752367 + r1752368;
        double r1752370 = r1752368 * r1752369;
        double r1752371 = fma(r1752367, r1752367, r1752370);
        double r1752372 = r1752364 / r1752371;
        return r1752372;
}

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

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))