Average Error: 29.6 → 0.5
Time: 18.5s
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 r1458276 = x;
        double r1458277 = 1.0;
        double r1458278 = r1458276 + r1458277;
        double r1458279 = cbrt(r1458278);
        double r1458280 = cbrt(r1458276);
        double r1458281 = r1458279 - r1458280;
        return r1458281;
}
↓
double f(double x) {
        double r1458282 = 1.0;
        double r1458283 = x;
        double r1458284 = r1458283 + r1458282;
        double r1458285 = cbrt(r1458284);
        double r1458286 = cbrt(r1458283);
        double r1458287 = r1458285 + r1458286;
        double r1458288 = r1458286 * r1458287;
        double r1458289 = fma(r1458285, r1458285, r1458288);
        double r1458290 = r1458282 / r1458289;
        return r1458290;
}

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. 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

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

  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 2019130 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))