Average Error: 29.4 → 0.5
Time: 17.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}
double f(double x) {
        double r2919531 = x;
        double r2919532 = 1.0;
        double r2919533 = r2919531 + r2919532;
        double r2919534 = cbrt(r2919533);
        double r2919535 = cbrt(r2919531);
        double r2919536 = r2919534 - r2919535;
        return r2919536;
}
↓
double f(double x) {
        double r2919537 = 1.0;
        double r2919538 = x;
        double r2919539 = cbrt(r2919538);
        double r2919540 = r2919538 + r2919537;
        double r2919541 = cbrt(r2919540);
        double r2919542 = r2919539 + r2919541;
        double r2919543 = r2919541 * r2919541;
        double r2919544 = fma(r2919539, r2919542, r2919543);
        double r2919545 = r2919537 / r2919544;
        return r2919545;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.4

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

  3. Applied flip3--29.3

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

  4. Simplified28.8

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

  5. Simplified28.8

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

  6. Taylor expanded around 0 0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))