Average Error: 29.4 → 0.5
Time: 18.6s
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 r2817799 = x;
        double r2817800 = 1.0;
        double r2817801 = r2817799 + r2817800;
        double r2817802 = cbrt(r2817801);
        double r2817803 = cbrt(r2817799);
        double r2817804 = r2817802 - r2817803;
        return r2817804;
}
↓
double f(double x) {
        double r2817805 = 1.0;
        double r2817806 = x;
        double r2817807 = cbrt(r2817806);
        double r2817808 = r2817806 + r2817805;
        double r2817809 = cbrt(r2817808);
        double r2817810 = r2817807 + r2817809;
        double r2817811 = r2817809 * r2817809;
        double r2817812 = fma(r2817807, r2817810, r2817811);
        double r2817813 = r2817805 / r2817812;
        return r2817813;
}

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)))