Average Error: 29.7 → 0.5
Time: 24.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}\]
\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]{x + 1} + \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r1824150 = x;
        double r1824151 = 1.0;
        double r1824152 = r1824150 + r1824151;
        double r1824153 = cbrt(r1824152);
        double r1824154 = cbrt(r1824150);
        double r1824155 = r1824153 - r1824154;
        return r1824155;
}
↓
double f(double x) {
        double r1824156 = 1.0;
        double r1824157 = x;
        double r1824158 = r1824157 + r1824156;
        double r1824159 = cbrt(r1824158);
        double r1824160 = cbrt(r1824157);
        double r1824161 = r1824159 + r1824160;
        double r1824162 = r1824160 * r1824161;
        double r1824163 = fma(r1824159, r1824159, r1824162);
        double r1824164 = r1824156 / r1824163;
        return r1824164;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.7

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  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

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))