Average Error: 29.7 → 0.5
Time: 32.1s
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 r7087661 = x;
        double r7087662 = 1.0;
        double r7087663 = r7087661 + r7087662;
        double r7087664 = cbrt(r7087663);
        double r7087665 = cbrt(r7087661);
        double r7087666 = r7087664 - r7087665;
        return r7087666;
}
↓
double f(double x) {
        double r7087667 = 1.0;
        double r7087668 = x;
        double r7087669 = r7087668 + r7087667;
        double r7087670 = cbrt(r7087669);
        double r7087671 = cbrt(r7087668);
        double r7087672 = r7087670 + r7087671;
        double r7087673 = r7087671 * r7087672;
        double r7087674 = fma(r7087670, r7087670, r7087673);
        double r7087675 = r7087667 / r7087674;
        return r7087675;
}

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

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

  7. Applied *-commutative0.5

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

  8. 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 2019128 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))