Average Error: 30.2 → 0.5
Time: 24.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, \sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, \sqrt[3]{x} \cdot \sqrt[3]{x}\right)}
double f(double x) {
        double r52727 = x;
        double r52728 = 1.0;
        double r52729 = r52727 + r52728;
        double r52730 = cbrt(r52729);
        double r52731 = cbrt(r52727);
        double r52732 = r52730 - r52731;
        return r52732;
}
↓
double f(double x) {
        double r52733 = 1.0;
        double r52734 = x;
        double r52735 = r52733 + r52734;
        double r52736 = cbrt(r52735);
        double r52737 = cbrt(r52734);
        double r52738 = r52736 + r52737;
        double r52739 = r52737 * r52737;
        double r52740 = fma(r52738, r52736, r52739);
        double r52741 = r52733 / r52740;
        return r52741;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.2

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

  3. Applied flip3--30.2

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

  4. Simplified0.5

    \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified33.7

    \[\leadsto \frac{0 + 1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}}\]
  6. Using strategy rm

  7. Applied sqr-pow33.7

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, \color{blue}{{x}^{\left(\frac{\frac{2}{3}}{2}\right)} \cdot {x}^{\left(\frac{\frac{2}{3}}{2}\right)}}\right)}\]

  8. Simplified33.4

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, \color{blue}{\sqrt[3]{x}} \cdot {x}^{\left(\frac{\frac{2}{3}}{2}\right)}\right)}\]

  9. Simplified0.5

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, \sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}}\right)}\]

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))