Average Error: 30.2 → 0.5
Time: 25.0s
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 r53092 = x;
        double r53093 = 1.0;
        double r53094 = r53092 + r53093;
        double r53095 = cbrt(r53094);
        double r53096 = cbrt(r53092);
        double r53097 = r53095 - r53096;
        return r53097;
}
↓
double f(double x) {
        double r53098 = 1.0;
        double r53099 = x;
        double r53100 = r53098 + r53099;
        double r53101 = cbrt(r53100);
        double r53102 = cbrt(r53099);
        double r53103 = r53101 + r53102;
        double r53104 = r53102 * r53102;
        double r53105 = fma(r53103, r53101, r53104);
        double r53106 = r53098 / r53105;
        return r53106;
}

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)))