Average Error: 30.0 → 0.5
Time: 21.7s
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 r36573 = x;
        double r36574 = 1.0;
        double r36575 = r36573 + r36574;
        double r36576 = cbrt(r36575);
        double r36577 = cbrt(r36573);
        double r36578 = r36576 - r36577;
        return r36578;
}
↓
double f(double x) {
        double r36579 = 1.0;
        double r36580 = x;
        double r36581 = r36579 + r36580;
        double r36582 = cbrt(r36581);
        double r36583 = cbrt(r36580);
        double r36584 = r36582 + r36583;
        double r36585 = r36583 * r36583;
        double r36586 = fma(r36584, r36582, r36585);
        double r36587 = r36579 / r36586;
        return r36587;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.0

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

  3. Applied flip3--30.0

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

  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. Simplified33.2

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

  7. Applied sqr-pow33.2

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

  8. Simplified33.0

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

  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 2019322 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))