Average Error: 30.4 → 0.5
Time: 38.0s
Precision: 64
\[\sqrt[3]{x + 1.0} - \sqrt[3]{x}\]
↓
\[\frac{1.0}{\mathsf{fma}\left(\sqrt[3]{x + 1.0}, \sqrt[3]{x + 1.0}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1.0} + \sqrt[3]{x}\right)\right)}\]
\sqrt[3]{x + 1.0} - \sqrt[3]{x}
↓
\frac{1.0}{\mathsf{fma}\left(\sqrt[3]{x + 1.0}, \sqrt[3]{x + 1.0}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1.0} + \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r3095685 = x;
        double r3095686 = 1.0;
        double r3095687 = r3095685 + r3095686;
        double r3095688 = cbrt(r3095687);
        double r3095689 = cbrt(r3095685);
        double r3095690 = r3095688 - r3095689;
        return r3095690;
}
↓
double f(double x) {
        double r3095691 = 1.0;
        double r3095692 = x;
        double r3095693 = r3095692 + r3095691;
        double r3095694 = cbrt(r3095693);
        double r3095695 = cbrt(r3095692);
        double r3095696 = r3095694 + r3095695;
        double r3095697 = r3095695 * r3095696;
        double r3095698 = fma(r3095694, r3095694, r3095697);
        double r3095699 = r3095691 / r3095698;
        return r3095699;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.4

    \[\sqrt[3]{x + 1.0} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied flip3--30.3

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + 1.0}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1.0} \cdot \sqrt[3]{x + 1.0} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1.0} \cdot \sqrt[3]{x}\right)}}\]

  4. Simplified29.8

    \[\leadsto \frac{\color{blue}{\left(1.0 + x\right) - x}}{\sqrt[3]{x + 1.0} \cdot \sqrt[3]{x + 1.0} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1.0} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified29.8

    \[\leadsto \frac{\left(1.0 + x\right) - x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{1.0 + x}, \sqrt[3]{1.0 + x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1.0 + x}\right)\right)}}\]

  6. Taylor expanded around 0 0.5

    \[\leadsto \frac{\color{blue}{1.0}}{\mathsf{fma}\left(\sqrt[3]{1.0 + x}, \sqrt[3]{1.0 + x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1.0 + x}\right)\right)}\]

  7. Final simplification0.5

    \[\leadsto \frac{1.0}{\mathsf{fma}\left(\sqrt[3]{x + 1.0}, \sqrt[3]{x + 1.0}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1.0} + \sqrt[3]{x}\right)\right)}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))