Average Error: 30.1 → 0.6
Time: 32.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}
double f(double x) {
        double r2582715 = x;
        double r2582716 = 1.0;
        double r2582717 = r2582715 + r2582716;
        double r2582718 = cbrt(r2582717);
        double r2582719 = cbrt(r2582715);
        double r2582720 = r2582718 - r2582719;
        return r2582720;
}
↓
double f(double x) {
        double r2582721 = 1.0;
        double r2582722 = x;
        double r2582723 = cbrt(r2582722);
        double r2582724 = r2582721 + r2582722;
        double r2582725 = cbrt(r2582724);
        double r2582726 = r2582723 + r2582725;
        double r2582727 = r2582725 * r2582725;
        double r2582728 = fma(r2582726, r2582723, r2582727);
        double r2582729 = r2582721 / r2582728;
        return r2582729;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.1

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

  3. Applied flip3--30.1

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

  4. Simplified29.4

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

  5. Simplified29.4

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

  6. Taylor expanded around 0 0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))