Average Error: 30.4 → 0.5
Time: 15.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)}
double f(double x) {
        double r899294 = x;
        double r899295 = 1.0;
        double r899296 = r899294 + r899295;
        double r899297 = cbrt(r899296);
        double r899298 = cbrt(r899294);
        double r899299 = r899297 - r899298;
        return r899299;
}
↓
double f(double x) {
        double r899300 = 1.0;
        double r899301 = x;
        double r899302 = r899301 + r899300;
        double r899303 = cbrt(r899302);
        double r899304 = cbrt(r899301);
        double r899305 = r899303 + r899304;
        double r899306 = r899304 * r899305;
        double r899307 = fma(r899303, r899303, r899306);
        double r899308 = r899300 / r899307;
        return r899308;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.4

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

  3. Applied flip3--30.3

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

  4. Simplified0.6

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))