Average Error: 30.2 → 0.5
Time: 15.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}
double f(double x) {
        double r855241 = x;
        double r855242 = 1.0;
        double r855243 = r855241 + r855242;
        double r855244 = cbrt(r855243);
        double r855245 = cbrt(r855241);
        double r855246 = r855244 - r855245;
        return r855246;
}
↓
double f(double x) {
        double r855247 = 1.0;
        double r855248 = x;
        double r855249 = cbrt(r855248);
        double r855250 = r855248 + r855247;
        double r855251 = cbrt(r855250);
        double r855252 = r855249 + r855251;
        double r855253 = r855251 * r855251;
        double r855254 = fma(r855252, r855249, r855253);
        double r855255 = r855247 / r855254;
        return r855255;
}

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--30.1

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

  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

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

  6. Taylor expanded around inf 0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))