Average Error: 29.7 → 0.5
Time: 13.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x + 1}\right), \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}\]
\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 \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r1278920 = x;
        double r1278921 = 1.0;
        double r1278922 = r1278920 + r1278921;
        double r1278923 = cbrt(r1278922);
        double r1278924 = cbrt(r1278920);
        double r1278925 = r1278923 - r1278924;
        return r1278925;
}
↓
double f(double x) {
        double r1278926 = 1.0;
        double r1278927 = x;
        double r1278928 = r1278927 + r1278926;
        double r1278929 = cbrt(r1278928);
        double r1278930 = cbrt(r1278927);
        double r1278931 = r1278930 * r1278930;
        double r1278932 = r1278929 * r1278930;
        double r1278933 = r1278931 + r1278932;
        double r1278934 = fma(r1278929, r1278929, r1278933);
        double r1278935 = r1278926 / r1278934;
        return r1278935;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.7

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

  3. Applied flip3--29.6

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

  4. Simplified29.1

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

  5. Simplified29.1

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

  6. Taylor expanded around inf 0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))