Average Error: 29.7 → 0.6
Time: 18.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}
double f(double x) {
        double r1436423 = x;
        double r1436424 = 1.0;
        double r1436425 = r1436423 + r1436424;
        double r1436426 = cbrt(r1436425);
        double r1436427 = cbrt(r1436423);
        double r1436428 = r1436426 - r1436427;
        return r1436428;
}
↓
double f(double x) {
        double r1436429 = 1.0;
        double r1436430 = x;
        double r1436431 = r1436430 + r1436429;
        double r1436432 = cbrt(r1436431);
        double r1436433 = cbrt(r1436430);
        double r1436434 = cbrt(r1436433);
        double r1436435 = r1436432 + r1436433;
        double r1436436 = r1436434 * r1436435;
        double r1436437 = r1436433 * r1436433;
        double r1436438 = cbrt(r1436437);
        double r1436439 = r1436436 * r1436438;
        double r1436440 = fma(r1436432, r1436432, r1436439);
        double r1436441 = r1436429 / r1436440;
        return r1436441;
}

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. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied add-cube-cbrt0.6

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

  8. Applied cbrt-prod0.6

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

  9. Applied associate-*l*0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))