Average Error: 29.9 → 0.6
Time: 17.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}\]
\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 \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r2997379 = x;
        double r2997380 = 1.0;
        double r2997381 = r2997379 + r2997380;
        double r2997382 = cbrt(r2997381);
        double r2997383 = cbrt(r2997379);
        double r2997384 = r2997382 - r2997383;
        return r2997384;
}
↓
double f(double x) {
        double r2997385 = 1.0;
        double r2997386 = x;
        double r2997387 = r2997386 + r2997385;
        double r2997388 = cbrt(r2997387);
        double r2997389 = cbrt(r2997386);
        double r2997390 = cbrt(r2997389);
        double r2997391 = r2997389 * r2997389;
        double r2997392 = cbrt(r2997391);
        double r2997393 = r2997390 * r2997392;
        double r2997394 = r2997388 + r2997389;
        double r2997395 = r2997393 * r2997394;
        double r2997396 = fma(r2997388, r2997388, r2997395);
        double r2997397 = r2997385 / r2997396;
        return r2997397;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.9

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

  3. Applied flip3--29.9

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

  4. Simplified0.5

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified0.5

    \[\leadsto \frac{1 + 0}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{1 + x}, \sqrt[3]{1 + x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right)}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.5

    \[\leadsto \frac{1 + 0}{\mathsf{fma}\left(\sqrt[3]{1 + x}, \sqrt[3]{1 + x}, \sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right)}\]

  8. Applied cbrt-prod0.6

    \[\leadsto \frac{1 + 0}{\mathsf{fma}\left(\sqrt[3]{1 + x}, \sqrt[3]{1 + x}, \color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right)}\]

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))