Average Error: 30.0 → 0.5
Time: 25.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\left(1\right)}{\mathsf{fma}\left(\sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{\left(1\right)}{\mathsf{fma}\left(\sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}
double f(double x) {
        double r827926 = x;
        double r827927 = 1.0;
        double r827928 = r827926 + r827927;
        double r827929 = cbrt(r827928);
        double r827930 = cbrt(r827926);
        double r827931 = r827929 - r827930;
        return r827931;
}
↓
double f(double x) {
        double r827932 = 1.0;
        double r827933 = /* ERROR: no posit support in C */;
        double r827934 = /* ERROR: no posit support in C */;
        double r827935 = x;
        double r827936 = r827935 + r827932;
        double r827937 = cbrt(r827936);
        double r827938 = cbrt(r827935);
        double r827939 = r827937 + r827938;
        double r827940 = r827937 * r827937;
        double r827941 = fma(r827939, r827938, r827940);
        double r827942 = r827934 / r827941;
        return r827942;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.0

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  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. Simplified29.3

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

  5. Simplified29.3

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

  7. Applied insert-posit1629.3

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))