Average Error: 30.1 → 0.5
Time: 24.5s
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 r1378992 = x;
        double r1378993 = 1.0;
        double r1378994 = r1378992 + r1378993;
        double r1378995 = cbrt(r1378994);
        double r1378996 = cbrt(r1378992);
        double r1378997 = r1378995 - r1378996;
        return r1378997;
}
↓
double f(double x) {
        double r1378998 = 1.0;
        double r1378999 = /* ERROR: no posit support in C */;
        double r1379000 = /* ERROR: no posit support in C */;
        double r1379001 = x;
        double r1379002 = r1379001 + r1378998;
        double r1379003 = cbrt(r1379002);
        double r1379004 = cbrt(r1379001);
        double r1379005 = r1379003 + r1379004;
        double r1379006 = r1379003 * r1379003;
        double r1379007 = fma(r1379005, r1379004, r1379006);
        double r1379008 = r1379000 / r1379007;
        return r1379008;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.1

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

  3. Applied flip3--30.0

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

  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 2019163 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))