Average Error: 29.4 → 0.5
Time: 23.4s
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 r1661120 = x;
        double r1661121 = 1.0;
        double r1661122 = r1661120 + r1661121;
        double r1661123 = cbrt(r1661122);
        double r1661124 = cbrt(r1661120);
        double r1661125 = r1661123 - r1661124;
        return r1661125;
}
↓
double f(double x) {
        double r1661126 = 1.0;
        double r1661127 = /* ERROR: no posit support in C */;
        double r1661128 = /* ERROR: no posit support in C */;
        double r1661129 = x;
        double r1661130 = r1661129 + r1661126;
        double r1661131 = cbrt(r1661130);
        double r1661132 = cbrt(r1661129);
        double r1661133 = r1661131 + r1661132;
        double r1661134 = r1661131 * r1661131;
        double r1661135 = fma(r1661133, r1661132, r1661134);
        double r1661136 = r1661128 / r1661135;
        return r1661136;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.4

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

  3. Applied flip3--29.4

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

  4. Simplified28.8

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

  5. Simplified28.8

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

  7. Applied insert-posit1628.8

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

  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 2019164 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))