Average Error: 30.0 → 0.5
Time: 35.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\left(1\right)}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{\left(1\right)}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}
double f(double x) {
        double r3461482 = x;
        double r3461483 = 1.0;
        double r3461484 = r3461482 + r3461483;
        double r3461485 = cbrt(r3461484);
        double r3461486 = cbrt(r3461482);
        double r3461487 = r3461485 - r3461486;
        return r3461487;
}
↓
double f(double x) {
        double r3461488 = 1.0;
        double r3461489 = /* ERROR: no posit support in C */;
        double r3461490 = /* ERROR: no posit support in C */;
        double r3461491 = x;
        double r3461492 = r3461491 + r3461488;
        double r3461493 = cbrt(r3461492);
        double r3461494 = cbrt(r3461491);
        double r3461495 = r3461493 + r3461494;
        double r3461496 = r3461495 * r3461494;
        double r3461497 = fma(r3461493, r3461493, r3461496);
        double r3461498 = r3461490 / r3461497;
        return r3461498;
}

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

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

  5. Simplified29.3

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

  7. Applied insert-posit1629.3

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))