Average Error: 29.3 → 0.5
Time: 20.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1}, \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\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 r3016128 = x;
        double r3016129 = 1.0;
        double r3016130 = r3016128 + r3016129;
        double r3016131 = cbrt(r3016130);
        double r3016132 = cbrt(r3016128);
        double r3016133 = r3016131 - r3016132;
        return r3016133;
}
↓
double f(double x) {
        double r3016134 = 1.0;
        double r3016135 = x;
        double r3016136 = r3016135 + r3016134;
        double r3016137 = cbrt(r3016136);
        double r3016138 = cbrt(r3016135);
        double r3016139 = r3016137 + r3016138;
        double r3016140 = r3016139 * r3016138;
        double r3016141 = fma(r3016137, r3016137, r3016140);
        double r3016142 = r3016134 / r3016141;
        return r3016142;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.3

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

  3. Applied flip3--29.2

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

  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

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

  6. Taylor expanded around 0 31.8

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))