Average Error: 30.1 → 0.6
Time: 1.2m
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}
double f(double x) {
        double r4691030 = x;
        double r4691031 = 1.0;
        double r4691032 = r4691030 + r4691031;
        double r4691033 = cbrt(r4691032);
        double r4691034 = cbrt(r4691030);
        double r4691035 = r4691033 - r4691034;
        return r4691035;
}
↓
double f(double x) {
        double r4691036 = 1.0;
        double r4691037 = x;
        double r4691038 = cbrt(r4691037);
        double r4691039 = r4691036 + r4691037;
        double r4691040 = cbrt(r4691039);
        double r4691041 = r4691038 + r4691040;
        double r4691042 = r4691040 * r4691040;
        double r4691043 = fma(r4691041, r4691038, r4691042);
        double r4691044 = r4691036 / r4691043;
        return r4691044;
}

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--30.1

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

  4. Simplified29.4

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

  5. Simplified29.4

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

  6. Taylor expanded around 0 0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))