Average Error: 29.5 → 0.5
Time: 3.5m
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}
double f(double x) {
        double r13515315 = x;
        double r13515316 = 1.0;
        double r13515317 = r13515315 + r13515316;
        double r13515318 = cbrt(r13515317);
        double r13515319 = cbrt(r13515315);
        double r13515320 = r13515318 - r13515319;
        return r13515320;
}
↓
double f(double x) {
        double r13515321 = 1.0;
        double r13515322 = x;
        double r13515323 = r13515322 + r13515321;
        double r13515324 = cbrt(r13515323);
        double r13515325 = cbrt(r13515322);
        double r13515326 = r13515324 * r13515325;
        double r13515327 = r13515325 * r13515325;
        double r13515328 = r13515326 + r13515327;
        double r13515329 = r13515324 * r13515324;
        double r13515330 = r13515328 + r13515329;
        double r13515331 = r13515321 / r13515330;
        return r13515331;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.5

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

  3. Applied flip3--29.5

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

  4. Taylor expanded around 0 0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))