Average Error: 29.7 → 0.6
Time: 3.5m
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x} + \sqrt[3]{\sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x} + \sqrt[3]{\sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r13462326 = x;
        double r13462327 = 1.0;
        double r13462328 = r13462326 + r13462327;
        double r13462329 = cbrt(r13462328);
        double r13462330 = cbrt(r13462326);
        double r13462331 = r13462329 - r13462330;
        return r13462331;
}
↓
double f(double x) {
        double r13462332 = 1.0;
        double r13462333 = x;
        double r13462334 = r13462333 + r13462332;
        double r13462335 = cbrt(r13462334);
        double r13462336 = r13462335 * r13462335;
        double r13462337 = cbrt(r13462333);
        double r13462338 = r13462335 * r13462337;
        double r13462339 = cbrt(r13462337);
        double r13462340 = r13462339 * r13462339;
        double r13462341 = r13462340 * r13462337;
        double r13462342 = r13462339 * r13462341;
        double r13462343 = r13462338 + r13462342;
        double r13462344 = r13462336 + r13462343;
        double r13462345 = r13462332 / r13462344;
        return r13462345;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.7

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

  3. Applied flip3--29.7

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

  4. Taylor expanded around -inf 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. Using strategy rm

  6. Applied add-cube-cbrt0.6

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

  7. Applied associate-*r*0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))