Average Error: 29.6 → 0.7
Time: 12.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)}
double f(double x) {
        double r840321 = x;
        double r840322 = 1.0;
        double r840323 = r840321 + r840322;
        double r840324 = cbrt(r840323);
        double r840325 = cbrt(r840321);
        double r840326 = r840324 - r840325;
        return r840326;
}
↓
double f(double x) {
        double r840327 = 1.0;
        double r840328 = x;
        double r840329 = r840328 + r840327;
        double r840330 = cbrt(r840329);
        double r840331 = r840330 * r840330;
        double r840332 = cbrt(r840328);
        double r840333 = cbrt(r840332);
        double r840334 = r840332 * r840332;
        double r840335 = cbrt(r840334);
        double r840336 = cbrt(r840335);
        double r840337 = cbrt(r840333);
        double r840338 = r840336 * r840337;
        double r840339 = r840333 * r840338;
        double r840340 = r840330 + r840332;
        double r840341 = r840339 * r840340;
        double r840342 = r840333 * r840341;
        double r840343 = r840331 + r840342;
        double r840344 = r840327 / r840343;
        return r840344;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.6

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

  3. Applied flip3--29.6

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

  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

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}}\]
  6. Using strategy rm

  7. Applied *-commutative0.5

    \[\leadsto \frac{1 + 0}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \color{blue}{\left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{x}}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.7

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

  10. Applied associate-*r*0.7

    \[\leadsto \frac{1 + 0}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \color{blue}{\left(\left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt0.7

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

  13. Applied cbrt-prod0.7

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

  14. Applied cbrt-prod0.7

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

  15. Final simplification0.7

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

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))