Average Error: 29.8 → 0.5
Time: 17.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\sqrt{1} \cdot \frac{\sqrt{1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\sqrt{1} \cdot \frac{\sqrt{1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}
double f(double x) {
        double r62297 = x;
        double r62298 = 1.0;
        double r62299 = r62297 + r62298;
        double r62300 = cbrt(r62299);
        double r62301 = cbrt(r62297);
        double r62302 = r62300 - r62301;
        return r62302;
}
↓
double f(double x) {
        double r62303 = 1.0;
        double r62304 = sqrt(r62303);
        double r62305 = x;
        double r62306 = r62305 + r62303;
        double r62307 = cbrt(r62306);
        double r62308 = r62307 * r62307;
        double r62309 = cbrt(r62305);
        double r62310 = r62309 + r62307;
        double r62311 = r62309 * r62310;
        double r62312 = r62308 + r62311;
        double r62313 = r62304 / r62312;
        double r62314 = r62304 * r62313;
        return r62314;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.8

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

  3. Applied add-cbrt-cube29.8

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

  4. Simplified29.8

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}^{3}}}\]
  5. Using strategy rm

  6. Applied flip3--29.7

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

  7. Simplified15.4

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

  8. Simplified15.4

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

  10. Applied *-un-lft-identity15.4

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

  11. Applied add-sqr-sqrt15.4

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

  12. Applied times-frac15.4

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

  13. Applied unpow-prod-down15.4

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

  14. Applied cbrt-prod15.4

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

  15. Simplified15.4

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

  16. Simplified0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2019305 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))