Average Error: 29.9 → 0.6
Time: 5.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}
double f(double x) {
        double r62347 = x;
        double r62348 = 1.0;
        double r62349 = r62347 + r62348;
        double r62350 = cbrt(r62349);
        double r62351 = cbrt(r62347);
        double r62352 = r62350 - r62351;
        return r62352;
}
↓
double f(double x) {
        double r62353 = 0.0;
        double r62354 = 1.0;
        double r62355 = r62353 + r62354;
        double r62356 = x;
        double r62357 = r62356 + r62354;
        double r62358 = cbrt(r62357);
        double r62359 = cbrt(r62356);
        double r62360 = r62358 + r62359;
        double r62361 = r62358 * r62360;
        double r62362 = 2.0;
        double r62363 = pow(r62359, r62362);
        double r62364 = cbrt(r62363);
        double r62365 = r62364 * r62364;
        double r62366 = r62365 * r62364;
        double r62367 = r62361 + r62366;
        double r62368 = r62355 / r62367;
        return r62368;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

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

  3. Applied flip3--29.9

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

  4. Simplified0.5

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

  5. Simplified32.8

    \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube40.0

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\sqrt[3]{\left({x}^{\frac{2}{3}} \cdot {x}^{\frac{2}{3}}\right) \cdot {x}^{\frac{2}{3}}}}}\]

  8. Simplified15.7

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{{x}^{2}}}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt15.7

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}}}\]

  11. Applied unpow-prod-down15.7

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}}}}\]

  12. Applied cbrt-prod8.1

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\sqrt[3]{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}}\]
  13. Using strategy rm

  14. Applied unpow-prod-down8.1

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]

  15. Applied cbrt-prod0.6

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]

  16. Final simplification0.6

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))