Average Error: 30.5 → 0.6
Time: 5.3s
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 code(double x) {
	return (cbrt((x + 1.0)) - cbrt(x));
}
↓
double code(double x) {
	return ((0.0 + 1.0) / ((cbrt((x + 1.0)) * (cbrt((x + 1.0)) + cbrt(x))) + ((cbrt(pow(cbrt(x), 2.0)) * cbrt(pow(cbrt(x), 2.0))) * cbrt(pow(cbrt(x), 2.0)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  3. Applied flip3--30.4

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

  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. Simplified33.4

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

  7. Applied add-cbrt-cube40.4

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

  8. Simplified16.0

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

  10. Applied add-cube-cbrt16.0

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

  11. Applied unpow-prod-down16.0

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

  12. Applied cbrt-prod8.5

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

  14. Applied unpow-prod-down8.5

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

  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 2020102 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))