Average Error: 29.8 → 0.7
Time: 3.8s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right)}^{2}}\right) + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right)}^{2}}\right) + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}
double code(double x) {
	return ((double) (((double) cbrt(((double) (x + 1.0)))) - ((double) cbrt(x))));
}
↓
double code(double x) {
	return ((double) (1.0 / ((double) (((double) (((double) pow(((double) cbrt(((double) cbrt(x)))), 5.0)) * ((double) (((double) cbrt(((double) cbrt(((double) cbrt(x)))))) * ((double) cbrt(((double) pow(((double) (((double) cbrt(((double) cbrt(((double) cbrt(x)))))) * ((double) (((double) cbrt(((double) cbrt(((double) cbrt(x)))))) * ((double) cbrt(((double) cbrt(((double) cbrt(x)))))))))), 2.0)))))))) + ((double) (((double) cbrt(((double) (1.0 + x)))) * ((double) (((double) cbrt(x)) + ((double) cbrt(((double) (1.0 + x))))))))))));
}

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 flip3--29.8

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

  4. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied cbrt-prod0.6

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

  9. Applied associate-*r*0.6

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

  10. Simplified0.7

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

  12. Applied add-cube-cbrt0.7

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  13. Applied cbrt-prod0.7

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  14. Simplified0.7

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]
  15. Using strategy rm

  16. Applied add-cube-cbrt0.7

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \left(\sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}}^{2}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  17. Final simplification0.7

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right)}^{2}}\right) + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))