Average Error: 29.7 → 0.6
Time: 7.4s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\left(\sqrt{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\left(\left(\sqrt{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}
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) (((double) (((double) sqrt(((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))) * ((double) sqrt(((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))))) * ((double) cbrt(((double) cbrt(x)))))) * ((double) (((double) (((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) cbrt(((double) cbrt(x)))))) + ((double) cbrt(((double) (x + 1.0)))))))) + ((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) cbrt(((double) (x + 1.0))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.7

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

  3. Applied add-cube-cbrt29.7

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

  4. Applied cbrt-prod29.8

    \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}\]
  5. Using strategy rm

  6. Applied flip3--29.8

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

  7. Simplified29.0

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

  8. Simplified29.0

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

  9. Taylor expanded around 0 0.6

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

  11. Applied add-sqr-sqrt0.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2020148 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))