Average Error: 29.5 → 0.6
Time: 3.0s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   (* (* (cbrt x) (cbrt (* (cbrt x) (cbrt x)))) (cbrt (cbrt x)))
   (* (cbrt (+ 1.0 x)) (+ (cbrt x) (cbrt (+ 1.0 x)))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	return 1.0 / (((cbrt(x) * cbrt(cbrt(x) * cbrt(x))) * cbrt(cbrt(x))) + (cbrt(1.0 + x) * (cbrt(x) + cbrt(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 29.5

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

  3. Applied flip3--_binary6429.4

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

  4. Simplified0.5

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

  5. Simplified33.2

    \[\leadsto \frac{1}{\color{blue}{{x}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt_binary6433.2

    \[\leadsto \frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]

  8. Applied unpow-prod-down_binary6433.2

    \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x}\right)}^{0.6666666666666666} \cdot {\left(\sqrt{x}\right)}^{0.6666666666666666}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]

  9. Simplified33.0

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x}} \cdot {\left(\sqrt{x}\right)}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]

  10. Simplified0.5

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

  12. Applied add-cube-cbrt_binary640.6

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

  13. Applied cbrt-prod_binary640.6

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

  14. Applied associate-*r*_binary640.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2021173 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))