Average Error: 29.5 → 0.6
Time: 2.4s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right)}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   (* (cbrt (+ 1.0 x)) (cbrt (+ 1.0 x)))
   (*
    (cbrt (* (cbrt x) (cbrt x)))
    (* (cbrt (cbrt x)) (+ (cbrt (+ 1.0 x)) (cbrt x)))))))
double code(double x) {
	return ((double) (((double) cbrt(((double) (x + 1.0)))) - ((double) cbrt(x))));
}
↓
double code(double x) {
	return (1.0 / ((double) (((double) (((double) cbrt(((double) (1.0 + x)))) * ((double) cbrt(((double) (1.0 + x)))))) + ((double) (((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) (((double) cbrt(((double) cbrt(x)))) * ((double) (((double) cbrt(((double) (1.0 + x)))) + ((double) cbrt(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 pow1/3_binary6431.0

    \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - \sqrt[3]{x}\]
  4. Using strategy rm

  5. Applied flip3--_binary6431.0

    \[\leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{0.3333333333333333}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + {\left(x + 1\right)}^{0.3333333333333333} \cdot \sqrt[3]{x}\right)}}\]

  6. Simplified17.0

    \[\leadsto \frac{\color{blue}{1}}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + {\left(x + 1\right)}^{0.3333333333333333} \cdot \sqrt[3]{x}\right)}\]

  7. Simplified16.8

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

  9. Applied sqr-pow_binary6416.8

    \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.6666666666666666}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{0.6666666666666666}{2}\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  10. Simplified16.6

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1}} \cdot {\left(x + 1\right)}^{\left(\frac{0.6666666666666666}{2}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  11. Simplified0.5

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

  13. Applied add-cube-cbrt_binary640.6

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

  14. Applied cbrt-prod_binary640.6

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

  15. Applied associate-*l*_binary640.6

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

  16. Simplified0.6

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

  17. Final simplification0.6

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))