Average Error: 30.9 → 0.6
Time: 3.6s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \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
  (+ (pow (cbrt x) 2.0) (* (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 / (pow(cbrt(x), 2.0) + (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 30.9

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

  3. Applied flip3--_binary64_42330.9

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

  4. Simplified0.6

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

  5. Simplified32.8

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

  7. Applied sqr-pow_binary64_39132.8

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

  8. Simplified32.6

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

  9. Simplified0.6

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

  11. Applied pow2_binary64_5000.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2020349 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))