Average Error: 30.1 → 0.5
Time: 8.9s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + t_0 \cdot \sqrt[3]{x}\right)} \end{array} \]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + t_0 \cdot \sqrt[3]{x}\right)}
\end{array}

(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ (pow t_0 2.0) (+ (* (cbrt x) (cbrt x)) (* t_0 (cbrt x)))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	double t_0 = cbrt(1.0 + x);
	return 1.0 / (pow(t_0, 2.0) + ((cbrt(x) * cbrt(x)) + (t_0 * 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 30.1

    \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]

  2. Applied flip3--_binary6430.0

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

  3. Taylor expanded in x around 0 0.5

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

  4. Applied *-un-lft-identity_binary640.5

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

  5. Applied cbrt-prod_binary640.5

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

  6. Applied *-un-lft-identity_binary640.5

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

  7. Applied cbrt-prod_binary640.5

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

  8. Applied swap-sqr_binary640.5

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

  9. Simplified0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))