Average Error: 29.5 → 0.5
Time: 5.8s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x} + t_0 \cdot \left(\sqrt[3]{x} + t_0\right)} \end{array} \]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x} + t_0 \cdot \left(\sqrt[3]{x} + t_0\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 (+ (* (cbrt x) (cbrt x)) (* t_0 (+ (cbrt x) t_0))))))
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 / ((cbrt(x) * cbrt(x)) + (t_0 * (cbrt(x) + t_0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.5

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

  2. Applied flip3--_binary6429.5

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

  3. Simplified0.5

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

  4. Simplified33.1

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

  5. Applied add-sqr-sqrt_binary6433.1

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

  6. Simplified32.9

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2021215 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))