Average Error: 30.1 → 8.9
Time: 5.1s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4880.516435323793:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\\ \mathbf{elif}\;x \leq 0.203611116240373:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x + -1}} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4880.516435323793:\\
\;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\\

\mathbf{elif}\;x \leq 0.203611116240373:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x + -1}} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\

\end{array}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -4880.516435323793)
   (+
    (-
     (* 0.3333333333333333 (cbrt (/ 1.0 (* x x))))
     (* 0.1111111111111111 (cbrt (/ 1.0 (pow x 5.0)))))
    (* 0.06172839506172839 (cbrt (/ 1.0 (pow x 8.0)))))
   (if (<= x 0.203611116240373)
     (-
      (/ (cbrt (+ (* x x) -1.0)) (cbrt (+ x -1.0)))
      (* (cbrt (* (cbrt x) (cbrt x))) (cbrt (cbrt x))))
     (/
      1.0
      (+
       (pow x 0.6666666666666666)
       (* (cbrt (+ x 1.0)) (+ (cbrt x) (cbrt (+ x 1.0)))))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -4880.516435323793) {
		tmp = ((0.3333333333333333 * cbrt(1.0 / (x * x))) - (0.1111111111111111 * cbrt(1.0 / pow(x, 5.0)))) + (0.06172839506172839 * cbrt(1.0 / pow(x, 8.0)));
	} else if (x <= 0.203611116240373) {
		tmp = (cbrt((x * x) + -1.0) / cbrt(x + -1.0)) - (cbrt(cbrt(x) * cbrt(x)) * cbrt(cbrt(x)));
	} else {
		tmp = 1.0 / (pow(x, 0.6666666666666666) + (cbrt(x + 1.0) * (cbrt(x) + cbrt(x + 1.0))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4880.5164353237933

    1. Initial program 60.3

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

    2. Taylor expanded around inf 44.4

      \[\leadsto \color{blue}{\left(0.06172839506172839 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{0.3333333333333333} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333}}\]

    3. Simplified31.2

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}}\]

    if -4880.5164353237933 < x < 0.203611116240373008

    1. Initial program 0.1

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

    3. Applied flip-+_binary640.1

      \[\leadsto \sqrt[3]{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \sqrt[3]{x}\]

    4. Applied cbrt-div_binary640.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}}} - \sqrt[3]{x}\]

    5. Simplified0.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{x \cdot x + -1}}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\]

    6. Simplified0.1

      \[\leadsto \frac{\sqrt[3]{x \cdot x + -1}}{\color{blue}{\sqrt[3]{x + -1}}} - \sqrt[3]{x}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt_binary640.1

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

    9. Applied cbrt-prod_binary640.1

      \[\leadsto \frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x + -1}} - \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}\]

    10. Applied cancel-sign-sub-inv_binary640.1

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

    if 0.203611116240373008 < x

    1. Initial program 59.5

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

    3. Applied flip3--_binary6459.3

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

    4. Simplified1.0

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

    5. Simplified4.5

      \[\leadsto \frac{1}{\color{blue}{{x}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4880.516435323793:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\\ \mathbf{elif}\;x \leq 0.203611116240373:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x + -1}} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))