Average Error: 29.7 → 11.7
Time: 5.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4991101183960429 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 2.21718745512740728 \cdot 10^{-10}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\sqrt[3]{x + 1}\right)}^{6}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4991101183960429 \cdot 10^{61}:\\
\;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\

\mathbf{elif}\;x \le 2.21718745512740728 \cdot 10^{-10}:\\
\;\;\;\;\sqrt[3]{{\left({\left(\sqrt[3]{x + 1}\right)}^{6}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double code(double x) {
	return (cbrt((x + 1.0)) - cbrt(x));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -4.499110118396043e+61)) {
		VAR = (((0.3333333333333333 * pow((1.0 / pow(x, 2.0)), 0.3333333333333333)) + (0.06172839506172839 * pow((1.0 / pow(x, 8.0)), 0.3333333333333333))) - (0.1111111111111111 * pow((1.0 / pow(x, 5.0)), 0.3333333333333333)));
	} else {
		double VAR_1;
		if ((x <= 2.2171874551274073e-10)) {
			VAR_1 = ((cbrt(pow(pow(cbrt((x + 1.0)), 6.0), 0.3333333333333333)) * cbrt(cbrt((x + 1.0)))) - cbrt(x));
		} else {
			VAR_1 = ((0.0 + 1.0) / ((cbrt((x + 1.0)) * (cbrt((x + 1.0)) + cbrt(x))) + pow(x, 0.6666666666666666)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -4.499110118396043e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 39.7

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    if -4.499110118396043e+61 < x < 2.2171874551274073e-10

    1. Initial program 5.0

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

    3. Applied add-cube-cbrt5.0

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

    4. Applied cbrt-prod5.0

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

    6. Applied pow1/36.1

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

    7. Applied pow1/36.1

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

    8. Applied pow-prod-down4.8

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

    9. Simplified4.8

      \[\leadsto \sqrt[3]{{\color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{6}\right)}}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\]

    if 2.2171874551274073e-10 < x

    1. Initial program 57.5

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

    3. Applied flip3--57.4

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

    4. Simplified1.0

      \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

    5. Simplified4.4

      \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4991101183960429 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 2.21718745512740728 \cdot 10^{-10}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\sqrt[3]{x + 1}\right)}^{6}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))