Average Error: 32.8 → 23.9
Time: 13.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.24942559102901105 \cdot 10^{-18}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\left({\left({x}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot {x}^{\frac{1}{3}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.41213401008201 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{x}^{\left(\frac{1}{n}\right)} \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -2.24942559102901105 \cdot 10^{-18}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\left({\left({x}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot {x}^{\frac{1}{3}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 2.41213401008201 \cdot 10^{-10}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{x}^{\left(\frac{1}{n}\right)} \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\\

\end{array}
double code(double x, double n) {
	return (pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)));
}
↓
double code(double x, double n) {
	double VAR;
	if (((1.0 / n) <= -2.249425591029011e-18)) {
		VAR = (pow((x + 1.0), (1.0 / n)) - (pow(((pow(pow(x, 0.3333333333333333), 0.6666666666666666) * pow(cbrt(x), 0.3333333333333333)) * pow(x, 0.3333333333333333)), (1.0 / n)) * pow(cbrt(x), (1.0 / n))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 2.412134010082013e-10)) {
			VAR_1 = (((1.0 / n) / x) - (((0.5 / n) / pow(x, 2.0)) - ((log(x) * 1.0) / (x * pow(n, 2.0)))));
		} else {
			VAR_1 = ((pow(pow((x + 1.0), (1.0 / n)), 3.0) - pow(pow(x, (1.0 / n)), 3.0)) / ((pow(x, (1.0 / n)) * (pow(x, (1.0 / n)) + pow((x + 1.0), (1.0 / n)))) + pow((x + 1.0), (2.0 * (1.0 / n)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -2.249425591029011e-18

    1. Initial program 4.4

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt4.4

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

    4. Applied unpow-prod-down4.5

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

    6. Applied pow1/34.5

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

    8. Applied add-cube-cbrt4.5

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

    9. Simplified4.4

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

    10. Simplified4.4

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

    if -2.249425591029011e-18 < (/ 1.0 n) < 2.412134010082013e-10

    1. Initial program 44.8

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

    2. Taylor expanded around inf 32.7

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

    3. Simplified32.1

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\]

    if 2.412134010082013e-10 < (/ 1.0 n)

    1. Initial program 5.8

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied flip3--5.9

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

    4. Simplified5.9

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

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.24942559102901105 \cdot 10^{-18}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\left({\left({x}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot {x}^{\frac{1}{3}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.41213401008201 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{x}^{\left(\frac{1}{n}\right)} \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))