Average Error: 32.8 → 23.6
Time: 13.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.2763059786227459 \cdot 10^{-15}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.1226015413190358 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot {x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {x}^{\left(2 \cdot \frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -2.2763059786227459 \cdot 10^{-15}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 1.1226015413190358 \cdot 10^{-6}:\\
\;\;\;\;\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(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot {x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {x}^{\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) <= -2.276305978622746e-15)) {
		VAR = (pow((x + 1.0), (1.0 / n)) - (pow(x, ((1.0 / n) / 2.0)) * pow(x, ((1.0 / n) / 2.0))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 1.1226015413190358e-06)) {
			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(x, (1.0 / n)) * pow(x, (2.0 * (1.0 / n)))) + pow(pow((x + 1.0), (1.0 / n)), 3.0)) / ((pow((x + 1.0), (2.0 * (1.0 / n))) + (pow(x, (2.0 * ((1.0 / n) / 2.0))) * pow((x + 1.0), (1.0 / n)))) + pow(x, (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) < -2.276305978622746e-15

    1. Initial program 3.9

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

    3. Applied sqr-pow4.1

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

    if -2.276305978622746e-15 < (/ 1.0 n) < 1.1226015413190358e-06

    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 32.3

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

    3. Simplified31.7

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

    if 1.1226015413190358e-06 < (/ 1.0 n)

    1. Initial program 5.5

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

    3. Applied sqr-pow5.5

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

    5. Applied flip3--5.5

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

    6. Simplified5.5

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

    7. Simplified5.5

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

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.2763059786227459 \cdot 10^{-15}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.1226015413190358 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot {x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {x}^{\left(2 \cdot \frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))