Average Error: 33.0 → 23.9
Time: 19.9s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -43883455996.7051315:\\ \;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \mathbf{elif}\;n \le 377.3545087368791:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{-1}{\frac{x \cdot {n}^{2}}{\log x}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -43883455996.7051315:\\
\;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\

\mathbf{elif}\;n \le 377.3545087368791:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{-1}{\frac{x \cdot {n}^{2}}{\log x}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\

\end{array}
double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n))))));
}
↓
double code(double x, double n) {
	double VAR;
	if ((n <= -43883455996.70513)) {
		VAR = ((double) (((double) (1.0 * ((double) (((double) (1.0 / ((double) (x * n)))) + ((double) (((double) log(x)) / ((double) (x * ((double) pow(n, 2.0)))))))))) - ((double) (((double) (0.5 / n)) / ((double) pow(x, 2.0))))));
	} else {
		double VAR_1;
		if ((n <= 377.3545087368791)) {
			VAR_1 = ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) cbrt(((double) pow(((double) pow(x, ((double) (1.0 / n)))), 3.0))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 / n)) / x)) + ((double) (1.0 * ((double) (0.0 - ((double) (-1.0 / ((double) (((double) (x * ((double) pow(n, 2.0)))) / ((double) log(x)))))))))))) - ((double) (((double) (0.5 / n)) / ((double) pow(x, 2.0))))));
		}
		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 n < -43883455996.7051315

    1. Initial program 44.5

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

    2. Taylor expanded around inf 32.0

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

    3. Simplified31.4

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}}\]

    4. Taylor expanded around 0 32.0

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

    5. Simplified32.0

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

    if -43883455996.7051315 < n < 377.3545087368791

    1. Initial program 3.4

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

    3. Applied add-cbrt-cube3.5

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

    4. Simplified3.5

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

    if 377.3545087368791 < n

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.6

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

    3. Simplified32.0

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}}\]
    4. Using strategy rm

    5. Applied inv-pow32.0

      \[\leadsto \left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \color{blue}{\left({x}^{-1}\right)}}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\]

    6. Applied log-pow32.0

      \[\leadsto \left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\color{blue}{-1 \cdot \log x}}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\]

    7. Applied associate-/l*32.0

      \[\leadsto \left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \color{blue}{\frac{-1}{\frac{x \cdot {n}^{2}}{\log x}}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -43883455996.7051315:\\ \;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \mathbf{elif}\;n \le 377.3545087368791:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{-1}{\frac{x \cdot {n}^{2}}{\log x}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))