Average Error: 32.5 → 23.8
Time: 12.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.99741516020183053 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.217177993717122 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.99741516020183053 \cdot 10^{-20}:\\
\;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.217177993717122 \cdot 10^{-6}:\\
\;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\

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

\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 ((((double) (1.0 / n)) <= -3.9974151602018305e-20)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))) * ((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))))) * ((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 1.2171779937171219e-06)) {
			VAR_1 = ((double) (((double) (1.0 * ((double) (((double) (1.0 / ((double) (x * n)))) - ((double) (((double) log(((double) (1.0 / x)))) / ((double) (x * ((double) pow(n, 2.0)))))))))) + ((double) (((double) -(0.5)) / ((double) (((double) pow(x, 2.0)) * n))))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))) + ((double) -(((double) pow(x, ((double) (2.0 * ((double) (1.0 / n)))))))))) / ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) + ((double) pow(x, ((double) (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) < -3.9974151602018305e-20

    1. Initial program 4.8

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

    3. Applied add-cube-cbrt4.8

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

    if -3.9974151602018305e-20 < (/ 1.0 n) < 1.2171779937171219e-06

    1. Initial program 44.7

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

    2. Taylor expanded around inf 32.2

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

    3. Simplified31.5

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

    5. Applied sub-neg31.5

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

    6. Applied distribute-lft-in31.5

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

    7. Simplified31.5

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

    8. Simplified31.5

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

    9. Taylor expanded around inf 32.2

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

    10. Simplified32.2

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

    if 1.2171779937171219e-06 < (/ 1.0 n)

    1. Initial program 5.6

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

    3. Applied flip--5.6

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

    4. Simplified5.6

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

  4. Final simplification23.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.99741516020183053 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.217177993717122 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))