Average Error: 33.0 → 24.2
Time: 15.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.16028312956267177:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.24288124722186634 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.16028312956267177:\\
\;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.24288124722186634 \cdot 10^{-23}:\\
\;\;\;\;\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}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{x}\right)}^{\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)) <= -0.16028312956267177)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) (((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))) * ((double) exp(((double) (((double) log(((double) sqrt(x)))) * ((double) (1.0 / n)))))))))))) * ((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) (((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))) * ((double) exp(((double) (((double) log(((double) sqrt(x)))) * ((double) (1.0 / n)))))))))))))) * ((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) (((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))) * ((double) exp(((double) (((double) log(((double) sqrt(x)))) * ((double) (1.0 / n))))))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 1.2428812472218663e-23)) {
			VAR_1 = ((double) (((double) (((double) (1.0 / n)) / x)) - ((double) (((double) (((double) (0.5 / n)) / ((double) pow(x, 2.0)))) - ((double) (((double) (((double) log(x)) * 1.0)) / ((double) (x * ((double) pow(n, 2.0))))))))));
		} else {
			VAR_1 = ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) (((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))) * ((double) pow(((double) sqrt(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) < -0.16028312956267177

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

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

    4. Applied unpow-prod-down0.3

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

    6. Applied add-exp-log0.3

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

    7. Applied pow-exp0.3

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

    9. Applied add-cube-cbrt0.3

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

    if -0.16028312956267177 < (/ 1.0 n) < 1.2428812472218663e-23

    1. Initial program 45.1

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

    2. Taylor expanded around inf 33.0

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

    3. Simplified32.4

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

    if 1.2428812472218663e-23 < (/ 1.0 n)

    1. Initial program 11.9

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

    3. Applied add-sqr-sqrt11.9

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

    4. Applied unpow-prod-down12.1

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

  4. Final simplification24.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.16028312956267177:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot e^{\log \left(\sqrt{x}\right) \cdot \frac{1}{n}}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.24288124722186634 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))