Average Error: 33.0 → 23.8
Time: 20.2s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.24462942643905788 \cdot 10^{-12}:\\ \;\;\;\;\left(\log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.75596332523924502 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -6.24462942643905788 \cdot 10^{-12}:\\
\;\;\;\;\left(\log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\

\mathbf{elif}\;\frac{1}{n} \le 2.75596332523924502 \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}:\\
\;\;\;\;\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\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)) <= -6.244629426439058e-12)) {
		VAR = ((double) (((double) (((double) log(((double) exp(((double) pow(((double) sqrt(((double) (x + 1.0)))), ((double) (1.0 / n)))))))) + ((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))))) * ((double) cbrt(((double) pow(((double) (((double) pow(((double) sqrt(((double) (x + 1.0)))), ((double) (1.0 / n)))) - ((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))))), 3.0))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 2.755963325239245e-10)) {
			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) (((double) (((double) pow(((double) sqrt(((double) sqrt(((double) (x + 1.0)))))), ((double) (1.0 / n)))) * ((double) pow(((double) sqrt(((double) sqrt(((double) (x + 1.0)))))), ((double) (1.0 / n)))))) + ((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))))) * ((double) (((double) (((double) pow(((double) sqrt(((double) sqrt(((double) (x + 1.0)))))), ((double) (1.0 / n)))) + ((double) sqrt(((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))))) * ((double) (((double) pow(((double) sqrt(((double) sqrt(((double) (x + 1.0)))))), ((double) (1.0 / n)))) - ((double) sqrt(((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) < -6.244629426439058e-12

    1. Initial program 1.8

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

    3. Applied add-sqr-sqrt1.9

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

    4. Applied add-sqr-sqrt1.9

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

    5. Applied unpow-prod-down1.8

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

    6. Applied difference-of-squares1.9

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

    8. Applied add-cbrt-cube2.0

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

    9. Simplified2.0

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

    11. Applied add-log-exp2.0

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

    if -6.244629426439058e-12 < (/ 1.0 n) < 2.755963325239245e-10

    1. Initial program 45.4

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

    2. Taylor expanded around inf 32.8

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

    3. Simplified32.2

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

    if 2.755963325239245e-10 < (/ 1.0 n)

    1. Initial program 7.0

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

    3. Applied add-sqr-sqrt7.1

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

    4. Applied add-sqr-sqrt7.1

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

    5. Applied unpow-prod-down7.1

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

    6. Applied difference-of-squares7.1

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

    8. Applied add-sqr-sqrt7.1

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

    9. Applied unpow-prod-down7.1

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

    10. Applied sqrt-prod7.1

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

    11. Applied add-sqr-sqrt7.1

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

    12. Applied sqrt-prod7.1

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

    13. Applied unpow-prod-down7.1

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

    14. Applied difference-of-squares7.1

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

    16. Applied add-sqr-sqrt7.1

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

    17. Applied sqrt-prod7.1

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

    18. Applied unpow-prod-down7.1

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

  4. Final simplification23.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.24462942643905788 \cdot 10^{-12}:\\ \;\;\;\;\left(\log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.75596332523924502 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))