Average Error: 33.0 → 23.6
Time: 17.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.5424017185802929 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.7895543843896953 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -6.5424017185802929 \cdot 10^{-7}:\\
\;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 4.7895543843896953 \cdot 10^{-16}:\\
\;\;\;\;\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}\\

\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)) <= -6.542401718580293e-07)) {
		VAR = ((double) (((double) (((double) sqrt(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))))) + ((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))) * ((double) log(((double) exp(((double) (((double) sqrt(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))))) - ((double) pow(((double) sqrt(x)), ((double) (1.0 / n))))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 4.789554384389695e-16)) {
			VAR_1 = ((double) (((double) (((double) (1.0 / x)) * ((double) (((double) (1.0 / n)) - ((double) (((double) log(((double) (1.0 / 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) < -6.542401718580293e-07

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied unpow-prod-down1.3

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

    5. Applied add-sqr-sqrt1.3

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

    6. Applied difference-of-squares1.3

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

    8. Applied add-log-exp1.5

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

    9. Applied add-log-exp1.5

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

    10. Applied diff-log1.5

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

    11. Simplified1.5

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

    if -6.542401718580293e-07 < (/ 1.0 n) < 4.789554384389695e-16

    1. Initial program 45.2

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified31.7

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

    if 4.789554384389695e-16 < (/ 1.0 n)

    1. Initial program 8.9

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

    3. Applied flip--8.9

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

    4. Simplified8.8

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

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.5424017185802929 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.7895543843896953 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))