Average Error: 29.7 → 22.2
Time: 9.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -24134035953.6544762:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{{x}^{2}}\right) - \left(-e^{\log \left(\frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right)\\ \mathbf{elif}\;n \le 134024401.69677079:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - e^{\log \left(\frac{\frac{0.5}{n}}{{x}^{2}}\right)}\right) - \left(-\frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -24134035953.6544762:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{{x}^{2}}\right) - \left(-e^{\log \left(\frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right)\\

\mathbf{elif}\;n \le 134024401.69677079:\\
\;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

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

\end{array}
double code(double x, double n) {
	return (pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)));
}
↓
double code(double x, double n) {
	double temp;
	if ((n <= -24134035953.654476)) {
		temp = ((((1.0 / n) / x) - ((0.5 / n) / pow(x, 2.0))) - -exp(log(((log(x) * 1.0) / (x * pow(n, 2.0))))));
	} else {
		double temp_1;
		if ((n <= 134024401.69677079)) {
			temp_1 = log(exp((pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)))));
		} else {
			temp_1 = ((((1.0 / n) / x) - exp(log(((0.5 / n) / pow(x, 2.0))))) - -((log(x) * 1.0) / (x * pow(n, 2.0))));
		}
		temp = temp_1;
	}
	return temp;
}

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 < -24134035953.654476

    1. Initial program 45.6

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

    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

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

    5. Applied sub-neg31.5

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

    6. Applied associate--r+31.5

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

    8. Applied add-exp-log64.0

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

    9. Applied pow-exp64.0

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

    10. Applied add-exp-log64.0

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

    11. Applied prod-exp64.0

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

    12. Applied add-exp-log64.0

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

    13. Applied add-exp-log64.0

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

    14. Applied prod-exp64.0

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

    15. Applied div-exp64.0

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

    16. Simplified31.6

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

    if -24134035953.654476 < n < 134024401.69677079

    1. Initial program 9.1

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

    3. Applied add-log-exp9.2

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

    4. Applied add-log-exp9.2

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

    5. Applied diff-log9.2

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

    6. Simplified9.2

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

    if 134024401.69677079 < n

    1. Initial program 44.4

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

    2. Taylor expanded around inf 32.6

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

    3. Simplified31.9

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

    5. Applied sub-neg31.9

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

    6. Applied associate--r+31.9

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

    8. Applied add-exp-log31.9

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

    9. Applied pow-exp31.9

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

    10. Applied add-exp-log31.9

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

    11. Applied add-exp-log31.9

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

    12. Applied div-exp31.9

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

    13. Applied div-exp31.8

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

    14. Simplified31.9

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -24134035953.6544762:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{{x}^{2}}\right) - \left(-e^{\log \left(\frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right)\\ \mathbf{elif}\;n \le 134024401.69677079:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - e^{\log \left(\frac{\frac{0.5}{n}}{{x}^{2}}\right)}\right) - \left(-\frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))