Average Error: 32.7 → 24.0
Time: 15.1s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -18.09077906127504 \lor \neg \left(\frac{1}{n} \le 0.001178309117642572\right):\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{\frac{\log x}{n \cdot n} + \frac{1}{n}}{\sqrt{x}}}{\sqrt{x}} - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -18.09077906127504 \lor \neg \left(\frac{1}{n} \le 0.001178309117642572\right):\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
}
↓
double code(double x, double n) {
	double VAR;
	if ((((1.0 / n) <= -18.090779061275043) || !((1.0 / n) <= 0.0011783091176425717))) {
		VAR = ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
	} else {
		VAR = ((double) (((double) (1.0 * ((((double) ((((double) log(x)) / ((double) (n * n))) + (1.0 / n))) / ((double) sqrt(x))) / ((double) sqrt(x))))) - (0.5 / ((double) log(((double) exp(((double) (((double) pow(x, 2.0)) * n)))))))));
	}
	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 2 regimes

  2. if (/ 1.0 n) < -18.09077906127504 or 0.001178309117642572 < (/ 1.0 n)

    1. Initial program 1.8

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

    if -18.09077906127504 < (/ 1.0 n) < 0.001178309117642572

    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 33.0

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

    3. Simplified32.4

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

    5. Applied div-inv32.4

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

    6. Applied associate-*l*32.4

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

    7. Simplified32.4

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{\log x}{n \cdot n} + \frac{1}{n}}{x}} - \frac{0.5}{{x}^{2} \cdot n}\]
    8. Using strategy rm

    9. Applied add-log-exp32.4

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

    11. Applied add-sqr-sqrt32.4

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

    12. Applied associate-/r*32.4

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -18.09077906127504 \lor \neg \left(\frac{1}{n} \le 0.001178309117642572\right):\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{\frac{\log x}{n \cdot n} + \frac{1}{n}}{\sqrt{x}}}{\sqrt{x}} - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))