Average Error: 32.6 → 24.3
Time: 9.8s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.0760340368957404 \cdot 10^{-20}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 6.011439729085773 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\sqrt[3]{{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1.0760340368957404 \cdot 10^{-20}:\\
\;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \leq 6.011439729085773 \cdot 10^{-10}:\\
\;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;{e}^{\left(\sqrt[3]{{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}\\

\end{array}
(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
↓
(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.0760340368957404e-20)
   (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))
   (if (<= (/ 1.0 n) 6.011439729085773e-10)
     (+ (- (/ 1.0 (* n x)) (/ 0.5 (* x (* n x)))) (/ (log x) (* x (* n n))))
     (pow
      E
      (cbrt
       (pow (log (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))) 3.0))))))
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 tmp;
	if ((1.0 / n) <= -1.0760340368957404e-20) {
		tmp = pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n));
	} else if ((1.0 / n) <= 6.011439729085773e-10) {
		tmp = ((1.0 / (n * x)) - (0.5 / (x * (n * x)))) + (log(x) / (x * (n * n)));
	} else {
		tmp = pow(((double) M_E), cbrt(pow(log(pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n))), 3.0)));
	}
	return tmp;
}

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 (/.f64 1 n) < -1.0760340368957404e-20

    1. Initial program 4.6

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

    if -1.0760340368957404e-20 < (/.f64 1 n) < 6.011439729085773e-10

    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 33.3

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

    3. Simplified33.2

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

    if 6.011439729085773e-10 < (/.f64 1 n)

    1. Initial program 5.4

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

    3. Applied add-exp-log_binary645.4

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

    5. Applied add-cbrt-cube_binary645.4

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

    6. Simplified5.4

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

    8. Applied *-un-lft-identity_binary645.4

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

    9. Applied exp-prod_binary645.4

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

    10. Simplified5.4

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

  4. Final simplification24.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.0760340368957404 \cdot 10^{-20}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 6.011439729085773 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\sqrt[3]{{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))