Average Error: 32.7 → 24.2
Time: 13.0s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0021522141285571558:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2.5126692412963367 \cdot 10^{-05}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -0.0021522141285571558:\\
\;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\

\mathbf{elif}\;\frac{1}{n} \leq 2.5126692412963367 \cdot 10^{-05}:\\
\;\;\;\;\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}:\\
\;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\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) -0.0021522141285571558)
   (cbrt (pow (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n))) 3.0))
   (if (<= (/ 1.0 n) 2.5126692412963367e-05)
     (+ (- (/ 1.0 (* n x)) (/ 0.5 (* x (* n x)))) (/ (log x) (* x (* n n))))
     (/
      (- (pow (+ 1.0 x) (/ 2.0 n)) (pow x (/ 2.0 n)))
      (+ (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))))
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) <= -0.0021522141285571558) {
		tmp = cbrt(pow((pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n))), 3.0));
	} else if ((1.0 / n) <= 2.5126692412963367e-05) {
		tmp = ((1.0 / (n * x)) - (0.5 / (x * (n * x)))) + (log(x) / (x * (n * n)));
	} else {
		tmp = (pow((1.0 + x), (2.0 / n)) - pow(x, (2.0 / n))) / (pow((1.0 + x), (1.0 / n)) + pow(x, (1.0 / n)));
	}
	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) < -0.0021522141285571558

    1. Initial program 0.4

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

    3. Applied add-cbrt-cube_binary64_1140.5

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

    4. Simplified0.5

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

    if -0.0021522141285571558 < (/.f64 1 n) < 2.5126692412963367e-5

    1. Initial program 44.5

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

    2. Taylor expanded around inf 32.9

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

    3. Simplified32.8

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

    if 2.5126692412963367e-5 < (/.f64 1 n)

    1. Initial program 5.9

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

    3. Applied flip--_binary64_535.9

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

    4. Simplified5.9

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

  4. Final simplification24.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0021522141285571558:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2.5126692412963367 \cdot 10^{-05}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))