Average Error: 33.3 → 24.7
Time: 13.2s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5.1540558658575505 \cdot 10^{-39}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.7832423444055043 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -5.1540558658575505 \cdot 10^{-39}:\\
\;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\right)}^{3}}\\

\mathbf{elif}\;\frac{1}{n} \leq 1.7832423444055043 \cdot 10^{-11}:\\
\;\;\;\;\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}:\\
\;\;\;\;\sqrt[3]{{\left(\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)}}\right)}^{3}}\\

\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) -5.1540558658575505e-39)
   (cbrt
    (pow
     (cbrt (pow (- (pow (+ 1.0 x) (/ 1.0 n)) (exp (/ (log x) n))) 3.0))
     3.0))
   (if (<= (/ 1.0 n) 1.7832423444055043e-11)
     (+ (- (/ 1.0 (* n x)) (/ 0.5 (* x (* n x)))) (/ (log x) (* x (* n n))))
     (cbrt
      (pow
       (/
        (- (pow (+ 1.0 x) (/ 2.0 n)) (pow x (/ 2.0 n)))
        (+ (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) <= -5.1540558658575505e-39) {
		tmp = cbrt(pow(cbrt(pow((pow((1.0 + x), (1.0 / n)) - exp(log(x) / n)), 3.0)), 3.0));
	} else if ((1.0 / n) <= 1.7832423444055043e-11) {
		tmp = ((1.0 / (n * x)) - (0.5 / (x * (n * x)))) + (log(x) / (x * (n * n)));
	} else {
		tmp = cbrt(pow(((pow((1.0 + x), (2.0 / n)) - pow(x, (2.0 / n))) / (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) < -5.1540558658575505e-39

    1. Initial program 9.9

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

    3. Applied add-cbrt-cube_binary64_44710.1

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

    4. Simplified10.1

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

    6. Applied pow-to-exp_binary64_48010.1

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

    7. Simplified10.1

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

    9. Applied add-cbrt-cube_binary64_44710.1

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

    10. Simplified10.1

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

    if -5.1540558658575505e-39 < (/.f64 1 n) < 1.78324234440550429e-11

    1. Initial program 45.1

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified32.3

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

    if 1.78324234440550429e-11 < (/.f64 1 n)

    1. Initial program 7.5

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

    3. Applied add-cbrt-cube_binary64_4477.5

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

    4. Simplified7.5

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

    6. Applied flip--_binary64_3897.5

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

    7. Simplified7.4

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

  4. Final simplification24.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5.1540558658575505 \cdot 10^{-39}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.7832423444055043 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))