Average Error: 33.1 → 11.5
Time: 27.2s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.1917113743328619 \cdot 10^{-32}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.1382942965891877 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(1 + x\right)}{n}\right)}^{3}\right) + \frac{\log \left(1 + x\right) - \log x}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{elif}\;\frac{1}{n} \leq 2.7004662572109683 \cdot 10^{-07}:\\ \;\;\;\;\frac{\left(\frac{1}{n} + 0.5 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{2}}{{n}^{3}}\right) - \left(0.16666666666666666 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{3}}{{n}^{4}} + \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1.1917113743328619 \cdot 10^{-32}:\\
\;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\

\mathbf{elif}\;\frac{1}{n} \leq 1.1382942965891877 \cdot 10^{-50}:\\
\;\;\;\;\left(\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(1 + x\right)}{n}\right)}^{3}\right) + \frac{\log \left(1 + x\right) - \log x}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\

\mathbf{elif}\;\frac{1}{n} \leq 2.7004662572109683 \cdot 10^{-07}:\\
\;\;\;\;\frac{\left(\frac{1}{n} + 0.5 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{2}}{{n}^{3}}\right) - \left(0.16666666666666666 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{3}}{{n}^{4}} + \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\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) -1.1917113743328619e-32)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1.1382942965891877e-50)
     (+
      (-
       (+
        (+
         (* 0.5 (/ (pow (log (+ 1.0 x)) 2.0) (* n n)))
         (* 0.16666666666666666 (pow (/ (log (+ 1.0 x)) n) 3.0)))
        (/ (- (log (+ 1.0 x)) (log x)) n))
       (* 0.16666666666666666 (pow (/ (log x) n) 3.0)))
      (* (/ (pow (log x) 2.0) (* n n)) -0.5))
     (if (<= (/ 1.0 n) 2.7004662572109683e-07)
       (/
        (-
         (+ (/ 1.0 n) (* 0.5 (/ (pow (log (/ 1.0 x)) 2.0) (pow n 3.0))))
         (+
          (* 0.16666666666666666 (/ (pow (log (/ 1.0 x)) 3.0) (pow n 4.0)))
          (/ (log (/ 1.0 x)) (pow n 2.0))))
        x)
       (/
        (- (pow (+ 1.0 x) (/ 2.0 n)) (pow x (/ 2.0 n)))
        (+ (pow x (/ 1.0 n)) (pow (+ 1.0 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) <= -1.1917113743328619e-32) {
		tmp = exp(log(x) / n) / (n * x);
	} else if ((1.0 / n) <= 1.1382942965891877e-50) {
		tmp = ((((0.5 * (pow(log(1.0 + x), 2.0) / (n * n))) + (0.16666666666666666 * pow((log(1.0 + x) / n), 3.0))) + ((log(1.0 + x) - log(x)) / n)) - (0.16666666666666666 * pow((log(x) / n), 3.0))) + ((pow(log(x), 2.0) / (n * n)) * -0.5);
	} else if ((1.0 / n) <= 2.7004662572109683e-07) {
		tmp = (((1.0 / n) + (0.5 * (pow(log(1.0 / x), 2.0) / pow(n, 3.0)))) - ((0.16666666666666666 * (pow(log(1.0 / x), 3.0) / pow(n, 4.0))) + (log(1.0 / x) / pow(n, 2.0)))) / x;
	} else {
		tmp = (pow((1.0 + x), (2.0 / n)) - pow(x, (2.0 / n))) / (pow(x, (1.0 / n)) + pow((1.0 + 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 4 regimes

  2. if (/.f64 1 n) < -1.19171137433286189e-32

    1. Initial program 8.1

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

    2. Taylor expanded around -inf 64.0

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

    3. Simplified5.0

      \[\leadsto \color{blue}{\frac{e^{\frac{0 + \log x}{n}}}{x \cdot n}}\]

    if -1.19171137433286189e-32 < (/.f64 1 n) < 1.1382942965891877e-50

    1. Initial program 43.9

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

    2. Taylor expanded around inf 12.9

      \[\leadsto \color{blue}{\left(\frac{\log \left(x + 1\right)}{n} + \left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + 0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}}\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)}\]

    3. Simplified12.9

      \[\leadsto \color{blue}{\left(\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(1 + x\right)}{n}\right)}^{3}\right) + \frac{\log \left(1 + x\right) - \log x}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5}\]

    if 1.1382942965891877e-50 < (/.f64 1 n) < 2.7004662572109683e-7

    1. Initial program 55.0

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

    2. Taylor expanded around inf 26.6

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + \left(\frac{\log \left(x + 1\right)}{n} + 0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)}\]

    3. Simplified26.6

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n \cdot n} + \left(0.041666666666666664 \cdot {\left(\frac{\log \left(1 + x\right)}{n}\right)}^{4} + \left(\frac{\log \left(1 + x\right)}{n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(1 + x\right)}{n}\right)}^{3}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{n \cdot n} + \left(0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3} + 0.041666666666666664 \cdot {\left(\frac{\log x}{n}\right)}^{4}\right)\right)\right)}\]

    4. Taylor expanded around inf 31.2

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{2}}{{n}^{3}} + \frac{1}{n}\right) - \left(0.16666666666666666 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{3}}{{n}^{4}} + \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}{x}}\]

    if 2.7004662572109683e-7 < (/.f64 1 n)

    1. Initial program 7.3

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

    3. Applied flip--_binary647.4

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

    4. Simplified7.4

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

    5. Simplified7.4

      \[\leadsto \frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{\color{blue}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.1917113743328619 \cdot 10^{-32}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.1382942965891877 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(1 + x\right)}{n}\right)}^{3}\right) + \frac{\log \left(1 + x\right) - \log x}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{elif}\;\frac{1}{n} \leq 2.7004662572109683 \cdot 10^{-07}:\\ \;\;\;\;\frac{\left(\frac{1}{n} + 0.5 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{2}}{{n}^{3}}\right) - \left(0.16666666666666666 \cdot \frac{{\log \left(\frac{1}{x}\right)}^{3}}{{n}^{4}} + \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021105 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))