Average Error: 33.0 → 24.5
Time: 15.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -44775.669105648325 \lor \neg \left(n \le 28170.0865394064494\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -44775.669105648325 \lor \neg \left(n \le 28170.0865394064494\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\

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

\end{array}
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 VAR;
	if (((n <= -44775.669105648325) || !(n <= 28170.08653940645))) {
		VAR = fma(1.0, (1.0 / (x * n)), -fma(0.5, (1.0 / (pow(x, 2.0) * n)), (1.0 * (log((1.0 / x)) / (x * pow(n, 2.0))))));
	} else {
		VAR = (log(exp((pow((x + 1.0), (1.0 / n)) - pow(sqrt(x), (2.0 * (1.0 / n)))))) + fma(pow(sqrt(x), (1.0 / n)), -pow(sqrt(x), (1.0 / n)), pow(sqrt(x), (2.0 * (1.0 / 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 n < -44775.669105648325 or 28170.08653940645 < n

    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 33.1

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

    3. Simplified33.1

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

    if -44775.669105648325 < n < 28170.08653940645

    1. Initial program 2.3

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

    3. Applied add-sqr-sqrt2.4

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

    4. Applied unpow-prod-down2.3

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

    5. Applied *-un-lft-identity2.3

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

    6. Applied unpow-prod-down2.3

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

    7. Applied prod-diff2.4

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

    8. Simplified2.4

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

    9. Simplified2.4

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right) + \color{blue}{\mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\]
    10. Using strategy rm

    11. Applied add-log-exp2.7

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

    12. Applied add-log-exp2.7

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

    13. Applied diff-log2.7

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

    14. Simplified2.7

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

  4. Final simplification24.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -44775.669105648325 \lor \neg \left(n \le 28170.0865394064494\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))