Average Error: 32.8 → 23.3
Time: 9.2s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \leq -19797961.77181827 \lor \neg \left(n \leq 829455.0002747169\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{0.5}{n \cdot x}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{e^{\frac{\log x}{n}}} \cdot \sqrt{e^{\frac{\log x}{n}}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \leq -19797961.77181827 \lor \neg \left(n \leq 829455.0002747169\right):\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{0.5}{n \cdot x}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\

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

\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 (or (<= n -19797961.77181827) (not (<= n 829455.0002747169)))
   (+ (* (/ 1.0 x) (- (/ 1.0 n) (/ 0.5 (* n x)))) (/ (log x) (* x (* n n))))
   (-
    (pow (+ 1.0 x) (/ 1.0 n))
    (* (sqrt (exp (/ (log x) n))) (sqrt (exp (/ (log x) 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 ((n <= -19797961.77181827) || !(n <= 829455.0002747169)) {
		tmp = ((1.0 / x) * ((1.0 / n) - (0.5 / (n * x)))) + (log(x) / (x * (n * n)));
	} else {
		tmp = pow((1.0 + x), (1.0 / n)) - (sqrt(exp(log(x) / n)) * sqrt(exp(log(x) / 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 2 regimes

  2. if n < -19797961.771818269 or 829455.00027471688 < 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 32.4

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

    3. Simplified32.3

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

    5. Applied *-un-lft-identity_binary6432.3

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

    6. Applied times-frac_binary6432.3

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

    7. Applied *-un-lft-identity_binary6432.3

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

    8. Applied times-frac_binary6431.7

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

    9. Applied distribute-lft-out--_binary6431.7

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

    if -19797961.771818269 < n < 829455.00027471688

    1. Initial program 2.8

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

    3. Applied add-exp-log_binary642.9

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

    4. Applied pow-exp_binary642.9

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

    5. Simplified2.9

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

    7. Applied add-sqr-sqrt_binary642.9

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

  4. Final simplification23.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -19797961.77181827 \lor \neg \left(n \leq 829455.0002747169\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{0.5}{n \cdot x}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{e^{\frac{\log x}{n}}} \cdot \sqrt{e^{\frac{\log x}{n}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))