Average Error: 32.9 → 7.1
Time: 15.8s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 502.9330801505896:\\ \;\;\;\;\frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 502.9330801505896:\\
\;\;\;\;\frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot 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 (<= x 502.9330801505896)
   (/
    (/ (- (* (log1p x) (log1p x)) (* (log x) (log x))) (+ (log1p x) (log x)))
    n)
   (/ (pow x (/ 1.0 n)) (* 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 (x <= 502.9330801505896) {
		tmp = (((log1p(x) * log1p(x)) - (log(x) * log(x))) / (log1p(x) + log(x))) / n;
	} else {
		tmp = pow(x, (1.0 / n)) / (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 x < 502.9330801505896

    1. Initial program 47.1

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

    2. Taylor expanded in n around inf 13.8

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

    3. Simplified13.7

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

    4. Applied flip--_binary6413.8

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

    5. Simplified13.8

      \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\color{blue}{\log x + \mathsf{log1p}\left(x\right)}}}{n} \]

    if 502.9330801505896 < x

    1. Initial program 20.8

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

    2. Taylor expanded in x around inf 1.4

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

    3. Simplified1.4

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 502.9330801505896:\\ \;\;\;\;\frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))