Average Error: 33.3 → 7.1
Time: 19.8s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 2623.9070389239055:\\ \;\;\;\;\begin{array}{l} t_0 := \log \left(x + 1\right)\\ \left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \left(0.16666666666666666 \cdot \frac{{t_0}^{3}}{{n}^{3}} + \frac{t_0}{n}\right)\right) - \left(0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 2623.9070389239055:\\
\;\;\;\;\begin{array}{l}
t_0 := \log \left(x + 1\right)\\
\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \left(0.16666666666666666 \cdot \frac{{t_0}^{3}}{{n}^{3}} + \frac{t_0}{n}\right)\right) - \left(0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\right)
\end{array}\\

\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 2623.9070389239055)
   (let* ((t_0 (log (+ x 1.0))))
     (-
      (+
       (* 0.5 (/ (pow t_0 2.0) (pow n 2.0)))
       (+ (* 0.16666666666666666 (/ (pow t_0 3.0) (pow n 3.0))) (/ t_0 n)))
      (+
       (* 0.16666666666666666 (/ (pow (log x) 3.0) (pow n 3.0)))
       (+ (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0))) (/ (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 <= 2623.9070389239055) {
		double t_0_1 = log(x + 1.0);
		tmp = ((0.5 * (pow(t_0_1, 2.0) / pow(n, 2.0))) + ((0.16666666666666666 * (pow(t_0_1, 3.0) / pow(n, 3.0))) + (t_0_1 / n))) - ((0.16666666666666666 * (pow(log(x), 3.0) / pow(n, 3.0))) + ((0.5 * (pow(log(x), 2.0) / pow(n, 2.0))) + (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 < 2623.90703892390547

    1. Initial program 47.2

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

    2. Taylor expanded in n around inf 13.4

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

    if 2623.90703892390547 < x

    1. Initial program 21.2

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

    2. Taylor expanded in x around inf 1.6

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

    3. Simplified1.6

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2623.9070389239055:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}} + \frac{\log \left(x + 1\right)}{n}\right)\right) - \left(0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))