Average Error: 32.7 → 7.0
Time: 36.2s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 9093778.054138303:\\ \;\;\;\;\left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(\frac{x + 1}{x}\right)}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 9093778.054138303:\\
\;\;\;\;\left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(\frac{x + 1}{x}\right)}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\

\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 9093778.054138303)
   (+
    (-
     (+
      (+
       (* 0.5 (/ (pow (log (+ x 1.0)) 2.0) (* n n)))
       (* 0.16666666666666666 (pow (/ (log (+ x 1.0)) n) 3.0)))
      (/ (log (/ (+ x 1.0) x)) n))
     (* 0.16666666666666666 (pow (/ (log x) n) 3.0)))
    (* (/ (pow (log x) 2.0) (* n n)) -0.5))
   (/ (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 <= 9093778.054138303) {
		tmp = ((((0.5 * (pow(log(x + 1.0), 2.0) / (n * n))) + (0.16666666666666666 * pow((log(x + 1.0) / n), 3.0))) + (log((x + 1.0) / x) / n)) - (0.16666666666666666 * pow((log(x) / n), 3.0))) + ((pow(log(x), 2.0) / (n * n)) * -0.5);
	} 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 < 9093778.054138303

    1. Initial program 46.8

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

    2. Taylor expanded around inf 13.9

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

    3. Simplified13.9

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

    5. Applied diff-log_binary64_17013.8

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

    if 9093778.054138303 < x

    1. Initial program 20.6

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

    2. Taylor expanded around inf 1.2

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

    3. Simplified1.2

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9093778.054138303:\\ \;\;\;\;\left(\left(\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{n \cdot n} + 0.16666666666666666 \cdot {\left(\frac{\log \left(x + 1\right)}{n}\right)}^{3}\right) + \frac{\log \left(\frac{x + 1}{x}\right)}{n}\right) - 0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))