Average Error: 31.8 → 11.0
Time: 18.8s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \leq -236843.41110673678:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + \left(\frac{\log \left(x + 1\right)}{n} + 0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1119850077.417114:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;n \leq 8.578898669612482 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(x + 1\right) - \log x}{n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \leq -236843.41110673678:\\
\;\;\;\;\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + \left(\frac{\log \left(x + 1\right)}{n} + 0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)\\

\mathbf{elif}\;n \leq 1119850077.417114:\\
\;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;n \leq 8.578898669612482 \cdot 10^{+57}:\\
\;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(x + 1\right) - \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 (<= n -236843.41110673678)
   (-
    (+
     (* 0.5 (/ (pow (log (+ x 1.0)) 2.0) (pow n 2.0)))
     (+
      (* 0.041666666666666664 (/ (pow (log (+ x 1.0)) 4.0) (pow n 4.0)))
      (+
       (/ (log (+ x 1.0)) n)
       (* 0.16666666666666666 (/ (pow (log (+ x 1.0)) 3.0) (pow n 3.0))))))
    (+
     (/ (log x) n)
     (+
      (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0)))
      (+
       (* 0.041666666666666664 (/ (pow (log x) 4.0) (pow n 4.0)))
       (* 0.16666666666666666 (/ (pow (log x) 3.0) (pow n 3.0)))))))
   (if (<= n 1119850077.417114)
     (*
      (cbrt (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
      (*
       (cbrt (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
       (cbrt (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))))
     (if (<= n 8.578898669612482e+57)
       (/ (/ (exp (/ (log x) n)) x) n)
       (/ (- (log (+ x 1.0)) (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 <= -236843.41110673678) {
		tmp = ((0.5 * (pow(log(x + 1.0), 2.0) / pow(n, 2.0))) + ((0.041666666666666664 * (pow(log(x + 1.0), 4.0) / pow(n, 4.0))) + ((log(x + 1.0) / n) + (0.16666666666666666 * (pow(log(x + 1.0), 3.0) / pow(n, 3.0)))))) - ((log(x) / n) + ((0.5 * (pow(log(x), 2.0) / pow(n, 2.0))) + ((0.041666666666666664 * (pow(log(x), 4.0) / pow(n, 4.0))) + (0.16666666666666666 * (pow(log(x), 3.0) / pow(n, 3.0))))));
	} else if (n <= 1119850077.417114) {
		tmp = cbrt(pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n))) * (cbrt(pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n))) * cbrt(pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n))));
	} else if (n <= 8.578898669612482e+57) {
		tmp = (exp(log(x) / n) / x) / n;
	} else {
		tmp = (log(x + 1.0) - 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 4 regimes

  2. if n < -236843.411106736778

    1. Initial program 44.1

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

    2. Taylor expanded around inf 14.0

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + \left(\frac{\log \left(x + 1\right)}{n} + 0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)}\]

    if -236843.411106736778 < n < 1119850077.41711402

    1. Initial program 2.9

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

    3. Applied add-cube-cbrt_binary642.9

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

    if 1119850077.41711402 < n < 8.578898669612482e57

    1. Initial program 54.7

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

    2. Taylor expanded around inf 31.4

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

    3. Simplified31.4

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}}\]
    4. Using strategy rm

    5. Applied associate-/r*_binary6430.7

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}}\]

    if 8.578898669612482e57 < n

    1. Initial program 42.8

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

    2. Taylor expanded around inf 12.1

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

    3. Simplified12.1

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -236843.41110673678:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + \left(\frac{\log \left(x + 1\right)}{n} + 0.16666666666666666 \cdot \frac{{\log \left(x + 1\right)}^{3}}{{n}^{3}}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1119850077.417114:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;n \leq 8.578898669612482 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(x + 1\right) - \log x}{n}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))