Average Error: 32.6 → 24.1
Time: 13.1s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -6.219281535401505 \cdot 10^{-12}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 1.006162847500964 \cdot 10^{-17}:\\ \;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\frac{0.5}{n \cdot x} - \left(\frac{0.25}{x \cdot \left(n \cdot x\right)} + \frac{-0.25}{x} \cdot \frac{\log x}{n \cdot n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -6.219281535401505 \cdot 10^{-12}:\\
\;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \leq 1.006162847500964 \cdot 10^{-17}:\\
\;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\frac{0.5}{n \cdot x} - \left(\frac{0.25}{x \cdot \left(n \cdot x\right)} + \frac{-0.25}{x} \cdot \frac{\log x}{n \cdot n}\right)\right)\\

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

\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 (<= (/ 1.0 n) -6.219281535401505e-12)
   (log (exp (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) 1.006162847500964e-17)
     (*
      (+ (pow x (/ 0.5 n)) (pow (+ 1.0 x) (/ 0.5 n)))
      (-
       (/ 0.5 (* n x))
       (+ (/ 0.25 (* x (* n x))) (* (/ -0.25 x) (/ (log x) (* n n))))))
     (exp (log (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 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 ((1.0 / n) <= -6.219281535401505e-12) {
		tmp = log(exp(pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n))));
	} else if ((1.0 / n) <= 1.006162847500964e-17) {
		tmp = (pow(x, (0.5 / n)) + pow((1.0 + x), (0.5 / n))) * ((0.5 / (n * x)) - ((0.25 / (x * (n * x))) + ((-0.25 / x) * (log(x) / (n * n)))));
	} else {
		tmp = exp(log(pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / 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 3 regimes

  2. if (/.f64 1 n) < -6.21928153540150479e-12

    1. Initial program 2.4

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

    3. Applied add-log-exp_binary643.1

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

    4. Applied add-log-exp_binary642.9

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

    5. Applied diff-log_binary642.9

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

    6. Simplified2.9

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

    if -6.21928153540150479e-12 < (/.f64 1 n) < 1.006162847500964e-17

    1. Initial program 45.3

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

    3. Applied sqr-pow_binary6445.3

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

    4. Applied sqr-pow_binary6445.3

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

    5. Applied difference-of-squares_binary6445.3

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

    6. Simplified45.3

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

    7. Simplified45.3

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

    8. Taylor expanded around inf 32.8

      \[\leadsto \left({x}^{\left(\frac{0.5}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

    9. Simplified32.7

      \[\leadsto \left({x}^{\left(\frac{0.5}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \color{blue}{\left(\frac{0.5}{x \cdot n} - \left(\frac{0.25}{x \cdot \left(x \cdot n\right)} + \frac{-0.25}{x} \cdot \frac{\log x}{n \cdot n}\right)\right)}\]

    if 1.006162847500964e-17 < (/.f64 1 n)

    1. Initial program 9.8

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

    3. Applied add-exp-log_binary649.8

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

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -6.219281535401505 \cdot 10^{-12}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 1.006162847500964 \cdot 10^{-17}:\\ \;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\frac{0.5}{n \cdot x} - \left(\frac{0.25}{x \cdot \left(n \cdot x\right)} + \frac{-0.25}{x} \cdot \frac{\log x}{n \cdot n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))