Average Error: 32.7 → 23.5
Time: 10.9s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5642061168275624 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 8.695048917784674 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\log x}{n \cdot n}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1.5642061168275624 \cdot 10^{-07}:\\
\;\;\;\;\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 8.695048917784674 \cdot 10^{-12}:\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\log x}{n \cdot n}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\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) -1.5642061168275624e-07)
   (log (exp (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) 8.695048917784674e-12)
     (- (* (/ 1.0 x) (+ (/ 1.0 n) (/ (log x) (* n n)))) (/ 0.5 (* x (* n x))))
     (/
      (- (pow (+ 1.0 x) (* (/ 1.0 n) 2.0)) (pow x (* (/ 1.0 n) 2.0)))
      (+ (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))))
double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
}
↓
double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -1.5642061168275624e-07)) {
		tmp = ((double) log(((double) exp(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))))))));
	} else {
		double tmp_1;
		if (((1.0 / n) <= 8.695048917784674e-12)) {
			tmp_1 = ((double) (((double) ((1.0 / x) * ((double) ((1.0 / n) + (((double) log(x)) / ((double) (n * n))))))) - (0.5 / ((double) (x * ((double) (n * x)))))));
		} else {
			tmp_1 = (((double) (((double) pow(((double) (1.0 + x)), ((double) ((1.0 / n) * 2.0)))) - ((double) pow(x, ((double) ((1.0 / n) * 2.0)))))) / ((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) + ((double) pow(x, (1.0 / n))))));
		}
		tmp = tmp_1;
	}
	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 1.0 n) < -1.56420611682756241e-7

    1. Initial program 1.4

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

    3. Applied add-log-exp_binary641.7

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

    4. Applied add-log-exp_binary641.6

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

    5. Applied diff-log_binary641.6

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

    6. Simplified1.6

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

    if -1.56420611682756241e-7 < (/.f64 1.0 n) < 8.69504891778467354e-12

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.5

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

    3. Simplified31.7

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

    if 8.69504891778467354e-12 < (/.f64 1.0 n)

    1. Initial program 8.5

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

    3. Applied flip--_binary648.5

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

    4. Simplified8.5

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5642061168275624 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 8.695048917784674 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} + \frac{\log x}{n \cdot n}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))