Average Error: 32.3 → 23.5
Time: 9.8s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -7.474757219983495:\\ \;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 3.0292317079672966 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -7.474757219983495:\\
\;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\

\mathbf{elif}\;\frac{1}{n} \leq 3.0292317079672966 \cdot 10^{-20}:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {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) -7.474757219983495)
   (exp (log (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) 3.0292317079672966e-20)
     (+ (- (/ (/ 1.0 x) n) (/ 0.5 (* x (* n x)))) (/ (log x) (* x (* n n))))
     (/
      (- (pow (pow (+ 1.0 x) (/ 1.0 n)) 3.0) (pow (pow x (/ 1.0 n)) 3.0))
      (+
       (pow (+ 1.0 x) (/ 2.0 n))
       (+
        (pow x (/ 2.0 n))
        (* (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) <= -7.474757219983495) {
		tmp = exp(log(pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n))));
	} else if ((1.0 / n) <= 3.0292317079672966e-20) {
		tmp = (((1.0 / x) / n) - (0.5 / (x * (n * x)))) + (log(x) / (x * (n * n)));
	} else {
		tmp = (pow(pow((1.0 + x), (1.0 / n)), 3.0) - pow(pow(x, (1.0 / n)), 3.0)) / (pow((1.0 + x), (2.0 / n)) + (pow(x, (2.0 / n)) + (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) < -7.4747572199834948

    1. Initial program 0.0

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

    3. Applied add-exp-log_binary64_1160.0

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

    if -7.4747572199834948 < (/.f64 1 n) < 3.02923170796729657e-20

    1. Initial program 43.4

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

    2. Taylor expanded around inf 31.8

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

    3. Simplified31.7

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

    5. Applied associate-/r*_binary64_2231.0

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

    if 3.02923170796729657e-20 < (/.f64 1 n)

    1. Initial program 11.4

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

    3. Applied flip3--_binary64_8211.4

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

    4. Simplified11.4

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -7.474757219983495:\\ \;\;\;\;e^{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 3.0292317079672966 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} + \left({x}^{\left(\frac{2}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))