Average Error: 32.1 → 23.4
Time: 12.0s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \leq -124899201.97469188:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \leq 8.437394065986364:\\ \;\;\;\;{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(n \cdot n\right) \cdot \sqrt[3]{x}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \leq -124899201.97469188:\\
\;\;\;\;\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{elif}\;n \leq 8.437394065986364:\\
\;\;\;\;{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(n \cdot n\right) \cdot \sqrt[3]{x}\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 (<= n -124899201.97469188)
   (+ (- (/ (/ 1.0 x) n) (/ 0.5 (* x (* n x)))) (/ (log x) (* x (* n n))))
   (if (<= n 8.437394065986364)
     (-
      (* (pow (sqrt (+ 1.0 x)) (/ 1.0 n)) (pow (sqrt (+ 1.0 x)) (/ 1.0 n)))
      (pow x (/ 1.0 n)))
     (+
      (- (/ 1.0 (* n x)) (/ 0.5 (* x (* n x))))
      (/ (log x) (* (* (cbrt x) (cbrt x)) (* (* n n) (cbrt x))))))))
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 <= -124899201.97469188) {
		tmp = (((1.0 / x) / n) - (0.5 / (x * (n * x)))) + (log(x) / (x * (n * n)));
	} else if (n <= 8.437394065986364) {
		tmp = (pow(sqrt(1.0 + x), (1.0 / n)) * pow(sqrt(1.0 + x), (1.0 / n))) - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / (n * x)) - (0.5 / (x * (n * x)))) + (log(x) / ((cbrt(x) * cbrt(x)) * ((n * n) * cbrt(x))));
	}
	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 n < -124899201.974691883

    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 32.7

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

    3. Simplified32.5

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

    5. Applied associate-/r*_binary64_2232.1

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

    if -124899201.974691883 < n < 8.4373940659863642

    1. Initial program 2.3

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

    3. Applied add-sqr-sqrt_binary64_1002.3

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

    4. Applied unpow-prod-down_binary64_1572.3

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

    if 8.4373940659863642 < n

    1. Initial program 44.6

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

    2. Taylor expanded around inf 32.8

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

    3. Simplified32.7

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

    5. Applied add-cube-cbrt_binary64_11332.7

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

    6. Applied associate-*l*_binary64_1932.7

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

    7. Simplified32.7

      \[\leadsto \left(\frac{1}{x \cdot n} - \frac{0.5}{x \cdot \left(x \cdot n\right)}\right) + \frac{\log x}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\left(n \cdot n\right) \cdot \sqrt[3]{x}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification23.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -124899201.97469188:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \leq 8.437394065986364:\\ \;\;\;\;{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(n \cdot n\right) \cdot \sqrt[3]{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020295 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))