Average Error: 32.7 → 8.6
Time: 16.1s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.9264718152456274 \cdot 10^{-256}:\\ \;\;\;\;\left(\left(\frac{x}{n} + \left(\left(1 + 0.5 \cdot \left(\frac{x}{n} \cdot \frac{x}{n}\right)\right) + 0.16666666666666666 \cdot {\left(\frac{x}{n}\right)}^{3}\right)\right) + \frac{{x}^{3}}{n} \cdot \left(0.3333333333333333 - \frac{0.5}{n}\right)\right) + \left(\frac{x \cdot x}{n} \cdot -0.5 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;x \leq 5.559606535285232:\\ \;\;\;\;\frac{\left(\log \left(x + 1\right) - \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) - \log \left(\sqrt[3]{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-\frac{\log \left(\frac{1}{x}\right)}{n}}}{x \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.9264718152456274 \cdot 10^{-256}:\\
\;\;\;\;\left(\left(\frac{x}{n} + \left(\left(1 + 0.5 \cdot \left(\frac{x}{n} \cdot \frac{x}{n}\right)\right) + 0.16666666666666666 \cdot {\left(\frac{x}{n}\right)}^{3}\right)\right) + \frac{{x}^{3}}{n} \cdot \left(0.3333333333333333 - \frac{0.5}{n}\right)\right) + \left(\frac{x \cdot x}{n} \cdot -0.5 - {x}^{\left(\frac{1}{n}\right)}\right)\\

\mathbf{elif}\;x \leq 5.559606535285232:\\
\;\;\;\;\frac{\left(\log \left(x + 1\right) - \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) - \log \left(\sqrt[3]{x}\right)}{n}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{-\frac{\log \left(\frac{1}{x}\right)}{n}}}{x \cdot 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 (<= x 1.9264718152456274e-256)
   (+
    (+
     (+
      (/ x n)
      (+
       (+ 1.0 (* 0.5 (* (/ x n) (/ x n))))
       (* 0.16666666666666666 (pow (/ x n) 3.0))))
     (* (/ (pow x 3.0) n) (- 0.3333333333333333 (/ 0.5 n))))
    (- (* (/ (* x x) n) -0.5) (pow x (/ 1.0 n))))
   (if (<= x 5.559606535285232)
     (/ (- (- (log (+ x 1.0)) (log (* (cbrt x) (cbrt x)))) (log (cbrt x))) n)
     (/ (exp (- (/ (log (/ 1.0 x)) n))) (* 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 (x <= 1.9264718152456274e-256) {
		tmp = (((x / n) + ((1.0 + (0.5 * ((x / n) * (x / n)))) + (0.16666666666666666 * pow((x / n), 3.0)))) + ((pow(x, 3.0) / n) * (0.3333333333333333 - (0.5 / n)))) + ((((x * x) / n) * -0.5) - pow(x, (1.0 / n)));
	} else if (x <= 5.559606535285232) {
		tmp = ((log(x + 1.0) - log(cbrt(x) * cbrt(x))) - log(cbrt(x))) / n;
	} else {
		tmp = exp(-(log(1.0 / x) / n)) / (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 3 regimes

  2. if x < 1.92647181524562741e-256

    1. Initial program 38.6

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

    2. Taylor expanded around 0 50.3

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{{x}^{3}}{n} + \left(\frac{x}{n} + \left(0.16666666666666666 \cdot \frac{{x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{x}^{2}}{{n}^{2}} + 1\right)\right)\right)\right) - \left(0.5 \cdot \frac{{x}^{3}}{{n}^{2}} + \left(0.5 \cdot \frac{{x}^{2}}{n} + e^{\frac{\log x}{n}}\right)\right)}\]

    3. Simplified38.0

      \[\leadsto \color{blue}{\left(\left(\frac{x}{n} + \left(\left(1 + 0.5 \cdot \left(\frac{x}{n} \cdot \frac{x}{n}\right)\right) + 0.16666666666666666 \cdot {\left(\frac{x}{n}\right)}^{3}\right)\right) + \frac{{x}^{3}}{n} \cdot \left(0.3333333333333333 - \frac{0.5}{n}\right)\right) + \left(\frac{x \cdot x}{n} \cdot -0.5 - {x}^{\left(\frac{1}{n}\right)}\right)}\]

    if 1.92647181524562741e-256 < x < 5.5596065352852317

    1. Initial program 48.6

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

    2. Taylor expanded around inf 12.2

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

    3. Simplified12.2

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

    5. Applied add-cube-cbrt_binary6412.2

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

    6. Applied log-prod_binary6412.3

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

    7. Applied associate--r+_binary6412.3

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

    if 5.5596065352852317 < x

    1. Initial program 20.9

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

    2. Taylor expanded around inf 1.6

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9264718152456274 \cdot 10^{-256}:\\ \;\;\;\;\left(\left(\frac{x}{n} + \left(\left(1 + 0.5 \cdot \left(\frac{x}{n} \cdot \frac{x}{n}\right)\right) + 0.16666666666666666 \cdot {\left(\frac{x}{n}\right)}^{3}\right)\right) + \frac{{x}^{3}}{n} \cdot \left(0.3333333333333333 - \frac{0.5}{n}\right)\right) + \left(\frac{x \cdot x}{n} \cdot -0.5 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;x \leq 5.559606535285232:\\ \;\;\;\;\frac{\left(\log \left(x + 1\right) - \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) - \log \left(\sqrt[3]{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-\frac{\log \left(\frac{1}{x}\right)}{n}}}{x \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))