Average Error: 33.2 → 24.4
Time: 11.9s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -9.123233094220298 \cdot 10^{-19}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({\left(e^{\frac{\log x}{n}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.439730988415116 \cdot 10^{-19}:\\ \;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\left(\frac{0.5}{n \cdot x} - \frac{0.25}{x \cdot \left(n \cdot x\right)}\right) + 0.25 \cdot \frac{\log x}{x \cdot \left(n \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -9.123233094220298 \cdot 10^{-19}:\\
\;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({\left(e^{\frac{\log x}{n}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\\

\mathbf{elif}\;\frac{1}{n} \leq 1.439730988415116 \cdot 10^{-19}:\\
\;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\left(\frac{0.5}{n \cdot x} - \frac{0.25}{x \cdot \left(n \cdot x\right)}\right) + 0.25 \cdot \frac{\log x}{x \cdot \left(n \cdot n\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\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) -9.123233094220298e-19)
   (-
    (pow (+ 1.0 x) (/ 1.0 n))
    (cbrt (pow (pow (exp (/ (log x) n)) (sqrt 3.0)) (sqrt 3.0))))
   (if (<= (/ 1.0 n) 1.439730988415116e-19)
     (*
      (+ (pow x (/ 0.5 n)) (pow (+ 1.0 x) (/ 0.5 n)))
      (+
       (- (/ 0.5 (* n x)) (/ 0.25 (* x (* n x))))
       (* 0.25 (/ (log x) (* x (* n n))))))
     (/
      (- (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) <= -9.123233094220298e-19) {
		tmp = pow((1.0 + x), (1.0 / n)) - cbrt(pow(pow(exp(log(x) / n), sqrt(3.0)), sqrt(3.0)));
	} else if ((1.0 / n) <= 1.439730988415116e-19) {
		tmp = (pow(x, (0.5 / n)) + pow((1.0 + x), (0.5 / n))) * (((0.5 / (n * x)) - (0.25 / (x * (n * x)))) + (0.25 * (log(x) / (x * (n * n)))));
	} else {
		tmp = (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) < -9.12323309422029831e-19

    1. Initial program 3.6

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

    3. Applied pow-to-exp_binary64_4803.6

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

    4. Simplified3.6

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

    6. Applied add-cbrt-cube_binary64_4474.0

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

    7. Simplified3.9

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{\color{blue}{{\left(e^{\frac{\log x}{n}}\right)}^{3}}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary64_4354.0

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

    10. Applied pow-unpow_binary64_4883.9

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

    if -9.12323309422029831e-19 < (/.f64 1 n) < 1.43973098841511593e-19

    1. Initial program 45.5

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

    3. Applied sqr-pow_binary64_38645.5

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

    4. Applied sqr-pow_binary64_38645.5

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

    5. Applied difference-of-squares_binary64_38345.5

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

    6. Simplified45.5

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

    7. Simplified45.5

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

    8. Taylor expanded around inf 32.7

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

    9. Simplified32.6

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

    if 1.43973098841511593e-19 < (/.f64 1 n)

    1. Initial program 10.1

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

    3. Applied flip--_binary64_38910.1

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

    4. Simplified10.0

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

  4. Final simplification24.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -9.123233094220298 \cdot 10^{-19}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({\left(e^{\frac{\log x}{n}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.439730988415116 \cdot 10^{-19}:\\ \;\;\;\;\left({x}^{\left(\frac{0.5}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{0.5}{n}\right)}\right) \cdot \left(\left(\frac{0.5}{n \cdot x} - \frac{0.25}{x \cdot \left(n \cdot x\right)}\right) + 0.25 \cdot \frac{\log x}{x \cdot \left(n \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))