Average Error: 32.4 → 23.5
Time: 11.6s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.0550413396632783 \cdot 10^{-05}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\log \left(e^{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2.4333254894340803 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{0.5}{n \cdot x}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1.0550413396632783 \cdot 10^{-05}:\\
\;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\log \left(e^{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \leq 2.4333254894340803 \cdot 10^{-12}:\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{0.5}{n \cdot x}\right) + \frac{\log x}{x \cdot \left(n \cdot n\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) -1.0550413396632783e-05)
   (-
    (pow (+ 1.0 x) (/ 1.0 n))
    (*
     (pow (* (cbrt x) (cbrt x)) (/ 1.0 n))
     (pow (log (exp (cbrt x))) (/ 1.0 n))))
   (if (<= (/ 1.0 n) 2.4333254894340803e-12)
     (+ (* (/ 1.0 x) (- (/ 1.0 n) (/ 0.5 (* n x)))) (/ (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) <= -1.0550413396632783e-05) {
		tmp = pow((1.0 + x), (1.0 / n)) - (pow((cbrt(x) * cbrt(x)), (1.0 / n)) * pow(log(exp(cbrt(x))), (1.0 / n)));
	} else if ((1.0 / n) <= 2.4333254894340803e-12) {
		tmp = ((1.0 / x) * ((1.0 / n) - (0.5 / (n * x)))) + (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) < -1.05504133966327833e-5

    1. Initial program 1.0

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

    3. Applied add-cube-cbrt_binary64_4461.0

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

    4. Applied unpow-prod-down_binary64_4901.0

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

    6. Applied add-log-exp_binary64_4502.1

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

    if -1.05504133966327833e-5 < (/.f64 1 n) < 2.43332548943408026e-12

    1. Initial program 44.8

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

    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. Simplified32.6

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

    5. Applied *-un-lft-identity_binary64_41432.6

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

    6. Applied times-frac_binary64_42032.6

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

    7. Applied *-un-lft-identity_binary64_41432.6

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

    8. Applied times-frac_binary64_42031.9

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

    9. Applied distribute-lft-out--_binary64_36831.9

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

    if 2.43332548943408026e-12 < (/.f64 1 n)

    1. Initial program 6.9

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

    3. Applied flip--_binary64_3896.9

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

    4. Simplified6.8

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.0550413396632783 \cdot 10^{-05}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\log \left(e^{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2.4333254894340803 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{0.5}{n \cdot x}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + x\right)}^{\left(\frac{2}{n}\right)} - {x}^{\left(\frac{2}{n}\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))