Average Error: 33.2 → 24.1
Time: 10.5s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.8014521313817948 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 6.165757944870841 \cdot 10^{-09}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \cdot \log \left(e^{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{0.5}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1.8014521313817948 \cdot 10^{-20}:\\
\;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \leq 6.165757944870841 \cdot 10^{-09}:\\
\;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \cdot \log \left(e^{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{0.5}{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) -1.8014521313817948e-20)
   (- (cbrt (pow (pow (+ 1.0 x) (/ 1.0 n)) 3.0)) (pow x (/ 1.0 n)))
   (if (<= (/ 1.0 n) 6.165757944870841e-09)
     (+ (- (/ 1.0 (* n x)) (/ 0.5 (* x (* n x)))) (/ (log x) (* x (* n n))))
     (*
      (+ (pow (sqrt (+ 1.0 x)) (/ 1.0 n)) (pow x (/ 0.5 n)))
      (log (exp (- (pow (sqrt (+ 1.0 x)) (/ 1.0 n)) (pow x (/ 0.5 n)))))))))
double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
}
↓
double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -1.8014521313817948e-20)) {
		tmp = ((double) (((double) cbrt(((double) pow(((double) pow(((double) (1.0 + x)), (1.0 / n))), 3.0)))) - ((double) pow(x, (1.0 / n)))));
	} else {
		double tmp_1;
		if (((1.0 / n) <= 6.165757944870841e-09)) {
			tmp_1 = ((double) (((double) ((1.0 / ((double) (n * x))) - (0.5 / ((double) (x * ((double) (n * x))))))) + (((double) log(x)) / ((double) (x * ((double) (n * n)))))));
		} else {
			tmp_1 = ((double) (((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), (1.0 / n))) + ((double) pow(x, (0.5 / n))))) * ((double) log(((double) exp(((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), (1.0 / n))) - ((double) pow(x, (0.5 / n)))))))))));
		}
		tmp = tmp_1;
	}
	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) < -1.8014521313817948e-20

    1. Initial program 4.9

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

    3. Applied add-cbrt-cube_binary645.0

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

    4. Simplified5.0

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

    if -1.8014521313817948e-20 < (/.f64 1 n) < 6.16575794487084098e-9

    1. Initial program 45.4

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified32.3

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

    if 6.16575794487084098e-9 < (/.f64 1 n)

    1. Initial program 7.2

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

    3. Applied sqr-pow_binary647.2

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

    4. Applied add-sqr-sqrt_binary647.2

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

    5. Applied unpow-prod-down_binary647.2

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

    6. Applied difference-of-squares_binary647.2

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

    7. Simplified7.2

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

    8. Simplified7.2

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

    10. Applied add-log-exp_binary647.2

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

    11. Applied add-log-exp_binary647.3

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

    12. Applied diff-log_binary647.3

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

    13. Simplified7.3

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

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.8014521313817948 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 6.165757944870841 \cdot 10^{-09}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right) \cdot \log \left(e^{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{0.5}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))