Average Error: 32.8 → 24.0
Time: 12.3s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -7.207699079219628 \cdot 10^{-09}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 0.010277233907577869:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -7.207699079219628 \cdot 10^{-09}:\\
\;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \leq 0.010277233907577869:\\
\;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\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) -7.207699079219628e-09)
   (log (exp (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) 0.010277233907577869)
     (+
      (/ 1.0 (* n x))
      (-
       (* (/ 1.0 (* n n)) (+ (/ (log 1.0) x) (/ (log x) x)))
       (/ 0.5 (* x (* n x)))))
     (*
      (sqrt (+ (pow (sqrt (+ 1.0 x)) (/ 1.0 n)) (pow (sqrt x) (/ 1.0 n))))
      (*
       (sqrt (+ (pow (sqrt (+ 1.0 x)) (/ 1.0 n)) (pow (sqrt x) (/ 1.0 n))))
       (- (pow (sqrt (+ 1.0 x)) (/ 1.0 n)) (pow (sqrt x) (/ 1.0 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 VAR;
	if (((1.0 / n) <= -7.207699079219628e-09)) {
		VAR = ((double) log(((double) exp(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))))))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 0.010277233907577869)) {
			VAR_1 = ((double) ((1.0 / ((double) (n * x))) + ((double) (((double) ((1.0 / ((double) (n * n))) * ((double) ((((double) log(1.0)) / x) + (((double) log(x)) / x))))) - (0.5 / ((double) (x * ((double) (n * x)))))))));
		} else {
			VAR_1 = ((double) (((double) sqrt(((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), (1.0 / n))) + ((double) pow(((double) sqrt(x)), (1.0 / n))))))) * ((double) (((double) sqrt(((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), (1.0 / n))) + ((double) pow(((double) sqrt(x)), (1.0 / n))))))) * ((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), (1.0 / n))) - ((double) pow(((double) sqrt(x)), (1.0 / n)))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 (/ 1.0 n) < -7.20769907921962817e-9

    1. Initial program 2.0

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

    3. Applied add-log-exp2.5

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

    4. Applied add-log-exp2.4

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

    5. Applied diff-log2.4

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

    6. Simplified2.4

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

    if -7.20769907921962817e-9 < (/ 1.0 n) < 0.010277233907577869

    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 32.4

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

    3. Simplified32.3

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

    if 0.010277233907577869 < (/ 1.0 n)

    1. Initial program 4.3

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

    3. Applied add-sqr-sqrt4.3

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

    4. Applied unpow-prod-down4.3

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

    5. Applied add-sqr-sqrt4.3

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

    6. Applied unpow-prod-down4.3

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

    7. Applied difference-of-squares4.3

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

    9. Applied add-sqr-sqrt4.3

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

    10. Applied associate-*l*4.3

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

    11. Simplified4.3

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -7.207699079219628 \cdot 10^{-09}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 0.010277233907577869:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))