Average Error: 33.2 → 7.1
Time: 23.5s
Precision: binary64
Cost: 7233
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 3749.2329152300117:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 3749.2329152300117:\\
\;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{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 3749.2329152300117)
   (/ (log (/ (+ x 1.0) x)) n)
   (/ (pow x (/ 1.0 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 <= 3749.2329152300117) {
		tmp = log((x + 1.0) / x) / n;
	} else {
		tmp = pow(x, (1.0 / 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

Alternatives

Alternative 1
Error12.6
Cost9093
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -824877.7140404872:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq -1.2258186981552663 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.1106135764503263 \cdot 10^{-118}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8.303479139423472 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 7.910911441440176 \cdot 10^{-37}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.03398736877701905:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 2
Error15.6
Cost7041
\[\begin{array}{l} \mathbf{if}\;x \leq 0.9894323734666607:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.144349130405594 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 3
Error15.8
Cost6977
\[\begin{array}{l} \mathbf{if}\;x \leq 0.5537129511135368:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.304932758506748 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 4
Error21.7
Cost1411
\[\begin{array}{l} \mathbf{if}\;n \leq -4.4521808829022955:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;n \leq 4.412972815379056 \cdot 10^{-285}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 29.133824801667156:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \end{array}\]
Alternative 5
Error23.6
Cost1283
\[\begin{array}{l} \mathbf{if}\;n \leq -3.2646001001743175:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 3.965136627028028 \cdot 10^{-281}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 29.133824801667156:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array}\]
Alternative 6
Error24.1
Cost1283
\[\begin{array}{l} \mathbf{if}\;n \leq -3.5850968805999273:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq 3.924502705116223 \cdot 10^{-280}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 29.133824801667156:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array}\]
Alternative 7
Error34.0
Cost706
\[\begin{array}{l} \mathbf{if}\;n \leq 4.602593217058747 \cdot 10^{-282}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 9.888380016423709 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 8
Error56.5
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if x < 3749.2329152300117

    1. Initial program 47.4

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

    2. Taylor expanded around inf 13.6

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

    4. Applied diff-log_binary64_17013.6

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

    5. Simplified13.6

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

    if 3749.2329152300117 < x

    1. Initial program 21.0

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

    2. Taylor expanded around inf 1.5

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

    3. Simplified1.5

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

    4. Simplified1.5

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3749.2329152300117:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))