Average Error: 32.6 → 7.0
Time: 59.9s
Precision: 64
Internal Precision: 1344
${\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}$
$\begin{array}{l} \mathbf{if}\;x \le 0.5096559954462845:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \left(\frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(\log x \cdot \log x\right) + \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{n}}{x}}{e^{\frac{\frac{1}{2}}{x}}} \cdot e^{\frac{\log x}{n}}\\ \end{array}$

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Results

 In Out
Enter valid numbers for all inputs

Derivation

1. Split input into 2 regimes
2. if x < 0.5096559954462845

1. Initial program 47.3

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

$\leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}$
4. Taylor expanded around inf 60.3

$\leadsto e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^{2}}{{n}^{2}}\right) - \frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right)}$
5. Applied simplify14.7

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

if 0.5096559954462845 < x

1. Initial program 20.3

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

$\leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}$
4. Taylor expanded around inf 43.4

$\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{x}\right) + \log \left(\frac{1}{n}\right)\right) - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{\log \left(\frac{1}{x}\right)}{n}\right)}}$
5. Applied simplify0.6

$\leadsto \color{blue}{\frac{\frac{\frac{1}{n}}{x}}{e^{\frac{\frac{1}{2}}{x}}} \cdot e^{\frac{\log x}{n}}}$
3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 59.9s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x n)
:name "pow(x+1,1/n)-pow(x,1/n)"
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))