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Derivation

1. Split input into 3 regimes

2. if (/ (* (- x (log x)) (+ (* (* 1 1/2) (+ (log x) x)) n)) (* n n)) < -0.09424485831697652 or 8.44226630658821e+177 < (/ (* (- x (log x)) (+ (* (* 1 1/2) (+ (log x) x)) n)) (* n n))

   1. Initial program 8.0

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

   3. Applied pow-to-exp8.1

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

   4. Applied simplify8.1

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

   if -0.09424485831697652 < (/ (* (- x (log x)) (+ (* (* 1 1/2) (+ (log x) x)) n)) (* n n)) < 7.560224877520036e-05

   1. Initial program 59.2

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

   2. Taylor expanded around inf 59.4

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

   3. Applied simplify14.5

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

   4. Taylor expanded around 0 13.6

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

   5. Applied simplify13.6

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

   if 7.560224877520036e-05 < (/ (* (- x (log x)) (+ (* (* 1 1/2) (+ (log x) x)) n)) (* n n)) < 8.44226630658821e+177

   1. Initial program 48.3

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

   2. Taylor expanded around inf 16.7

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

   3. Applied simplify16.7

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

4. Applied simplify10.9

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\left(n + \left(\log x + x\right) \cdot \frac{1}{2}\right) \cdot \left(x - \log x\right)}{n \cdot n} \le -0.09424485831697652:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\\ \mathbf{if}\;\frac{\left(n + \left(\log x + x\right) \cdot \frac{1}{2}\right) \cdot \left(x - \log x\right)}{n \cdot n} \le 7.560224877520036 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right) + \left(\frac{x}{n} - \frac{\log x}{n}\right)\\ \mathbf{if}\;\frac{\left(n + \left(\log x + x\right) \cdot \frac{1}{2}\right) \cdot \left(x - \log x\right)}{n \cdot n} \le 8.44226630658821 \cdot 10^{+177}:\\ \;\;\;\;\frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))