Error

Derivation

1. Split input into 2 regimes.

2. if n < -4088346716.1158066 or 49343267539819.87 < n

   1. Initial program 43.9

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

   2. Applied taylor 9.6

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

   3. Taylor expanded around inf 9.6

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

   4. Applied simplify 0.9

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

   if -4088346716.1158066 < n < 49343267539819.87

   1. Initial program 5.2

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

   3. Applied add-sqr-sqrt 5.2

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

   4. Applied add-sqr-sqrt 5.2

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

   5. Applied unpow-prod-down 5.3

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

   6. Applied difference-of-squares 5.3

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

   7. Applied taylor 2.6

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

   8. Taylor expanded around 0 2.6

      \[\leadsto \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(1 - \sqrt{e^{\frac{\log x}{n}}}\right)}\]
3. Recombined 2 regimes into one program.
4. Removed slow pow expressions

Runtime

Time bar (total: 1.2m) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1068028399 4028058041 2917032441 2563479541 765645300 1132738916)'
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))