Average Error: 33.0 → 22.9
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\log x}{n} - {x}^{\left(\frac{1}{n}\right)}\right) + \left(\frac{\frac{1}{n}}{x} + 1\right) \le -2921.216722044893:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\left(\frac{\log x}{n} - {x}^{\left(\frac{1}{n}\right)}\right) + \left(\frac{\frac{1}{n}}{x} + 1\right) \le 1.5343359613837087 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

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 2 regimes

  2. if (+ (+ 1 (/ (/ 1 n) x)) (+ 0 (- (/ (log x) n) (pow x (/ 1 n))))) < -2921.216722044893 or 1.5343359613837087e-13 < (+ (+ 1 (/ (/ 1 n) x)) (+ 0 (- (/ (log x) n) (pow x (/ 1 n)))))

    1. Initial program 23.7

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

    3. Applied add-sqr-sqrt23.8

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

    4. Applied add-sqr-sqrt23.8

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

    5. Applied unpow-prod-down23.8

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

    6. Applied difference-of-squares23.8

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

    if -2921.216722044893 < (+ (+ 1 (/ (/ 1 n) x)) (+ 0 (- (/ (log x) n) (pow x (/ 1 n))))) < 1.5343359613837087e-13

    1. Initial program 41.1

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify22.1

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

    4. Taylor expanded around -inf 63.0

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

    5. Applied simplify22.1

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

  4. Applied simplify22.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\frac{\log x}{n} - {x}^{\left(\frac{1}{n}\right)}\right) + \left(\frac{\frac{1}{n}}{x} + 1\right) \le -2921.216722044893:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\left(\frac{\log x}{n} - {x}^{\left(\frac{1}{n}\right)}\right) + \left(\frac{\frac{1}{n}}{x} + 1\right) \le 1.5343359613837087 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018201 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))