Average Error: 32.8 → 23.8
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le -1.0895631794344091 \cdot 10^{-07}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le 2.354398128069095 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

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 (* n x)) (+ 1 (/ (log x) n))) (pow x (/ 1 n))) < -1.0895631794344091e-07 or 2.354398128069095e-306 < (- (+ (/ 1 (* n x)) (+ 1 (/ (log x) n))) (pow x (/ 1 n)))

    1. Initial program 25.8

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

    2. Initial simplification25.8

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

    4. Applied add-log-exp26.0

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

    6. Applied add-sqr-sqrt26.1

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

    7. Applied log-prod26.0

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

    if -1.0895631794344091e-07 < (- (+ (/ 1 (* n x)) (+ 1 (/ (log x) n))) (pow x (/ 1 n))) < 2.354398128069095e-306

    1. Initial program 39.2

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

    2. Initial simplification39.2

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

    4. Applied add-log-exp39.2

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

    5. Taylor expanded around inf 21.7

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

    6. Simplified21.7

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

  4. Final simplification23.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le -1.0895631794344091 \cdot 10^{-07}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\left(\left(1 + \frac{\log x}{n}\right) + \frac{1}{x \cdot n}\right) - {x}^{\left(\frac{1}{n}\right)} \le 2.354398128069095 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))