Average Error: 33.2 → 23.7
Time: 34.6s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1744025376359.6016 \lor \neg \left(n \le 15134501.209798293\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

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 n < -1744025376359.6016 or 15134501.209798293 < n

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 63.0

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

    3. Simplified32.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]
    4. Using strategy rm

    5. Applied div-inv32.3

      \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\color{blue}{\frac{1}{x} \cdot \frac{1}{n}} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\]
    6. Using strategy rm

    7. Applied associate-*l/32.3

      \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\color{blue}{\frac{1 \cdot \frac{1}{n}}{x}} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\]

    8. Simplified32.3

      \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\color{blue}{\frac{1}{n}}}{x} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\]

    if -1744025376359.6016 < n < 15134501.209798293

    1. Initial program 4.2

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

    3. Applied add-exp-log4.2

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

    4. Applied pow-exp4.2

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

    5. Simplified2.9

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

  4. Final simplification23.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1744025376359.6016 \lor \neg \left(n \le 15134501.209798293\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 34.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.823.722.110.884.5%
herbie shell --seed 2018274 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))