Average Error: 29.4 → 19.2
Time: 1.2m
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.691515613026755 \cdot 10^{-18}:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\log \left(e^{\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.5313165778492556 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

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 3 regimes

  2. if (/ 1 n) < -3.691515613026755e-18

    1. Initial program 2.2

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

    2. Initial simplification2.2

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

    4. Applied add-cbrt-cube2.2

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

    6. Applied add-log-exp2.4

      \[\leadsto \sqrt[3]{\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \color{blue}{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube2.4

      \[\leadsto \sqrt[3]{\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \log \left(e^{\color{blue}{\sqrt[3]{\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}}\right)\right) \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\]

    if -3.691515613026755e-18 < (/ 1 n) < 3.5313165778492556e-05

    1. Initial program 44.6

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

    2. Initial simplification44.6

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

    3. Taylor expanded around -inf 63.0

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

    4. Simplified31.7

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

    if 3.5313165778492556e-05 < (/ 1 n)

    1. Initial program 23.6

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

    2. Initial simplification23.6

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

    4. Applied add-exp-log23.6

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

    5. Applied pow-exp23.6

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

    6. Simplified1.1

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

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.691515613026755 \cdot 10^{-18}:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\log \left(e^{\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.5313165778492556 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))