Average Error: 32.8 → 24.3
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.1482662732671587 \cdot 10^{-08}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.017369047176291 \cdot 10^{-07}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(\frac{-\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -2.1482662732671587e-08

    1. Initial program 1.2

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

    3. Applied add-log-exp1.6

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

    4. Applied add-log-exp1.5

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

    5. Applied diff-log1.5

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

    6. Simplified1.5

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

    8. Applied add-sqr-sqrt1.5

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

    9. Applied log-prod1.5

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

    if -2.1482662732671587e-08 < (/ 1 n) < 5.017369047176291e-07

    1. Initial program 44.7

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

    2. Taylor expanded around inf 33.1

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

    3. Simplified33.1

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

    if 5.017369047176291e-07 < (/ 1 n)

    1. Initial program 5.8

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

    3. Applied add-exp-log5.8

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

    4. Applied pow-exp5.8

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

    5. Simplified2.7

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

  4. Final simplification24.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.1482662732671587 \cdot 10^{-08}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.017369047176291 \cdot 10^{-07}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(\frac{-\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

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