Average Error: 32.5 → 22.2
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -0.16923893252320515:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 1.961457645701062 \cdot 10^{-299}:\\ \;\;\;\;{\left(\frac{1}{x}\right)}^{\left(\frac{1}{3} \cdot 3\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(\frac{1}{3} \cdot 3\right)} + \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) \cdot \left({\left(\frac{1}{x}\right)}^{\left(\frac{1}{3} \cdot 3\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(\frac{1}{3} \cdot 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

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 (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n))) < -0.16923893252320515

    1. Initial program 18.7

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

    3. Applied add-log-exp19.0

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

    4. Applied add-log-exp18.9

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

    5. Applied diff-log18.9

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

    6. Simplified18.9

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

    if -0.16923893252320515 < (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n))) < 1.961457645701062e-299

    1. Initial program 40.2

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

    3. Applied add-cbrt-cube40.2

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

    4. Taylor expanded around inf 43.3

      \[\leadsto \color{blue}{e^{\frac{1}{3} \cdot \left(3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{n}\right)\right)} - \left(\frac{\log \left(\frac{1}{x}\right) \cdot e^{\frac{1}{3} \cdot \left(3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{n}\right)\right)}}{n} + \frac{1}{2} \cdot \frac{e^{\frac{1}{3} \cdot \left(3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{n}\right)\right)}}{x}\right)}\]

    5. Simplified21.1

      \[\leadsto \color{blue}{\left({\left(\frac{1}{x}\right)}^{\left(\frac{1}{3} \cdot 3\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(\frac{1}{3} \cdot 3\right)}\right) \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) + {\left(\frac{1}{x}\right)}^{\left(\frac{1}{3} \cdot 3\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(\frac{1}{3} \cdot 3\right)}}\]

    if 1.961457645701062e-299 < (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n)))

    1. Initial program 31.5

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

    3. Applied add-exp-log31.5

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

    4. Applied pow-exp31.5

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

    5. Simplified29.8

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -0.16923893252320515:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 1.961457645701062 \cdot 10^{-299}:\\ \;\;\;\;{\left(\frac{1}{x}\right)}^{\left(\frac{1}{3} \cdot 3\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(\frac{1}{3} \cdot 3\right)} + \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) \cdot \left({\left(\frac{1}{x}\right)}^{\left(\frac{1}{3} \cdot 3\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(\frac{1}{3} \cdot 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))