Average Error: 32.9 → 14.6
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le -5180411914392.576:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le -5.948555254464106 \cdot 10^{-95}:\\ \;\;\;\;\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \log x}{\frac{n \cdot n}{\frac{1}{2}}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le 3.3749424080124432 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right)\\ \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le 1.4124048183076218 \cdot 10^{+176}:\\ \;\;\;\;\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \log x}{\frac{n \cdot n}{\frac{1}{2}}}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}\]

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 x) n) (- (+ (+ 1 0) (/ (log x) n)) (pow x (/ 1 n)))) < -5180411914392.576 or 1.4124048183076218e+176 < (+ (/ (/ 1 x) n) (- (+ (+ 1 0) (/ (log x) n)) (pow x (/ 1 n))))

    1. Initial program 16.5

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

    3. Applied add-cbrt-cube16.5

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

    4. Applied simplify16.5

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

    if -5180411914392.576 < (+ (/ (/ 1 x) n) (- (+ (+ 1 0) (/ (log x) n)) (pow x (/ 1 n)))) < -5.948555254464106e-95 or 3.3749424080124432e-118 < (+ (/ (/ 1 x) n) (- (+ (+ 1 0) (/ (log x) n)) (pow x (/ 1 n)))) < 1.4124048183076218e+176

    1. Initial program 54.9

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

    3. Applied add-log-exp55.1

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

    4. Applied add-log-exp55.0

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

    5. Applied diff-log55.0

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

    6. Applied simplify55.0

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

    7. Taylor expanded around inf 60.4

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

    8. Applied simplify14.3

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

    if -5.948555254464106e-95 < (+ (/ (/ 1 x) n) (- (+ (+ 1 0) (/ (log x) n)) (pow x (/ 1 n)))) < 3.3749424080124432e-118

    1. Initial program 34.4

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

    3. Applied add-cbrt-cube34.4

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

    4. Applied simplify34.4

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

    5. Taylor expanded around inf 34.4

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

    6. Applied simplify13.1

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

  4. Applied simplify14.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le -5180411914392.576:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le -5.948555254464106 \cdot 10^{-95}:\\ \;\;\;\;\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \log x}{\frac{n \cdot n}{\frac{1}{2}}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le 3.3749424080124432 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right)\\ \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{\frac{1}{x}}{n} \le 1.4124048183076218 \cdot 10^{+176}:\\ \;\;\;\;\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \log x}{\frac{n \cdot n}{\frac{1}{2}}}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018296 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))