Average Error: 32.3 → 12.1
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + {\left(e^{\sqrt[3]{\log \left(\log x\right) - \log \left(\left(n \cdot x\right) \cdot n\right)} \cdot \sqrt[3]{\log \left(\log x\right) - \log \left(\left(n \cdot x\right) \cdot n\right)}}\right)}^{\left(\sqrt[3]{\log \left(\log x\right) - \log \left(\left(n \cdot x\right) \cdot n\right)}\right)} \le 4.375868223365995 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + {\left(e^{\sqrt[3]{\log \left(\log x\right) - \log \left(\left(n \cdot x\right) \cdot n\right)} \cdot \sqrt[3]{\log \left(\log x\right) - \log \left(\left(n \cdot x\right) \cdot n\right)}}\right)}^{\left(\sqrt[3]{\log \left(\log x\right) - \log \left(\left(n \cdot x\right) \cdot n\right)}\right)} \le 1.7815982055454507 \cdot 10^{+308}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \end{array}\]

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) x) (/ (/ 1/2 n) (* x x))) (pow (exp (* (cbrt (- (log (log x)) (log (* (* n x) n)))) (cbrt (- (log (log x)) (log (* (* n x) n)))))) (cbrt (- (log (log x)) (log (* (* n x) n)))))) < 4.375868223365995e-83

    1. Initial program 28.8

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify5.4

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

    4. Taylor expanded around -inf 63.0

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

    5. Applied simplify5.3

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

    if 4.375868223365995e-83 < (+ (- (/ (/ 1 n) x) (/ (/ 1/2 n) (* x x))) (pow (exp (* (cbrt (- (log (log x)) (log (* (* n x) n)))) (cbrt (- (log (log x)) (log (* (* n x) n)))))) (cbrt (- (log (log x)) (log (* (* n x) n)))))) < 1.7815982055454507e+308

    1. Initial program 2.7

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

    3. Applied add-cbrt-cube2.7

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

    4. Applied simplify2.7

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

    if 1.7815982055454507e+308 < (+ (- (/ (/ 1 n) x) (/ (/ 1/2 n) (* x x))) (pow (exp (* (cbrt (- (log (log x)) (log (* (* n x) n)))) (cbrt (- (log (log x)) (log (* (* n x) n)))))) (cbrt (- (log (log x)) (log (* (* n x) n))))))

    1. Initial program 42.5

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

    2. Taylor expanded around inf 60.2

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

    3. Applied simplify19.7

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))