Average Error: 32.8 → 16.7
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.0033910930127292:\\ \;\;\;\;\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\frac{\frac{1}{2}}{n} \cdot \log x}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 261320414792.23853:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\\ \mathbf{if}\;x \le 3.5357319956524395 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{if}\;x \le 2.2864678593371913 \cdot 10^{+108}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\\ \mathbf{if}\;x \le 5.924180328777098 \cdot 10^{+143}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{if}\;x \le 2.36932382819333 \cdot 10^{+166} \lor \neg \left(x \le 2.0894740047193565 \cdot 10^{+210}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x}\\ \end{array}\]

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 5 regimes

  2. if x < 1.0033910930127292

    1. Initial program 46.7

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

    2. Taylor expanded around inf 60.0

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

    3. Applied simplify15.1

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

    if 1.0033910930127292 < x < 261320414792.23853 or 5.924180328777098e+143 < x < 2.36932382819333e+166 or 2.0894740047193565e+210 < x

    1. Initial program 11.3

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

    3. Applied pow-to-exp11.4

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

    4. Applied simplify11.4

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

    6. Applied add-cbrt-cube11.4

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

    7. Applied simplify11.4

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

    if 261320414792.23853 < x < 3.5357319956524395e+75 or 2.2864678593371913e+108 < x < 5.924180328777098e+143

    1. Initial program 32.6

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

    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 simplify21.3

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

    if 3.5357319956524395e+75 < x < 2.2864678593371913e+108

    1. Initial program 27.3

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

    3. Applied pow-to-exp27.3

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

    4. Applied simplify27.3

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

    6. Applied add-cbrt-cube27.3

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

    7. Applied simplify27.3

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

    if 2.36932382819333e+166 < x < 2.0894740047193565e+210

    1. Initial program 17.2

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

    2. Taylor expanded around inf 23.4

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

    3. Applied simplify23.4

      \[\leadsto \color{blue}{\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x}}\]
  3. Recombined 5 regimes into one program.

  4. Applied simplify16.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 1.0033910930127292:\\ \;\;\;\;\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\frac{\frac{1}{2}}{n} \cdot \log x}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 261320414792.23853:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\\ \mathbf{if}\;x \le 3.5357319956524395 \cdot 10^{+75}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{if}\;x \le 2.2864678593371913 \cdot 10^{+108}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\\ \mathbf{if}\;x \le 5.924180328777098 \cdot 10^{+143}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{if}\;x \le 2.36932382819333 \cdot 10^{+166} \lor \neg \left(x \le 2.0894740047193565 \cdot 10^{+210}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x}\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018199 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))