Average Error: 33.2 → 24.1
Time: 30.0s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.966910209523465 \cdot 10^{-13} \lor \neg \left(\frac{1}{n} \le 6.520844947716273 \cdot 10^{-08}\right):\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{\left(x \cdot n\right) \cdot n} + \frac{1}{n} \cdot \left(\frac{\frac{\frac{-1}{2}}{x}}{x} + \frac{1}{x}\right)\\ \end{array}\]

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 2 regimes

  2. if (/ 1 n) < -6.966910209523465e-13 or 6.520844947716273e-08 < (/ 1 n)

    1. Initial program 4.1

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

    3. Applied add-sqr-sqrt4.1

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

    if -6.966910209523465e-13 < (/ 1 n) < 6.520844947716273e-08

    1. Initial program 45.1

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

    3. Applied add-sqr-sqrt45.1

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

    4. Taylor expanded around inf 32.9

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

    5. Simplified32.8

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

    7. Applied associate-/r*32.3

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

    9. Applied div-inv32.3

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

    10. Applied *-un-lft-identity32.3

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

    11. Applied times-frac32.3

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

    12. Applied distribute-lft-out32.3

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

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.966910209523465 \cdot 10^{-13} \lor \neg \left(\frac{1}{n} \le 6.520844947716273 \cdot 10^{-08}\right):\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{\left(x \cdot n\right) \cdot n} + \frac{1}{n} \cdot \left(\frac{\frac{\frac{-1}{2}}{x}}{x} + \frac{1}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 30.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes33.224.122.410.884.1%
herbie shell --seed 2018274 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))