Average Error: 29.5 → 21.7
Time: 42.4s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.0851174143905235:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.6860430671809155 \cdot 10^{-18}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\log x}{x} \cdot \frac{\frac{\frac{1}{16}}{n}}{n} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot n} - \frac{\frac{\frac{1}{4}}{x}}{n}\right)\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

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 (/ 1 n) < -0.0851174143905235

    1. Initial program 0.3

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

    2. Initial simplification0.3

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied add-sqr-sqrt0.3

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

    6. Applied difference-of-squares0.3

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

    8. Applied add-sqr-sqrt0.3

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

    9. Applied sqrt-prod0.3

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

    10. Applied add-sqr-sqrt0.3

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

    11. Applied difference-of-squares0.3

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

    if -0.0851174143905235 < (/ 1 n) < 4.6860430671809155e-18

    1. Initial program 44.6

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

    2. Initial simplification44.6

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

    4. Applied add-sqr-sqrt44.7

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

    5. Applied add-sqr-sqrt44.6

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

    6. Applied difference-of-squares44.6

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

    8. Applied add-sqr-sqrt44.6

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

    9. Applied sqrt-prod44.7

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

    10. Applied add-sqr-sqrt44.6

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

    11. Applied difference-of-squares44.6

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

    12. Taylor expanded around inf 31.6

      \[\leadsto \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{16} \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{8} \cdot \frac{1}{{x}^{2} \cdot n}\right)\right)}\right)\]

    13. Simplified31.0

      \[\leadsto \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \color{blue}{\left(\frac{\frac{\frac{1}{16}}{n}}{n} \cdot \frac{\log x}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot n} - \frac{\frac{\frac{1}{4}}{x}}{n}\right)\right)}\right)\]

    if 4.6860430671809155e-18 < (/ 1 n)

    1. Initial program 27.0

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

    2. Initial simplification27.0

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

    4. Applied add-sqr-sqrt27.0

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

    5. Applied add-sqr-sqrt27.0

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

    6. Applied difference-of-squares27.0

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

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.0851174143905235:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.6860430671809155 \cdot 10^{-18}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\log x}{x} \cdot \frac{\frac{\frac{1}{16}}{n}}{n} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot n} - \frac{\frac{\frac{1}{4}}{x}}{n}\right)\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 42.4s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))