Average Error: 32.8 → 19.4
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -203214826676021.62:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -8.74616837982526 \cdot 10^{-40}:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) - \left(\frac{\log x}{n} \cdot \frac{\log x}{\frac{n}{\frac{1}{2}}} - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right)\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -5.65341283901179 \cdot 10^{-174}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -8.279323975862342 \cdot 10^{-187}:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) - \left(\frac{\log x}{n} \cdot \frac{\log x}{\frac{n}{\frac{1}{2}}} - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right)\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -5.041134567104183 \cdot 10^{-213}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le 1.9033695602191532 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le 6.220115550801002 \cdot 10^{+185}:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) - \left(\frac{\log x}{n} \cdot \frac{\log x}{\frac{n}{\frac{1}{2}}} - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (- (- (/ (/ 1 n) x))) < -203214826676021.62 or -8.74616837982526e-40 < (- (- (/ (/ 1 n) x))) < -5.65341283901179e-174 or -8.279323975862342e-187 < (- (- (/ (/ 1 n) x))) < -5.041134567104183e-213

    1. Initial program 33.0

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

    3. Applied add-sqr-sqrt33.0

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

    4. Applied add-sqr-sqrt33.0

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

    5. Applied unpow-prod-down33.0

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

    6. Applied difference-of-squares33.0

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

    if -203214826676021.62 < (- (- (/ (/ 1 n) x))) < -8.74616837982526e-40 or -5.65341283901179e-174 < (- (- (/ (/ 1 n) x))) < -8.279323975862342e-187 or 1.9033695602191532e-193 < (- (- (/ (/ 1 n) x))) < 6.220115550801002e+185

    1. Initial program 49.3

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

    2. Taylor expanded around inf 60.5

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

    3. Applied simplify21.4

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

    if -5.041134567104183e-213 < (- (- (/ (/ 1 n) x))) < 1.9033695602191532e-193

    1. Initial program 25.4

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

    2. Taylor expanded around inf 27.1

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

    3. Applied simplify56.8

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

    4. Taylor expanded around inf 26.6

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

    5. Applied simplify7.2

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

    if 6.220115550801002e+185 < (- (- (/ (/ 1 n) x)))

    1. Initial program 9.3

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

    3. Applied add-sqr-sqrt9.3

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

    4. Applied add-sqr-sqrt9.3

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

    5. Applied unpow-prod-down9.3

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

    6. Applied difference-of-squares9.4

      \[\leadsto \color{blue}{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify19.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -203214826676021.62:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -8.74616837982526 \cdot 10^{-40}:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) - \left(\frac{\log x}{n} \cdot \frac{\log x}{\frac{n}{\frac{1}{2}}} - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right)\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -5.65341283901179 \cdot 10^{-174}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -8.279323975862342 \cdot 10^{-187}:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) - \left(\frac{\log x}{n} \cdot \frac{\log x}{\frac{n}{\frac{1}{2}}} - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right)\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le -5.041134567104183 \cdot 10^{-213}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le 1.9033695602191532 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{if}\;\frac{\frac{1}{n}}{x} \le 6.220115550801002 \cdot 10^{+185}:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) - \left(\frac{\log x}{n} \cdot \frac{\log x}{\frac{n}{\frac{1}{2}}} - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))