Average Error: 32.8 → 3.3
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le -0.17898575465010072:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le -9.293554416118733 \cdot 10^{-269}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le 1.1564829395514594 \cdot 10^{-305}:\\ \;\;\;\;\left(\frac{\frac{\log x}{x}}{n \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))) < -0.17898575465010072

    1. Initial program 1.3

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

    3. Applied add-exp-log1.3

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

    4. Applied pow-exp1.3

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

    5. Applied simplify0.4

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

    if -0.17898575465010072 < (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))) < -9.293554416118733e-269

    1. Initial program 59.2

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

    3. Applied add-exp-log59.2

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

    4. Applied pow-exp59.2

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

    5. Applied simplify59.2

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

    6. Taylor expanded around inf 59.7

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

    7. Applied simplify4.1

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

    9. Applied div-inv4.2

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

    if -9.293554416118733e-269 < (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n))) < 1.1564829395514594e-305

    1. Initial program 30.5

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

    3. Applied add-exp-log30.5

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

    4. Applied pow-exp30.5

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

    5. Applied simplify30.5

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

    7. Applied expm1-log1p-u30.5

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

    8. Taylor expanded around -inf 63.0

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

    9. Applied simplify4.2

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{n}}{x} + 0\right) + \frac{\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}}\]

    if 1.1564829395514594e-305 < (- (expm1 (/ (log1p x) n)) (fma (/ (log x) n) (/ (log x) (/ n 1/2)) (/ (log x) n)))

    1. Initial program 58.0

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

    3. Applied add-exp-log58.1

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

    4. Applied pow-exp58.0

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

    5. Applied simplify58.1

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

    6. Taylor expanded around inf 59.3

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

    7. Applied simplify4.9

      \[\leadsto \color{blue}{(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify3.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le -0.17898575465010072:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le -9.293554416118733 \cdot 10^{-269}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_* \le 1.1564829395514594 \cdot 10^{-305}:\\ \;\;\;\;\left(\frac{\frac{\log x}{x}}{n \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{\frac{n}{\frac{1}{2}}}\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

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