Average Error: 32.2 → 11.6
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{1}{\frac{n}{\log x}}\right))_* \le -2.074005852959405 \cdot 10^{-17}:\\ \;\;\;\;(e^{\log_* (1 + \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right))} - 1)^*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{1}{\frac{n}{\log x}}\right))_* \le -2.318477557482887 \cdot 10^{-305}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{1}{\frac{n}{\log x}}\right))_* \le 9.479777899109092 \cdot 10^{-294}:\\ \;\;\;\;(\left(\frac{\log x}{n}\right) \cdot \left(\frac{\log x}{n}\right) + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ 1 (/ n (log x))))) < -2.074005852959405e-17

    1. Initial program 3.2

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

    3. Applied add-exp-log3.2

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

    4. Applied pow-exp3.2

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

    5. Applied simplify1.9

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

    7. Applied expm1-log1p-u2.4

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

    if -2.074005852959405e-17 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ 1 (/ n (log x))))) < -2.318477557482887e-305 or 9.479777899109092e-294 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ 1 (/ n (log x)))))

    1. Initial program 53.7

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

    3. Applied add-exp-log53.7

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

    4. Applied pow-exp53.7

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

    5. Applied simplify53.7

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

    6. Taylor expanded around inf 54.0

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

    7. Applied simplify7.5

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

    9. Applied div-inv9.8

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

    if -2.318477557482887e-305 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ 1 (/ n (log x))))) < 9.479777899109092e-294

    1. Initial program 27.5

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

    3. Applied add-exp-log27.5

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

    4. Applied pow-exp27.5

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

    5. Applied simplify27.5

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

    6. Taylor expanded around inf 27.5

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

    7. Applied simplify25.5

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

    8. Taylor expanded around inf 24.5

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

    9. Applied simplify25.2

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))