Average Error: 32.3 → 2.3
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -2.7929830070619843:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -1.1868816666533937 \cdot 10^{-299}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 1.5921410751477714 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))) < -2.7929830070619843

    1. Initial program 1.7

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

    3. Applied add-exp-log1.7

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

    4. Applied pow-exp1.7

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

    5. Applied simplify0.4

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

    if -2.7929830070619843 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))) < -1.1868816666533937e-299

    1. Initial program 58.4

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

    3. Applied add-exp-log58.4

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

    4. Applied pow-exp58.4

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

    5. Applied simplify58.4

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

    6. Taylor expanded around inf 59.0

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

    7. Applied simplify3.7

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

    if -1.1868816666533937e-299 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))) < 1.5921410751477714e-308

    1. Initial program 29.1

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

    2. Taylor expanded around inf 2.3

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

    3. Applied simplify2.3

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

    if 1.5921410751477714e-308 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 58.6

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

    3. Applied add-exp-log58.6

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

    4. Applied pow-exp58.6

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

    5. Applied simplify58.6

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

    6. Taylor expanded around inf 59.4

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

    7. Applied simplify3.9

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

    9. Applied div-inv4.0

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018198 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))