Average Error: 32.2 → 2.4
Time: 2.2m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -1.5779069424576656 \cdot 10^{-15}:\\ \;\;\;\;(e^{\log_* (1 + \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right))} - 1)^*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -2.8978052857401013 \cdot 10^{-308}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{1}{\frac{n}{\log x}}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 2.126219926370099 \cdot 10^{-304}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(n \cdot x\right)} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\log x \cdot \frac{1}{n}\right))_*\\ \end{array}\]

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))) < -1.5779069424576656e-15

    1. Initial program 3.0

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

    3. Applied add-exp-log3.0

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

    4. Applied pow-exp3.0

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

    5. Applied simplify1.6

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

    7. Applied expm1-log1p-u2.1

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

    if -1.5779069424576656e-15 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < -2.8978052857401013e-308

    1. Initial program 60.4

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

    3. Applied add-exp-log60.4

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

    4. Applied pow-exp60.4

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

    5. Applied simplify60.4

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

    6. Taylor expanded around inf 60.4

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

    7. Applied simplify1.8

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

    9. Applied clear-num1.9

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

    if -2.8978052857401013e-308 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n))) < 2.126219926370099e-304

    1. Initial program 26.8

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

    2. Taylor expanded around inf 1.7

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

    3. Applied simplify1.7

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

    if 2.126219926370099e-304 < (- (expm1 (/ (log1p x) n)) (fma (/ 1/2 (* n n)) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 58.5

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

    3. Applied add-exp-log58.5

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

    4. Applied pow-exp58.5

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

    5. Applied simplify58.5

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

    6. Taylor expanded around inf 59.3

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

    7. Applied simplify4.2

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

    9. Applied div-inv4.3

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

Runtime

Time bar (total: 2.2m)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))))