Average Error: 32.8 → 2.3
Time: 2.5m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -5.985105191138487 \cdot 10^{-17}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\frac{\log_* (1 + x)}{n}} \cdot \sqrt[3]{\frac{\log_* (1 + x)}{n}}}\right)}^{\left(\sqrt[3]{\frac{\log_* (1 + x)}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le -1.5808670951946372 \cdot 10^{-302}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_* \le 0.0:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log_* (1 + x)}{n}} - 1)^* - (\left(\frac{\frac{\frac{1}{2}}{n}}{n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \end{array}\]

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)) (/ (log x) n))) < -5.985105191138487e-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-log3.3

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

    4. Applied pow-exp3.3

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

    5. Applied simplify1.8

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

    7. Applied add-cube-cbrt1.8

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\frac{\log_* (1 + x)}{n}} \cdot \sqrt[3]{\frac{\log_* (1 + x)}{n}}\right) \cdot \sqrt[3]{\frac{\log_* (1 + x)}{n}}}} - {x}^{\left(\frac{1}{n}\right)}\]

    8. Applied exp-prod1.8

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\frac{\log_* (1 + x)}{n}} \cdot \sqrt[3]{\frac{\log_* (1 + x)}{n}}}\right)}^{\left(\sqrt[3]{\frac{\log_* (1 + x)}{n}}\right)}} - {x}^{\left(\frac{1}{n}\right)}\]

    if -5.985105191138487e-17 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))) < -1.5808670951946372e-302 or 0.0 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n)))

    1. Initial program 59.4

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

    3. Applied add-exp-log59.4

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

    4. Applied pow-exp59.4

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

    5. Applied simplify59.4

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

    6. Taylor expanded around inf 59.8

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

    7. Applied simplify3.4

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

    if -1.5808670951946372e-302 < (- (expm1 (/ (log1p x) n)) (fma (/ (/ 1/2 n) n) (* (log x) (log x)) (/ (log x) n))) < 0.0

    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 1.5

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

    3. Applied simplify1.5

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))