Average Error: 32.8 → 7.1
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.5101828990122546:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{\frac{\log x}{n}}} \cdot \sqrt{e^{\frac{\log x}{n}}}\right) \cdot \frac{\frac{\frac{1}{n}}{x}}{e^{\frac{\frac{1}{2}}{x}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if x < 0.5101828990122546

    1. Initial program 47.0

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

    2. Taylor expanded around inf 60.0

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

    3. Applied simplify15.2

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

    if 0.5101828990122546 < x

    1. Initial program 21.5

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

    3. Applied add-exp-log21.5

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

    4. Taylor expanded around inf 42.4

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

    5. Applied simplify0.6

      \[\leadsto \color{blue}{e^{\frac{\log x}{n}} \cdot \frac{\frac{\frac{1}{n}}{x}}{e^{\frac{\frac{1}{2}}{x}}}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.7

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

Runtime

Time bar (total: 1.3m)Debug logProfile

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