Average Error: 33.1 → 15.9
Time: 3.9m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.6494336565169249:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 7.038456018210188 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{if}\;x \le 9.493223571370615 \cdot 10^{+183}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \le 9.520693218546424 \cdot 10^{+210}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if x < 0.6494336565169249

    1. Initial program 47.6

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

    2. Taylor expanded around inf 59.9

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

    3. Applied simplify14.4

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

    if 0.6494336565169249 < x < 7.038456018210188e+135 or 9.493223571370615e+183 < x < 9.520693218546424e+210

    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 63.0

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

    3. Applied simplify22.7

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

    if 7.038456018210188e+135 < x < 9.493223571370615e+183

    1. Initial program 19.5

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

    3. Applied add-cube-cbrt19.5

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

    if 9.520693218546424e+210 < x

    1. Initial program 6.7

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

    3. Applied add-cbrt-cube6.7

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

    4. Applied simplify6.7

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube6.7

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

    7. Applied simplify6.7

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\right)}^{3}}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube6.7

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

    10. Applied simplify6.7

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\right)}^{3}}\right)}^{3}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))