Average Error: 32.3 → 16.4
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.8385424682349487:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 5.984857460889397 \cdot 10^{+68}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x}\\ \mathbf{if}\;x \le 5.538161835309689 \cdot 10^{+77}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}}\right) \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{e^{\frac{\log x}{n}}}\right) \cdot \log \left(e^{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{e^{\frac{\log x}{n}}}}\right)}\\ \mathbf{if}\;x \le 7.394087600696336 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(n \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}}\right) \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{e^{\frac{\log x}{n}}}\right) \cdot \log \left(e^{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{e^{\frac{\log x}{n}}}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if x < 0.8385424682349487

    1. Initial program 46.4

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

    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 simplify15.3

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

    if 0.8385424682349487 < x < 5.984857460889397e+68 or 5.538161835309689e+77 < x < 7.394087600696336e+134

    1. Initial program 31.7

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

    2. Taylor expanded around inf 23.4

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

    3. Applied simplify23.4

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

    if 5.984857460889397e+68 < x < 5.538161835309689e+77 or 7.394087600696336e+134 < x

    1. Initial program 13.0

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

    3. Applied pow-to-exp13.1

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

    4. Applied simplify13.1

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

    6. Applied add-cube-cbrt13.1

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

    8. Applied add-sqr-sqrt13.1

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

    9. Applied add-sqr-sqrt13.1

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

    10. Applied unpow-prod-down13.1

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

    11. Applied difference-of-squares13.1

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

    13. Applied add-log-exp13.1

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

    14. Applied add-log-exp13.0

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

    15. Applied diff-log13.0

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

    16. Applied simplify13.0

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))