Average Error: 32.8 → 23.5
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -7.088671140842874 \cdot 10^{-08}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 2.3769939386456005 \cdot 10^{-216}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(\frac{-\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n))) < -7.088671140842874e-08

    1. Initial program 21.2

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

    2. Initial simplification21.2

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

    4. Applied add-exp-log21.2

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

    5. Applied pow-exp21.2

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

    6. Simplified21.2

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

    8. Applied add-log-exp21.5

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

    10. Applied add-sqr-sqrt21.5

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

    11. Applied log-prod21.5

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

    if -7.088671140842874e-08 < (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n))) < 2.3769939386456005e-216

    1. Initial program 39.2

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

    2. Initial simplification39.2

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

    3. Taylor expanded around inf 21.7

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

    4. Simplified21.7

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

    if 2.3769939386456005e-216 < (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n)))

    1. Initial program 32.6

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

    2. Initial simplification32.6

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

    4. Applied add-exp-log32.6

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

    5. Applied pow-exp32.6

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

    6. Simplified31.1

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

    8. Applied add-log-exp31.6

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

    10. Applied add-cube-cbrt31.6

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

    11. Applied exp-prod31.6

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

    12. Applied log-pow31.6

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

    14. Applied add-sqr-sqrt31.6

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

    15. Applied log-prod31.6

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -7.088671140842874 \cdot 10^{-08}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 2.3769939386456005 \cdot 10^{-216}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(\frac{-\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))