Average Error: 32.8 → 22.5
Time: 55.0s
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} + \frac{\log x}{n}\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right) \le -0.5632357087630802:\\ \;\;\;\;\log \left(\log_* (1 + (e^{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} - 1)^*)\right)\\ \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} + \frac{\log x}{n}\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right) \le 7.78415512267959 \cdot 10^{-06}:\\ \;\;\;\;\left(\log 1 + \frac{\frac{\log x}{x}}{n \cdot n}\right) - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{1}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({e}^{\left(\sqrt{\frac{\log_* (1 + x)}{n}}\right)}\right)}^{\left(\sqrt{\frac{\log_* (1 + x)}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n)))) < -0.5632357087630802

    1. Initial program 19.9

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

    3. Applied add-log-exp20.2

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

    4. Applied add-log-exp20.2

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

    5. Applied diff-log20.2

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

    6. Applied simplify20.2

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

    8. Applied log1p-expm1-u20.2

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

    if -0.5632357087630802 < (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n)))) < 7.78415512267959e-06

    1. Initial program 40.1

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

    3. Applied add-log-exp40.1

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

    4. Applied add-log-exp40.1

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

    5. Applied diff-log40.1

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

    6. Applied simplify40.1

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

    7. Taylor expanded around -inf 63.0

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

    8. Applied simplify21.6

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

    if 7.78415512267959e-06 < (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n))))

    1. Initial program 30.7

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

    3. Applied add-exp-log30.7

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

    4. Applied pow-exp30.7

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

    5. Applied simplify28.8

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

    7. Applied *-un-lft-identity28.8

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

    8. Applied exp-prod28.8

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

    9. Applied simplify28.8

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

    11. Applied add-sqr-sqrt28.8

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

    12. Applied pow-unpow28.8

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

Runtime

Time bar (total: 55.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))