Average Error: 32.8 → 22.8
Time: 1.1m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(1 - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{1}{x \cdot n}\right) + \frac{\log x}{n} \le -18.023303851257182:\\ \;\;\;\;\log \left(\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{if}\;\left(\left(1 - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{1}{x \cdot n}\right) + \frac{\log x}{n} \le 8.491805588372093 \cdot 10^{-06}:\\ \;\;\;\;\left(\log 1 + \frac{\frac{\log x}{n \cdot n}}{x}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (+ (- 1 (pow x (/ 1 n))) (/ 1 (* x n))) (/ (log x) n)) < -18.023303851257182

    1. Initial program 19.6

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

    3. Applied add-log-exp19.9

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

    4. Applied add-log-exp19.9

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

    5. Applied diff-log19.9

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

    6. Applied simplify19.9

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

    8. Applied sub-neg19.9

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

    9. Applied exp-sum19.9

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

    11. Applied add-sqr-sqrt19.9

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

    12. Applied associate-*r*19.5

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

    if -18.023303851257182 < (+ (+ (- 1 (pow x (/ 1 n))) (/ 1 (* x n))) (/ (log x) n)) < 8.491805588372093e-06

    1. Initial program 40.2

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

    3. Applied add-log-exp40.2

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

    4. Applied add-log-exp40.2

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

    5. Applied diff-log40.2

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

    6. Applied simplify40.2

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

    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 simplify21.8

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

    if 8.491805588372093e-06 < (+ (+ (- 1 (pow x (/ 1 n))) (/ 1 (* x n))) (/ (log x) 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 flip--30.7

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

Runtime

Time bar (total: 1.1m)Debug logProfile

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