Average Error: 32.8 → 12.5
Time: 2.2m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(x - \log x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(\log x \cdot \log x\right) \cdot \left(\log x \cdot \log x\right) + \left(x \cdot x\right) \cdot \left(\log x \cdot \log x\right)\right)\right) + n \cdot \left(\frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left({\left(x \cdot x\right)}^{3} - {\left(\log x \cdot \log x\right)}^{3}\right)\right)}{\left(n \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x + \log x \cdot \log x\right) + \left(n \cdot \log x\right) \cdot {\left(\log x\right)}^{3}} \le -0.0005657820419688625:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;\frac{\left(x - \log x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(\log x \cdot \log x\right) \cdot \left(\log x \cdot \log x\right) + \left(x \cdot x\right) \cdot \left(\log x \cdot \log x\right)\right)\right) + n \cdot \left(\frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left({\left(x \cdot x\right)}^{3} - {\left(\log x \cdot \log x\right)}^{3}\right)\right)}{\left(n \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x + \log x \cdot \log x\right) + \left(n \cdot \log x\right) \cdot {\left(\log x\right)}^{3}} \le 3.3882049535259356 \cdot 10^{-05}:\\ \;\;\;\;\left(\frac{x}{n} - \frac{\log x}{n}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (- x (log x)) (+ (* (* x x) (* x x)) (+ (* (* (log x) (log x)) (* (log x) (log x))) (* (* x x) (* (log x) (log x)))))) (* n (* (/ (/ 1/2 n) n) (- (pow (* x x) 3) (pow (* (log x) (log x)) 3))))) (+ (* (* n (* x x)) (+ (* x x) (* (log x) (log x)))) (* (* n (log x)) (pow (log x) 3)))) < -0.0005657820419688625 or 3.3882049535259356e-05 < (/ (+ (* (- x (log x)) (+ (* (* x x) (* x x)) (+ (* (* (log x) (log x)) (* (log x) (log x))) (* (* x x) (* (log x) (log x)))))) (* n (* (/ (/ 1/2 n) n) (- (pow (* x x) 3) (pow (* (log x) (log x)) 3))))) (+ (* (* n (* x x)) (+ (* x x) (* (log x) (log x)))) (* (* n (log x)) (pow (log x) 3))))

    1. Initial program 14.8

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

    3. Applied add-log-exp15.0

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

    4. Applied add-log-exp14.9

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

    5. Applied diff-log14.9

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

    6. Applied simplify14.9

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

    if -0.0005657820419688625 < (/ (+ (* (- x (log x)) (+ (* (* x x) (* x x)) (+ (* (* (log x) (log x)) (* (log x) (log x))) (* (* x x) (* (log x) (log x)))))) (* n (* (/ (/ 1/2 n) n) (- (pow (* x x) 3) (pow (* (log x) (log x)) 3))))) (+ (* (* n (* x x)) (+ (* x x) (* (log x) (log x)))) (* (* n (log x)) (pow (log x) 3)))) < 3.3882049535259356e-05

    1. Initial program 58.7

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

    2. Taylor expanded around inf 58.7

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

    3. Applied simplify9.7

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

    4. Taylor expanded around 0 9.0

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

    5. Applied simplify9.0

      \[\leadsto \color{blue}{\left(\frac{x}{n} - \frac{\log x}{n}\right) + \frac{\frac{\frac{1}{2}}{n}}{n} \cdot \left(x \cdot x - \log x \cdot \log x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))