Average Error: 32.4 → 13.8
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + e^{\log \left(\frac{\log x}{\left(n \cdot x\right) \cdot n}\right)} \le -2.3087914140683312 \cdot 10^{-171}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + e^{\log \left(\frac{\log x}{\left(n \cdot x\right) \cdot n}\right)} \le 9.805524243971042 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}}{n \cdot x} \cdot \frac{\sqrt[3]{\log x}}{n}\\ \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + e^{\log \left(\frac{\log x}{\left(n \cdot x\right) \cdot n}\right)} \le +\infty:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (/ (/ 1 n) x) (/ (/ 1/2 n) (* x x))) (exp (log (/ (log x) (* (* n x) n))))) < -2.3087914140683312e-171 or +inf.0 < (+ (- (/ (/ 1 n) x) (/ (/ 1/2 n) (* x x))) (exp (log (/ (log x) (* (* n x) n)))))

    1. Initial program 45.9

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

    2. Taylor expanded around inf 60.1

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

    3. Applied simplify19.9

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

    if -2.3087914140683312e-171 < (+ (- (/ (/ 1 n) x) (/ (/ 1/2 n) (* x x))) (exp (log (/ (log x) (* (* n x) n))))) < 9.805524243971042e-111

    1. Initial program 28.4

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify7.9

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

    5. Applied add-cube-cbrt7.9

      \[\leadsto \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\color{blue}{\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}}}{\left(n \cdot x\right) \cdot n}\]

    6. Applied times-frac7.8

      \[\leadsto \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \color{blue}{\frac{\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}}{n \cdot x} \cdot \frac{\sqrt[3]{\log x}}{n}}\]

    if 9.805524243971042e-111 < (+ (- (/ (/ 1 n) x) (/ (/ 1/2 n) (* x x))) (exp (log (/ (log x) (* (* n x) n))))) < +inf.0

    1. Initial program 12.1

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

    3. Applied add-log-exp12.1

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

    4. Applied add-log-exp12.1

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

    5. Applied diff-log12.1

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

    6. Applied simplify12.1

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

    8. Applied add-cube-cbrt12.1

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

    9. Applied log-prod12.1

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))