Error

Derivation

1. Split input into 3 regimes

2. if (+ (* (+ 1/3 1/3) (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n))))) (log (cbrt (exp (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))))) < -0.0008261050182977736

   1. Initial program 19.2

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

   3. Applied add-log-exp19.5

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

   4. Applied add-log-exp19.5

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

   5. Applied diff-log19.5

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

   6. Applied simplify19.5

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

   8. Applied add-cube-cbrt19.6

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

   9. Applied log-prod19.6

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

   11. Applied pow1/319.5

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

   12. Applied pow1/319.5

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

   13. Applied pow-prod-up19.5

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

   14. Applied log-pow19.5

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

   15. Applied simplify19.5

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

   17. Applied add-sqr-sqrt19.5

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

   18. Applied cbrt-prod19.5

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

   19. Applied log-prod19.5

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

   if -0.0008261050182977736 < (+ (* (+ 1/3 1/3) (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n))))) (log (cbrt (exp (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))))) < 6.325270227888274e-17

   1. Initial program 40.2

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

   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 simplify20.9

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

   if 6.325270227888274e-17 < (+ (* (+ 1/3 1/3) (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n))))) (log (cbrt (exp (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))))))

   1. Initial program 29.4

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

   3. Applied add-log-exp29.4

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

   4. Applied add-log-exp29.4

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

   5. Applied diff-log29.5

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

   6. Applied simplify29.4

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

   8. Applied add-cube-cbrt30.6

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

   9. Applied log-prod30.6

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

   11. Applied pow1/330.4

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

   12. Applied pow1/330.0

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

   13. Applied pow-prod-up29.9

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

   14. Applied log-pow29.9

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

   15. Applied simplify29.9

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

   17. Applied sub-neg29.9

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

   18. Applied exp-sum29.9

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

   19. Applied cbrt-prod29.4

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

   20. Applied log-prod29.4

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

Runtime

Time bar (total: 56.7s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))