Error

Try it out

Derivation

1. Split input into 3 regimes

2. if x < -1.003492093787603

   1. Initial program 59.7

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

   2. Initial simplification59.7

      \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]

   3. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(\frac{5}{81} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{3}} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}}\]

   4. Simplified0.9

      \[\leadsto \color{blue}{\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \frac{\sqrt[3]{x}}{x}}\]
   5. Using strategy rm

   6. Applied div-inv0.9

      \[\leadsto \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{x}\right)}\]

   7. Applied associate-*r*0.9

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x}}\]

   if -1.003492093787603 < x < 0.9050262628886326

   1. Initial program 0.0

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

   2. Initial simplification0.0

      \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]

   3. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\left(\frac{1}{3} \cdot x + 1\right) - \frac{1}{9} \cdot {x}^{2}\right)} - \sqrt[3]{x}\]

   if 0.9050262628886326 < x

   1. Initial program 59.7

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

   2. Initial simplification59.7

      \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]

   3. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(\frac{5}{81} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{3}} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}}\]

   4. Simplified1.0

      \[\leadsto \color{blue}{\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \frac{\sqrt[3]{x}}{x}}\]
   5. Using strategy rm

   6. Applied div-inv1.0

      \[\leadsto \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{x}\right)}\]

   7. Applied associate-*r*1.0

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x}}\]
   8. Using strategy rm

   9. Applied add-cbrt-cube1.0

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}} \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x}\]

   10. Applied cbrt-unprod0.9

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot x}} \cdot \frac{1}{x}\]
   11. Using strategy rm

   12. Applied flip-+0.9

       \[\leadsto \sqrt[3]{\left(\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \color{blue}{\frac{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}{\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}}\right) \cdot x} \cdot \frac{1}{x}\]

   13. Applied flip3-+1.1

       \[\leadsto \sqrt[3]{\left(\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \color{blue}{\frac{{\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}}{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}}\right) \cdot \frac{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}{\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}\right) \cdot x} \cdot \frac{1}{x}\]

   14. Applied flip3-+1.1

       \[\leadsto \sqrt[3]{\left(\left(\color{blue}{\frac{{\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}}{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}} \cdot \frac{{\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}}{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}\right) \cdot \frac{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}{\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}\right) \cdot x} \cdot \frac{1}{x}\]

   15. Applied frac-times1.1

       \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{\left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right) \cdot \left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right)}{\left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)}} \cdot \frac{\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}{\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}\right) \cdot x} \cdot \frac{1}{x}\]

   16. Applied frac-times1.4

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right) \cdot \left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}} \cdot x} \cdot \frac{1}{x}\]

   17. Applied associate-*l/1.3

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\left(\left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right) \cdot \left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot x}{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}}} \cdot \frac{1}{x}\]

   18. Applied cbrt-div1.0

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(\left(\left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right) \cdot \left({\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3} + {\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot x}}{\sqrt[3]{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{\frac{5}{81}}{x}}{x} \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}}} \cdot \frac{1}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.003492093787603:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right) \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 0.9050262628886326:\\ \;\;\;\;\left(\left(x \cdot \frac{1}{3} + 1\right) - {x}^{2} \cdot \frac{1}{9}\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot \left(\left(\left({\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3} + {\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3}\right) \cdot \left({\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)}^{3} + {\left(\frac{\frac{\frac{5}{81}}{x}}{x}\right)}^{3}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)\right)}}{\sqrt[3]{\left(\left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} \cdot \frac{\frac{\frac{5}{81}}{x}}{x} + \left(\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right)\right) \cdot \left(\frac{\frac{\frac{5}{81}}{x}}{x} - \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right)\right)}} \cdot \frac{1}{x}\\ \end{array}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))