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Target

Derivation

1. Split input into 2 regimes

2. if n < -2.020135719333177e+24 or 4.6995232979681e-310 < n

   1. Initial program 50.1

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied flip3--50.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt50.1

      \[\leadsto 100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}{\frac{i}{n}}\]

   6. Applied cube-mult50.1

      \[\leadsto 100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - \color{blue}{1 \cdot \left(1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

   7. Applied *-un-lft-identity50.1

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{1 \cdot {\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1 \cdot \left(1 \cdot 1\right)}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

   8. Applied distribute-lft-out--50.1

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{1 \cdot \left({\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

   9. Applied times-frac50.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}} \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}{\frac{i}{n}}\]

   10. Applied associate-/l*50.1

       \[\leadsto 100 \cdot \color{blue}{\frac{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{\frac{i}{n}}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}\]

   11. Simplified50.1

       \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}}{\frac{\frac{i}{n}}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}\]

   12. Taylor expanded around 0 16.6

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\color{blue}{\left(\frac{1}{2} \cdot \frac{i}{n \cdot \sqrt{3}} + \left(\frac{1}{6} \cdot \frac{i \cdot \sqrt{3}}{{n}^{2}} + \frac{1}{3} \cdot \frac{\sqrt{3}}{n}\right)\right) - \frac{1}{2} \cdot \frac{i \cdot \sqrt{3}}{n}}}\]

   13. Simplified11.7

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\color{blue}{\frac{\frac{1}{3}}{n} \cdot \sqrt{3} + \left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{\frac{i}{n}}} + \left(\frac{\frac{1}{6}}{n} - \frac{1}{2}\right) \cdot \left(\frac{i}{n} \cdot \sqrt{3}\right)\right)}}\]

   if -2.020135719333177e+24 < n < 4.6995232979681e-310

   1. Initial program 17.2

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied flip3--17.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt17.2

      \[\leadsto 100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}{\frac{i}{n}}\]

   6. Applied cube-mult17.2

      \[\leadsto 100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - \color{blue}{1 \cdot \left(1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

   7. Applied *-un-lft-identity17.2

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{1 \cdot {\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1 \cdot \left(1 \cdot 1\right)}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

   8. Applied distribute-lft-out--17.2

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{1 \cdot \left({\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

   9. Applied times-frac17.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}} \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}{\frac{i}{n}}\]

   10. Applied associate-/l*17.2

       \[\leadsto 100 \cdot \color{blue}{\frac{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{\frac{i}{n}}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}\]

   11. Simplified17.2

       \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}}{\frac{\frac{i}{n}}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\frac{\frac{i}{n}}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\sqrt{\color{blue}{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}}\]

   14. Applied sqrt-prod17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\frac{\frac{i}{n}}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1 \cdot 1}{\color{blue}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}} \cdot \sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}}\]

   15. Applied add-sqr-sqrt17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\frac{\frac{i}{n}}{\frac{\color{blue}{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} \cdot \sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}}} - 1 \cdot 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}} \cdot \sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}\]

   16. Applied difference-of-squares17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\frac{\frac{i}{n}}{\frac{\color{blue}{\left(\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} + 1\right) \cdot \left(\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1\right)}}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}} \cdot \sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}\]

   17. Applied times-frac17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\frac{\frac{i}{n}}{\color{blue}{\frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} + 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}} \cdot \frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}}\]

   18. Applied div-inv17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\frac{\color{blue}{i \cdot \frac{1}{n}}}{\frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} + 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}} \cdot \frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}\]

   19. Applied times-frac17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\color{blue}{\frac{i}{\frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} + 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}} \cdot \frac{\frac{1}{n}}{\frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}}\]

   20. Simplified17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\color{blue}{\left(\frac{i}{1 + \sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}}} \cdot \left|\sqrt[3]{\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}}\right|\right)} \cdot \frac{\frac{1}{n}}{\frac{\sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - 1}{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}}}\]

   21. Simplified17.2

       \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}}{\left(\frac{i}{1 + \sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}}} \cdot \left|\sqrt[3]{\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}}\right|\right) \cdot \color{blue}{\frac{\sqrt{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}}{n \cdot \sqrt{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}} - n}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.020135719333177 \cdot 10^{+24} \lor \neg \left(n \le 4.6995232979681 \cdot 10^{-310}\right):\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right)}}}{\left(\left(\frac{\frac{1}{6}}{n} - \frac{1}{2}\right) \cdot \left(\frac{i}{n} \cdot \sqrt{3}\right) + \frac{\frac{1}{2}}{\frac{\sqrt{3}}{\frac{i}{n}}}\right) + \frac{\frac{1}{3}}{n} \cdot \sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right)}}}{\left(\frac{i}{\sqrt{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3}} + 1} \cdot \left|\sqrt[3]{\left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right) + {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}}\right|\right) \cdot \frac{\sqrt{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right)}}}{\sqrt{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3}} \cdot n - n}} \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))

Details

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