Compound Interest

Average Error: 42.9 → 23.7

Time: 39.9s

Precision: 64

Math

\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.813114393619137495994961909777216500149 \cdot 10^{175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, i \cdot 1\right)\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \le -1.421989441319127851613951560151701461699 \cdot 10^{58}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}, 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le -1.995072885870678103259479030384682118893:\\ \;\;\;\;\mathsf{fma}\left(\left(n \cdot n\right) \cdot \frac{\log 1}{i}, 100, \mathsf{fma}\left(50, i \cdot n, 100 \cdot n\right)\right) - \left(i \cdot n\right) \cdot \left(50 \cdot \log 1\right)\\ \mathbf{elif}\;n \le -4.586926552796907742204996293991418623461 \cdot 10^{-267}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}, 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le 8.765996428721520261943646244564954658781 \cdot 10^{-223}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, i \cdot 1\right)\right)}{i}}{\frac{1}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r7582780 = 100.0;
        double r7582781 = 1.0;
        double r7582782 = i;
        double r7582783 = n;
        double r7582784 = r7582782 / r7582783;
        double r7582785 = r7582781 + r7582784;
        double r7582786 = pow(r7582785, r7582783);
        double r7582787 = r7582786 - r7582781;
        double r7582788 = r7582787 / r7582784;
        double r7582789 = r7582780 * r7582788;
        return r7582789;
}

↓

double f(double i, double n) {
        double r7582790 = n;
        double r7582791 = -2.8131143936191375e+175;
        bool r7582792 = r7582790 <= r7582791;
        double r7582793 = 1.0;
        double r7582794 = log(r7582793);
        double r7582795 = i;
        double r7582796 = r7582795 * r7582795;
        double r7582797 = 0.5;
        double r7582798 = r7582797 * r7582794;
        double r7582799 = r7582797 - r7582798;
        double r7582800 = r7582795 * r7582793;
        double r7582801 = fma(r7582796, r7582799, r7582800);
        double r7582802 = fma(r7582794, r7582790, r7582801);
        double r7582803 = 100.0;
        double r7582804 = r7582803 * r7582790;
        double r7582805 = r7582802 * r7582804;
        double r7582806 = r7582805 / r7582795;
        double r7582807 = -1.4219894413191279e+58;
        bool r7582808 = r7582790 <= r7582807;
        double r7582809 = r7582803 / r7582795;
        double r7582810 = r7582795 / r7582790;
        double r7582811 = r7582810 + r7582793;
        double r7582812 = pow(r7582811, r7582790);
        double r7582813 = r7582812 * r7582812;
        double r7582814 = r7582813 * r7582812;
        double r7582815 = r7582793 * r7582793;
        double r7582816 = r7582793 * r7582815;
        double r7582817 = r7582814 - r7582816;
        double r7582818 = r7582793 + r7582812;
        double r7582819 = fma(r7582818, r7582793, r7582813);
        double r7582820 = r7582817 / r7582819;
        double r7582821 = 1.0;
        double r7582822 = r7582821 / r7582790;
        double r7582823 = r7582820 / r7582822;
        double r7582824 = r7582809 * r7582823;
        double r7582825 = -1.995072885870678;
        bool r7582826 = r7582790 <= r7582825;
        double r7582827 = r7582790 * r7582790;
        double r7582828 = r7582794 / r7582795;
        double r7582829 = r7582827 * r7582828;
        double r7582830 = 50.0;
        double r7582831 = r7582795 * r7582790;
        double r7582832 = fma(r7582830, r7582831, r7582804);
        double r7582833 = fma(r7582829, r7582803, r7582832);
        double r7582834 = r7582830 * r7582794;
        double r7582835 = r7582831 * r7582834;
        double r7582836 = r7582833 - r7582835;
        double r7582837 = -4.586926552796908e-267;
        bool r7582838 = r7582790 <= r7582837;
        double r7582839 = 8.76599642872152e-223;
        bool r7582840 = r7582790 <= r7582839;
        double r7582841 = fma(r7582793, r7582795, r7582821);
        double r7582842 = fma(r7582790, r7582794, r7582841);
        double r7582843 = r7582842 - r7582793;
        double r7582844 = r7582843 / r7582810;
        double r7582845 = r7582803 * r7582844;
        double r7582846 = fma(r7582790, r7582794, r7582801);
        double r7582847 = r7582803 * r7582846;
        double r7582848 = r7582847 / r7582795;
        double r7582849 = r7582848 / r7582822;
        double r7582850 = r7582840 ? r7582845 : r7582849;
        double r7582851 = r7582838 ? r7582824 : r7582850;
        double r7582852 = r7582826 ? r7582836 : r7582851;
        double r7582853 = r7582808 ? r7582824 : r7582852;
        double r7582854 = r7582792 ? r7582806 : r7582853;
        return r7582854;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	42.9
Herbie	23.7

\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

1. Split input into 5 regimes

2. if n < -2.8131143936191375e+175

   1. Initial program 53.6

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

   3. Applied div-inv 53.6

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

   4. Applied *-un-lft-identity 53.6

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

   5. Applied times-frac 53.1

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

   6. Applied associate-*r* 53.1

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

   7. Simplified 53.1

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

   8. Taylor expanded around 0 25.5

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}}{\frac{1}{n}}\]

   9. Simplified 25.5

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 26.1

       \[\leadsto \frac{100}{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}} \cdot \frac{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}{\frac{1}{n}}\]

   12. Applied *-un-lft-identity 26.1

       \[\leadsto \frac{\color{blue}{1 \cdot 100}}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}} \cdot \frac{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}{\frac{1}{n}}\]

   13. Applied times-frac 26.1

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \frac{100}{\sqrt[3]{i}}\right)} \cdot \frac{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}{\frac{1}{n}}\]

   14. Applied associate-*l* 25.9

       \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{100}{\sqrt[3]{i}} \cdot \frac{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}{\frac{1}{n}}\right)}\]

   15. Simplified 25.9

       \[\leadsto \frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, 1 \cdot i\right)\right) \cdot n\right)}{\sqrt[3]{i}}}\]
   16. Using strategy rm

   17. Applied frac-times 25.9

       \[\leadsto \color{blue}{\frac{1 \cdot \left(100 \cdot \left(\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, 1 \cdot i\right)\right) \cdot n\right)\right)}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}\]

   18. Simplified 25.9

       \[\leadsto \frac{\color{blue}{\left(100 \cdot n\right) \cdot \mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i \cdot i, 0.5 - \log 1 \cdot 0.5, i \cdot 1\right)\right)}}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}\]

   19. Simplified 25.1

       \[\leadsto \frac{\left(100 \cdot n\right) \cdot \mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i \cdot i, 0.5 - \log 1 \cdot 0.5, i \cdot 1\right)\right)}{\color{blue}{i}}\]

   if -2.8131143936191375e+175 < n < -1.4219894413191279e+58 or -1.995072885870678 < n < -4.586926552796908e-267

   1. Initial program 25.6

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

   3. Applied div-inv 25.7

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

   4. Applied *-un-lft-identity 25.7

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

   5. Applied times-frac 25.9

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

   6. Applied associate-*r* 26.0

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

   7. Simplified 26.0

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

   9. Applied flip3-- 26.0

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

   10. Simplified 26.0

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

   11. Simplified 26.0

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

   if -1.4219894413191279e+58 < n < -1.995072885870678

   1. Initial program 35.7

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

   3. Applied div-inv 35.7

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

   4. Applied *-un-lft-identity 35.7

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

   5. Applied times-frac 35.7

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

   6. Applied associate-*r* 35.7

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

   7. Simplified 35.7

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

   8. Taylor expanded around 0 28.8

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}}{\frac{1}{n}}\]

   9. Simplified 28.8

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}}{\frac{1}{n}}\]

   10. Taylor expanded around 0 28.1

       \[\leadsto \color{blue}{\left(100 \cdot \frac{\log 1 \cdot {n}^{2}}{i} + \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right) - 50 \cdot \left(\log 1 \cdot \left(i \cdot n\right)\right)}\]

   11. Simplified 28.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log 1}{i} \cdot \left(n \cdot n\right), 100, \mathsf{fma}\left(50, n \cdot i, n \cdot 100\right)\right) - \left(n \cdot i\right) \cdot \left(\log 1 \cdot 50\right)}\]

   if -4.586926552796908e-267 < n < 8.76599642872152e-223

   1. Initial program 29.9

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

   2. Taylor expanded around 0 14.8

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

   3. Simplified 14.8

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right)} - 1}{\frac{i}{n}}\]

   if 8.76599642872152e-223 < n

   1. Initial program 57.6

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

   3. Applied div-inv 57.6

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

   4. Applied *-un-lft-identity 57.6

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

   5. Applied times-frac 57.3

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

   6. Applied associate-*r* 57.3

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

   7. Simplified 57.3

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

   8. Taylor expanded around 0 27.7

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}}{\frac{1}{n}}\]

   9. Simplified 27.7

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied associate-*r/ 22.6

       \[\leadsto \color{blue}{\frac{\frac{100}{i} \cdot \mathsf{fma}\left(\log 1, n, \left(0.5 \cdot i\right) \cdot i + \left(i \cdot 1 - \left(\left(0.5 \cdot i\right) \cdot i\right) \cdot \log 1\right)\right)}{\frac{1}{n}}}\]

   12. Simplified 22.3

       \[\leadsto \frac{\color{blue}{\frac{100 \cdot \mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, 1 \cdot i\right)\right)}{i}}}{\frac{1}{n}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.813114393619137495994961909777216500149 \cdot 10^{175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, i \cdot 1\right)\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \le -1.421989441319127851613951560151701461699 \cdot 10^{58}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}, 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le -1.995072885870678103259479030384682118893:\\ \;\;\;\;\mathsf{fma}\left(\left(n \cdot n\right) \cdot \frac{\log 1}{i}, 100, \mathsf{fma}\left(50, i \cdot n, 100 \cdot n\right)\right) - \left(i \cdot n\right) \cdot \left(50 \cdot \log 1\right)\\ \mathbf{elif}\;n \le -4.586926552796907742204996293991418623461 \cdot 10^{-267}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}, 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le 8.765996428721520261943646244564954658781 \cdot 10^{-223}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5 - 0.5 \cdot \log 1, i \cdot 1\right)\right)}{i}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))