Compound Interest

Average Error: 42.6 → 21.2

Time: 34.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.09568466101685804692245085334434406831861:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.636828096634822172894426601649853756376 \cdot 10^{-231}:\\ \;\;\;\;\left(\left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \frac{100}{i}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 2.421886781167336491233076084341835921918 \cdot 10^{-182}:\\ \;\;\;\;\frac{100}{i} \cdot \sqrt[3]{{\left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}^{3}}\\ \mathbf{elif}\;i \le 6.433403710498657594142250154269082661313 \cdot 10^{46}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 1.353418716849568981322833055613312096611 \cdot 10^{183}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} - 1}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r122023 = 100.0;
        double r122024 = 1.0;
        double r122025 = i;
        double r122026 = n;
        double r122027 = r122025 / r122026;
        double r122028 = r122024 + r122027;
        double r122029 = pow(r122028, r122026);
        double r122030 = r122029 - r122024;
        double r122031 = r122030 / r122027;
        double r122032 = r122023 * r122031;
        return r122032;
}

↓

double f(double i, double n) {
        double r122033 = i;
        double r122034 = -0.09568466101685805;
        bool r122035 = r122033 <= r122034;
        double r122036 = 100.0;
        double r122037 = n;
        double r122038 = r122033 / r122037;
        double r122039 = pow(r122038, r122037);
        double r122040 = 1.0;
        double r122041 = r122039 - r122040;
        double r122042 = r122036 * r122041;
        double r122043 = r122042 / r122038;
        double r122044 = 4.636828096634822e-231;
        bool r122045 = r122033 <= r122044;
        double r122046 = 0.5;
        double r122047 = 2.0;
        double r122048 = pow(r122033, r122047);
        double r122049 = log(r122040);
        double r122050 = r122049 * r122037;
        double r122051 = fma(r122046, r122048, r122050);
        double r122052 = fma(r122040, r122033, r122051);
        double r122053 = r122048 * r122049;
        double r122054 = r122046 * r122053;
        double r122055 = r122052 - r122054;
        double r122056 = cbrt(r122055);
        double r122057 = r122056 * r122056;
        double r122058 = r122036 / r122033;
        double r122059 = r122057 * r122058;
        double r122060 = 1.0;
        double r122061 = r122060 / r122037;
        double r122062 = r122056 / r122061;
        double r122063 = r122059 * r122062;
        double r122064 = 2.4218867811673365e-182;
        bool r122065 = r122033 <= r122064;
        double r122066 = r122055 * r122037;
        double r122067 = 3.0;
        double r122068 = pow(r122066, r122067);
        double r122069 = cbrt(r122068);
        double r122070 = r122058 * r122069;
        double r122071 = 6.433403710498658e+46;
        bool r122072 = r122033 <= r122071;
        double r122073 = r122055 / r122033;
        double r122074 = r122073 * r122037;
        double r122075 = r122036 * r122074;
        double r122076 = 1.353418716849569e+183;
        bool r122077 = r122033 <= r122076;
        double r122078 = r122033 / r122041;
        double r122079 = r122037 / r122078;
        double r122080 = r122036 * r122079;
        double r122081 = fma(r122049, r122037, r122060);
        double r122082 = fma(r122040, r122033, r122081);
        double r122083 = r122082 - r122040;
        double r122084 = r122083 / r122038;
        double r122085 = r122036 * r122084;
        double r122086 = r122077 ? r122080 : r122085;
        double r122087 = r122072 ? r122075 : r122086;
        double r122088 = r122065 ? r122070 : r122087;
        double r122089 = r122045 ? r122063 : r122088;
        double r122090 = r122035 ? r122043 : r122089;
        return r122090;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.6
Target	42.1
Herbie	21.2

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

Derivation

1. Split input into 6 regimes

2. if i < -0.09568466101685805

   1. Initial program 28.6

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 18.7

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

   if -0.09568466101685805 < i < 4.636828096634822e-231

   1. Initial program 49.2

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

   3. Applied div-inv 49.2

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

   4. Applied *-un-lft-identity 49.2

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

   5. Applied times-frac 48.8

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

   6. Applied associate-*r* 48.9

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

   7. Simplified 48.9

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

   8. Taylor expanded around 0 17.2

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

   9. Simplified 17.2

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 17.2

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

   12. Applied add-sqr-sqrt 17.2

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

   13. Applied times-frac 17.2

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

   14. Applied add-cube-cbrt 17.8

       \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{n}}\]

   15. Applied times-frac 17.8

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(\frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{1}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{n}}\right)}\]

   16. Applied associate-*r* 18.6

       \[\leadsto \color{blue}{\left(\frac{100}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{1}}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{n}}}\]

   17. Simplified 18.6

       \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \frac{100}{i}\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{n}}\]

   if 4.636828096634822e-231 < i < 2.4218867811673365e-182

   1. Initial program 48.3

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

   3. Applied div-inv 48.3

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

   4. Applied *-un-lft-identity 48.3

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

   5. Applied times-frac 47.8

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

   6. Applied associate-*r* 47.8

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

   7. Simplified 47.8

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

   8. Taylor expanded around 0 16.9

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

   9. Simplified 16.9

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 39.9

       \[\leadsto \frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{\color{blue}{\sqrt[3]{\left(n \cdot n\right) \cdot n}}}}\]

   12. Applied add-cbrt-cube 39.9

       \[\leadsto \frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(n \cdot n\right) \cdot n}}}\]

   13. Applied cbrt-undiv 39.9

       \[\leadsto \frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(n \cdot n\right) \cdot n}}}}\]

   14. Applied add-cbrt-cube 48.3

       \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\sqrt[3]{\left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(n \cdot n\right) \cdot n}}}\]

   15. Applied cbrt-undiv 48.3

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\sqrt[3]{\frac{\left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(n \cdot n\right) \cdot n}}}}\]

   16. Simplified 24.7

       \[\leadsto \frac{100}{i} \cdot \sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}^{3}}}\]

   if 2.4218867811673365e-182 < i < 6.433403710498658e+46

   1. Initial program 50.3

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

   3. Applied div-inv 50.3

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

   4. Applied *-un-lft-identity 50.3

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

   5. Applied times-frac 50.2

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

   6. Applied associate-*r* 50.2

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

   7. Simplified 50.2

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

   8. Taylor expanded around 0 19.3

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

   9. Simplified 19.3

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied div-inv 19.3

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

   12. Applied associate-*l* 19.2

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

   13. Simplified 21.5

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

   if 6.433403710498658e+46 < i < 1.353418716849569e+183

   1. Initial program 32.0

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

   2. Taylor expanded around inf 29.2

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

   3. Simplified 32.0

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

   if 1.353418716849569e+183 < i

   1. Initial program 33.1

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

   2. Taylor expanded around 0 32.9

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

   3. Simplified 32.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 21.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.09568466101685804692245085334434406831861:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.636828096634822172894426601649853756376 \cdot 10^{-231}:\\ \;\;\;\;\left(\left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \frac{100}{i}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 2.421886781167336491233076084341835921918 \cdot 10^{-182}:\\ \;\;\;\;\frac{100}{i} \cdot \sqrt[3]{{\left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}^{3}}\\ \mathbf{elif}\;i \le 6.433403710498657594142250154269082661313 \cdot 10^{46}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 1.353418716849568981322833055613312096611 \cdot 10^{183}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} - 1}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))