Compound Interest

Average Error: 42.9 → 32.7

Time: 7.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -8.97495351037423816126988860777680675991 \cdot 10^{177}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -40962607978263219673797282402861580288:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8.676387546643350025026434985864840262105 \cdot 10^{-297}:\\ \;\;\;\;100 \cdot \frac{\sqrt[3]{\frac{\frac{\left|{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right| \cdot \left|{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right| + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}\\ \mathbf{elif}\;n \le 1.496434530671250065211050155721986673116 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{\frac{\mathsf{fma}\left(2, i, \mathsf{fma}\left(2, \log 1 \cdot n, 1\right)\right) + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.480459864121622227072210143558391846685 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}} - \frac{\frac{\frac{\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r1637025 = 100.0;
        double r1637026 = 1.0;
        double r1637027 = i;
        double r1637028 = n;
        double r1637029 = r1637027 / r1637028;
        double r1637030 = r1637026 + r1637029;
        double r1637031 = pow(r1637030, r1637028);
        double r1637032 = r1637031 - r1637026;
        double r1637033 = r1637032 / r1637029;
        double r1637034 = r1637025 * r1637033;
        return r1637034;
}

↓

double f(double i, double n) {
        double r1637035 = n;
        double r1637036 = -8.974953510374238e+177;
        bool r1637037 = r1637035 <= r1637036;
        double r1637038 = 100.0;
        double r1637039 = 1.0;
        double r1637040 = i;
        double r1637041 = r1637040 / r1637035;
        double r1637042 = r1637039 + r1637041;
        double r1637043 = pow(r1637042, r1637035);
        double r1637044 = r1637043 - r1637039;
        double r1637045 = r1637044 / r1637040;
        double r1637046 = r1637038 * r1637045;
        double r1637047 = r1637046 * r1637035;
        double r1637048 = -4.096260797826322e+37;
        bool r1637049 = r1637035 <= r1637048;
        double r1637050 = 0.5;
        double r1637051 = 2.0;
        double r1637052 = pow(r1637040, r1637051);
        double r1637053 = log(r1637039);
        double r1637054 = r1637053 * r1637035;
        double r1637055 = fma(r1637050, r1637052, r1637054);
        double r1637056 = r1637052 * r1637053;
        double r1637057 = r1637050 * r1637056;
        double r1637058 = r1637055 - r1637057;
        double r1637059 = fma(r1637040, r1637039, r1637058);
        double r1637060 = r1637059 / r1637041;
        double r1637061 = r1637038 * r1637060;
        double r1637062 = -8.67638754664335e-297;
        bool r1637063 = r1637035 <= r1637062;
        double r1637064 = r1637051 * r1637035;
        double r1637065 = pow(r1637042, r1637064);
        double r1637066 = fabs(r1637065);
        double r1637067 = r1637066 * r1637066;
        double r1637068 = r1637039 * r1637039;
        double r1637069 = -r1637068;
        double r1637070 = r1637069 * r1637069;
        double r1637071 = -r1637070;
        double r1637072 = r1637067 + r1637071;
        double r1637073 = r1637065 + r1637068;
        double r1637074 = r1637072 / r1637073;
        double r1637075 = r1637043 + r1637039;
        double r1637076 = r1637074 / r1637075;
        double r1637077 = cbrt(r1637076);
        double r1637078 = r1637051 * r1637064;
        double r1637079 = pow(r1637042, r1637078);
        double r1637080 = r1637079 + r1637071;
        double r1637081 = r1637080 / r1637073;
        double r1637082 = r1637081 / r1637075;
        double r1637083 = cbrt(r1637082);
        double r1637084 = r1637077 * r1637083;
        double r1637085 = r1637041 / r1637083;
        double r1637086 = r1637084 / r1637085;
        double r1637087 = r1637038 * r1637086;
        double r1637088 = 1.49643453067125e-224;
        bool r1637089 = r1637035 <= r1637088;
        double r1637090 = 2.0;
        double r1637091 = 1.0;
        double r1637092 = fma(r1637051, r1637054, r1637091);
        double r1637093 = fma(r1637090, r1637040, r1637092);
        double r1637094 = r1637093 + r1637069;
        double r1637095 = r1637094 / r1637075;
        double r1637096 = r1637095 / r1637041;
        double r1637097 = r1637038 * r1637096;
        double r1637098 = 1.4804598641216222e-197;
        bool r1637099 = r1637035 <= r1637098;
        double r1637100 = r1637079 / r1637073;
        double r1637101 = r1637100 / r1637075;
        double r1637102 = r1637101 / r1637041;
        double r1637103 = r1637070 / r1637073;
        double r1637104 = r1637103 / r1637075;
        double r1637105 = r1637104 / r1637041;
        double r1637106 = r1637102 - r1637105;
        double r1637107 = r1637038 * r1637106;
        double r1637108 = r1637099 ? r1637107 : r1637061;
        double r1637109 = r1637089 ? r1637097 : r1637108;
        double r1637110 = r1637063 ? r1637087 : r1637109;
        double r1637111 = r1637049 ? r1637061 : r1637110;
        double r1637112 = r1637037 ? r1637047 : r1637111;
        return r1637112;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	43.1
Herbie	32.7

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

Derivation

1. Split input into 5 regimes

2. if n < -8.974953510374238e+177

   1. Initial program 53.2

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

   3. Applied associate-/r/ 52.7

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

   4. Applied associate-*r* 52.7

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

   if -8.974953510374238e+177 < n < -4.096260797826322e+37 or 1.4804598641216222e-197 < n

   1. Initial program 53.1

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

   2. Taylor expanded around 0 36.4

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

   3. Simplified 36.4

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

   if -4.096260797826322e+37 < n < -8.67638754664335e-297

   1. Initial program 19.0

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

   3. Applied flip-- 19.0

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

   4. Simplified 19.0

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

   6. Applied flip-+ 19.0

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

   7. Simplified 19.0

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

   8. Simplified 19.0

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

   10. Applied add-cube-cbrt 19.0

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

   11. Applied associate-/l* 19.0

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

   13. Applied add-sqr-sqrt 19.0

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

   14. Simplified 19.0

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

   15. Simplified 19.0

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

   if -8.67638754664335e-297 < n < 1.49643453067125e-224

   1. Initial program 34.6

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

   3. Applied flip-- 34.6

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

   4. Simplified 34.6

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

   5. Taylor expanded around 0 15.1

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

   6. Simplified 15.1

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

   if 1.49643453067125e-224 < n < 1.4804598641216222e-197

   1. Initial program 47.4

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

   3. Applied flip-- 47.4

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

   4. Simplified 47.4

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

   6. Applied flip-+ 47.4

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

   7. Simplified 47.4

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

   8. Simplified 47.4

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

   10. Applied unsub-neg 47.4

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

   11. Applied div-sub 47.4

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

   12. Applied div-sub 47.4

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

   13. Applied div-sub 47.4

       \[\leadsto 100 \cdot \color{blue}{\left(\frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}} - \frac{\frac{\frac{\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 32.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.97495351037423816126988860777680675991 \cdot 10^{177}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -40962607978263219673797282402861580288:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8.676387546643350025026434985864840262105 \cdot 10^{-297}:\\ \;\;\;\;100 \cdot \frac{\sqrt[3]{\frac{\frac{\left|{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right| \cdot \left|{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right| + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}\\ \mathbf{elif}\;n \le 1.496434530671250065211050155721986673116 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{\frac{\mathsf{fma}\left(2, i, \mathsf{fma}\left(2, \log 1 \cdot n, 1\right)\right) + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.480459864121622227072210143558391846685 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}} - \frac{\frac{\frac{\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))