Compound Interest

Average Error: 42.1 → 27.4

Time: 21.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -2.2920532433751285 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)}}{i} \cdot \left(\sqrt[3]{\mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)} \cdot n\right)\\ \mathbf{elif}\;i \le -6.547047229389803 \cdot 10^{-148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\mathsf{fma}\left(\frac{50}{3}, \left(i \cdot \left(i \cdot i\right)\right), \left(50 \cdot \left(i \cdot i\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -9.429451325520378 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(100, \left({\left(\frac{i}{n} + 1\right)}^{n}\right), -100\right)}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.448607663119347:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\mathsf{fma}\left(\frac{50}{3}, \left(i \cdot \left(i \cdot i\right)\right), \left(50 \cdot \left(i \cdot i\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.7000146131065657 \cdot 10^{+237}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log i\right)\right), \left(\mathsf{fma}\left(100, \left(n \cdot \log i\right), \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{100}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\log n \cdot \left(\log n \cdot \log n\right)\right) \cdot \left(n \cdot \left(n \cdot n\right)\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\log n \cdot n\right) \cdot 100\right)\right)\right)\right)\right)\right)\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.7504058567965248 \cdot 10^{+288}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(100, \left({\left(\frac{i}{n} + 1\right)}^{n}\right), -100\right)}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log i\right)\right), \left(\mathsf{fma}\left(100, \left(n \cdot \log i\right), \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{100}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\log n \cdot \left(\log n \cdot \log n\right)\right) \cdot \left(n \cdot \left(n \cdot n\right)\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\log n \cdot n\right) \cdot 100\right)\right)\right)\right)\right)\right)\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log n\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r2550010 = 100.0;
        double r2550011 = 1.0;
        double r2550012 = i;
        double r2550013 = n;
        double r2550014 = r2550012 / r2550013;
        double r2550015 = r2550011 + r2550014;
        double r2550016 = pow(r2550015, r2550013);
        double r2550017 = r2550016 - r2550011;
        double r2550018 = r2550017 / r2550014;
        double r2550019 = r2550010 * r2550018;
        return r2550019;
}

↓

double f(double i, double n) {
        double r2550020 = i;
        double r2550021 = -2.2920532433751285e-92;
        bool r2550022 = r2550020 <= r2550021;
        double r2550023 = 100.0;
        double r2550024 = n;
        double r2550025 = r2550020 / r2550024;
        double r2550026 = log1p(r2550025);
        double r2550027 = r2550026 * r2550024;
        double r2550028 = exp(r2550027);
        double r2550029 = -100.0;
        double r2550030 = fma(r2550023, r2550028, r2550029);
        double r2550031 = r2550030 * r2550030;
        double r2550032 = cbrt(r2550031);
        double r2550033 = r2550032 / r2550020;
        double r2550034 = cbrt(r2550030);
        double r2550035 = r2550034 * r2550024;
        double r2550036 = r2550033 * r2550035;
        double r2550037 = -6.547047229389803e-148;
        bool r2550038 = r2550020 <= r2550037;
        double r2550039 = 16.666666666666668;
        double r2550040 = r2550020 * r2550020;
        double r2550041 = r2550020 * r2550040;
        double r2550042 = 50.0;
        double r2550043 = r2550042 * r2550040;
        double r2550044 = fma(r2550039, r2550041, r2550043);
        double r2550045 = fma(r2550020, r2550023, r2550044);
        double r2550046 = r2550045 / r2550025;
        double r2550047 = -9.429451325520378e-161;
        bool r2550048 = r2550020 <= r2550047;
        double r2550049 = 1.0;
        double r2550050 = r2550025 + r2550049;
        double r2550051 = pow(r2550050, r2550024);
        double r2550052 = fma(r2550023, r2550051, r2550029);
        double r2550053 = exp(r2550052);
        double r2550054 = log(r2550053);
        double r2550055 = r2550054 / r2550025;
        double r2550056 = 2.448607663119347;
        bool r2550057 = r2550020 <= r2550056;
        double r2550058 = 2.7000146131065657e+237;
        bool r2550059 = r2550020 <= r2550058;
        double r2550060 = r2550024 * r2550024;
        double r2550061 = r2550024 * r2550060;
        double r2550062 = log(r2550020);
        double r2550063 = r2550062 * r2550062;
        double r2550064 = r2550063 * r2550062;
        double r2550065 = r2550061 * r2550064;
        double r2550066 = r2550024 * r2550062;
        double r2550067 = 33.333333333333336;
        double r2550068 = r2550061 * r2550062;
        double r2550069 = log(r2550024);
        double r2550070 = r2550069 * r2550069;
        double r2550071 = r2550068 * r2550070;
        double r2550072 = r2550060 * r2550042;
        double r2550073 = r2550072 * r2550063;
        double r2550074 = fma(r2550039, r2550071, r2550073);
        double r2550075 = fma(r2550067, r2550071, r2550074);
        double r2550076 = fma(r2550023, r2550066, r2550075);
        double r2550077 = fma(r2550039, r2550065, r2550076);
        double r2550078 = r2550063 * r2550069;
        double r2550079 = r2550061 * r2550078;
        double r2550080 = r2550069 * r2550062;
        double r2550081 = r2550072 * r2550080;
        double r2550082 = r2550081 + r2550081;
        double r2550083 = r2550069 * r2550070;
        double r2550084 = r2550083 * r2550061;
        double r2550085 = r2550069 * r2550024;
        double r2550086 = r2550085 * r2550023;
        double r2550087 = fma(r2550039, r2550079, r2550086);
        double r2550088 = fma(r2550039, r2550084, r2550087);
        double r2550089 = r2550082 + r2550088;
        double r2550090 = fma(r2550067, r2550079, r2550089);
        double r2550091 = r2550077 - r2550090;
        double r2550092 = r2550072 * r2550070;
        double r2550093 = r2550091 + r2550092;
        double r2550094 = r2550093 / r2550025;
        double r2550095 = 1.7504058567965248e+288;
        bool r2550096 = r2550020 <= r2550095;
        double r2550097 = r2550096 ? r2550055 : r2550094;
        double r2550098 = r2550059 ? r2550094 : r2550097;
        double r2550099 = r2550057 ? r2550046 : r2550098;
        double r2550100 = r2550048 ? r2550055 : r2550099;
        double r2550101 = r2550038 ? r2550046 : r2550100;
        double r2550102 = r2550022 ? r2550036 : r2550101;
        return r2550102;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.1
Target	42.2
Herbie	27.4

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

Derivation

1. Split input into 4 regimes

2. if i < -2.2920532433751285e-92

   1. Initial program 33.6

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

   2. Simplified 33.5

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

   4. Applied add-exp-log 33.5

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

   5. Simplified 16.4

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}}\right), -100\right)}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied add-cbrt-cube 16.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}}{\frac{i}{n}}\]
   8. Using strategy rm

   9. Applied div-inv 16.4

      \[\leadsto \frac{\sqrt[3]{\left(\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}{\color{blue}{i \cdot \frac{1}{n}}}\]

   10. Applied cbrt-prod 16.4

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)} \cdot \sqrt[3]{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}}{i \cdot \frac{1}{n}}\]

   11. Applied times-frac 17.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}{\frac{1}{n}}}\]

   12. Simplified 17.0

       \[\leadsto \frac{\sqrt[3]{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}{i} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)} \cdot n\right)}\]

   if -2.2920532433751285e-92 < i < -6.547047229389803e-148 or -9.429451325520378e-161 < i < 2.448607663119347

   1. Initial program 49.7

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

   2. Simplified 49.7

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

   3. Taylor expanded around 0 34.5

      \[\leadsto \frac{\color{blue}{100 \cdot i + \left(50 \cdot {i}^{2} + \frac{50}{3} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   4. Simplified 34.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(i, 100, \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(i \cdot i\right) \cdot i\right), \left(50 \cdot \left(i \cdot i\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   if -6.547047229389803e-148 < i < -9.429451325520378e-161 or 2.7000146131065657e+237 < i < 1.7504058567965248e+288

   1. Initial program 35.4

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

   2. Simplified 35.3

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

   4. Applied add-log-exp 35.5

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

   if 2.448607663119347 < i < 2.7000146131065657e+237 or 1.7504058567965248e+288 < i

   1. Initial program 31.0

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

   2. Simplified 30.9

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

   4. Applied add-exp-log 30.9

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

   5. Simplified 47.7

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}}\right), -100\right)}{\frac{i}{n}}\]

   6. Taylor expanded around 0 18.5

      \[\leadsto \frac{\color{blue}{\left(50 \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(100 \cdot \left(n \cdot \log i\right) + \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + 50 \cdot \left({n}^{2} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log i \cdot \log n\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + 100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   7. Simplified 18.5

      \[\leadsto \frac{\color{blue}{\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n\right) + \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\log i \cdot \left(\log i \cdot \log i\right)\right) \cdot \left(\left(n \cdot n\right) \cdot n\right)\right), \left(\mathsf{fma}\left(100, \left(n \cdot \log i\right), \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(\left(n \cdot n\right) \cdot n\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(n \cdot n\right) \cdot n\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{100}{3}, \left(\left(\log n \cdot \left(\log i \cdot \log i\right)\right) \cdot \left(\left(n \cdot n\right) \cdot n\right)\right), \left(\left(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log i\right) + \left(50 \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log i\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\log n \cdot \log n\right) \cdot \log n\right) \cdot \left(\left(n \cdot n\right) \cdot n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\log n \cdot \left(\log i \cdot \log i\right)\right) \cdot \left(\left(n \cdot n\right) \cdot n\right)\right), \left(\left(n \cdot \log n\right) \cdot 100\right)\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 27.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.2920532433751285 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right) \cdot \mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)}}{i} \cdot \left(\sqrt[3]{\mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)} \cdot n\right)\\ \mathbf{elif}\;i \le -6.547047229389803 \cdot 10^{-148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\mathsf{fma}\left(\frac{50}{3}, \left(i \cdot \left(i \cdot i\right)\right), \left(50 \cdot \left(i \cdot i\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -9.429451325520378 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(100, \left({\left(\frac{i}{n} + 1\right)}^{n}\right), -100\right)}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.448607663119347:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\mathsf{fma}\left(\frac{50}{3}, \left(i \cdot \left(i \cdot i\right)\right), \left(50 \cdot \left(i \cdot i\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.7000146131065657 \cdot 10^{+237}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log i\right)\right), \left(\mathsf{fma}\left(100, \left(n \cdot \log i\right), \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{100}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\log n \cdot \left(\log n \cdot \log n\right)\right) \cdot \left(n \cdot \left(n \cdot n\right)\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\log n \cdot n\right) \cdot 100\right)\right)\right)\right)\right)\right)\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.7504058567965248 \cdot 10^{+288}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(100, \left({\left(\frac{i}{n} + 1\right)}^{n}\right), -100\right)}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log i\right)\right), \left(\mathsf{fma}\left(100, \left(n \cdot \log i\right), \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right), \left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{100}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log i\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\log n \cdot \left(\log n \cdot \log n\right)\right) \cdot \left(n \cdot \left(n \cdot n\right)\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log n\right)\right), \left(\left(\log n \cdot n\right) \cdot 100\right)\right)\right)\right)\right)\right)\right) + \left(\left(n \cdot n\right) \cdot 50\right) \cdot \left(\log n \cdot \log n\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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))))