Compound Interest

Average Error: 40.9 → 32.8

Time: 34.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.67216965178613575:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{\sqrt[3]{{\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)} \cdot \sqrt[3]{{\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)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \frac{\frac{\frac{\sqrt[3]{{\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)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{1}{n}}\right)\\ \mathbf{elif}\;i \le 5.4848294551835708 \cdot 10^{145}:\\ \;\;\;\;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}\;i \le 3.45170537215142098 \cdot 10^{222}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 2.74199199550633987 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(4, \log 1 \cdot n, 1\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}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n\right)\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r284205 = 100.0;
        double r284206 = 1.0;
        double r284207 = i;
        double r284208 = n;
        double r284209 = r284207 / r284208;
        double r284210 = r284206 + r284209;
        double r284211 = pow(r284210, r284208);
        double r284212 = r284211 - r284206;
        double r284213 = r284212 / r284209;
        double r284214 = r284205 * r284213;
        return r284214;
}

↓

double f(double i, double n) {
        double r284215 = i;
        double r284216 = -0.6721696517861357;
        bool r284217 = r284215 <= r284216;
        double r284218 = 100.0;
        double r284219 = 1.0;
        double r284220 = n;
        double r284221 = r284215 / r284220;
        double r284222 = r284219 + r284221;
        double r284223 = 2.0;
        double r284224 = r284223 * r284220;
        double r284225 = r284223 * r284224;
        double r284226 = pow(r284222, r284225);
        double r284227 = r284219 * r284219;
        double r284228 = -r284227;
        double r284229 = r284228 * r284228;
        double r284230 = -r284229;
        double r284231 = r284226 + r284230;
        double r284232 = cbrt(r284231);
        double r284233 = r284232 * r284232;
        double r284234 = pow(r284222, r284224);
        double r284235 = r284234 + r284227;
        double r284236 = sqrt(r284235);
        double r284237 = r284233 / r284236;
        double r284238 = pow(r284222, r284220);
        double r284239 = r284238 + r284219;
        double r284240 = cbrt(r284239);
        double r284241 = r284240 * r284240;
        double r284242 = r284237 / r284241;
        double r284243 = r284242 / r284215;
        double r284244 = r284232 / r284236;
        double r284245 = r284244 / r284240;
        double r284246 = 1.0;
        double r284247 = r284246 / r284220;
        double r284248 = r284245 / r284247;
        double r284249 = r284243 * r284248;
        double r284250 = r284218 * r284249;
        double r284251 = 5.484829455183571e+145;
        bool r284252 = r284215 <= r284251;
        double r284253 = 0.5;
        double r284254 = pow(r284215, r284223);
        double r284255 = log(r284219);
        double r284256 = r284255 * r284220;
        double r284257 = fma(r284253, r284254, r284256);
        double r284258 = r284254 * r284255;
        double r284259 = r284253 * r284258;
        double r284260 = r284257 - r284259;
        double r284261 = fma(r284215, r284219, r284260);
        double r284262 = r284261 / r284221;
        double r284263 = r284218 * r284262;
        double r284264 = 3.451705372151421e+222;
        bool r284265 = r284215 <= r284264;
        double r284266 = r284234 + r284228;
        double r284267 = cbrt(r284266);
        double r284268 = r284267 * r284267;
        double r284269 = sqrt(r284239);
        double r284270 = r284268 / r284269;
        double r284271 = r284270 / r284215;
        double r284272 = r284267 / r284269;
        double r284273 = r284272 * r284220;
        double r284274 = r284271 * r284273;
        double r284275 = r284218 * r284274;
        double r284276 = 2.74199199550634e+289;
        bool r284277 = r284215 <= r284276;
        double r284278 = 4.0;
        double r284279 = 4.0;
        double r284280 = fma(r284279, r284256, r284246);
        double r284281 = fma(r284278, r284215, r284280);
        double r284282 = r284281 + r284230;
        double r284283 = r284282 / r284235;
        double r284284 = r284283 / r284239;
        double r284285 = r284284 / r284221;
        double r284286 = r284218 * r284285;
        double r284287 = r284277 ? r284286 : r284275;
        double r284288 = r284265 ? r284275 : r284287;
        double r284289 = r284252 ? r284263 : r284288;
        double r284290 = r284217 ? r284250 : r284289;
        return r284290;
}

Error

Bits error versus i

Bits error versus n

Target

Original	40.9
Target	41.1
Herbie	32.8

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

Derivation

1. Split input into 4 regimes

2. if i < -0.6721696517861357

   1. Initial program 27.5

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

   3. Applied flip-- 27.5

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

   4. Simplified 27.4

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

   6. Applied flip-+ 27.4

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

   7. Simplified 27.4

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

   8. Simplified 27.4

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

   10. Applied div-inv 27.4

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

   11. Applied add-cube-cbrt 27.4

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

   12. Applied add-sqr-sqrt 27.4

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

   13. Applied add-cube-cbrt 27.4

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

   14. Applied times-frac 27.4

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

   15. Applied times-frac 27.4

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

   16. Applied times-frac 27.9

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

   if -0.6721696517861357 < i < 5.484829455183571e+145

   1. Initial program 47.1

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

   2. Taylor expanded around 0 34.6

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

   3. Simplified 34.6

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

   if 5.484829455183571e+145 < i < 3.451705372151421e+222 or 2.74199199550634e+289 < i

   1. Initial program 32.7

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

   3. Applied flip-- 32.7

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

   4. Simplified 32.7

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

   6. Applied div-inv 32.7

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

   7. Applied add-sqr-sqrt 32.7

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

   8. Applied add-cube-cbrt 32.7

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

   9. Applied times-frac 32.8

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

   10. Applied times-frac 32.8

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

   11. Simplified 32.7

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

   if 3.451705372151421e+222 < i < 2.74199199550634e+289

   1. Initial program 31.9

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

   3. Applied flip-- 31.9

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

   4. Simplified 31.9

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

   6. Applied flip-+ 31.9

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

   7. Simplified 31.8

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

   8. Simplified 31.8

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

   9. Taylor expanded around 0 33.8

      \[\leadsto 100 \cdot \frac{\frac{\frac{\color{blue}{\left(4 \cdot i + \left(4 \cdot \left(\log 1 \cdot n\right) + 1\right)\right)} + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]

   10. Simplified 33.8

       \[\leadsto 100 \cdot \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(4, i, \mathsf{fma}\left(4, \log 1 \cdot n, 1\right)\right)} + \left(-\left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 32.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.67216965178613575:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{\sqrt[3]{{\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)} \cdot \sqrt[3]{{\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)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \frac{\frac{\frac{\sqrt[3]{{\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)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{1}{n}}\right)\\ \mathbf{elif}\;i \le 5.4848294551835708 \cdot 10^{145}:\\ \;\;\;\;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}\;i \le 3.45170537215142098 \cdot 10^{222}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 2.74199199550633987 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(4, \log 1 \cdot n, 1\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}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))))